Probability & Statistics
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Acknowledgement We at first bow our heads before Allah Almighty, who has bestowed His countless blessings upon ourselves, guided us towards the way of success, blessed us with the courage of facing the problems and obstacles and finally enabled us to accomplish this project work. We wish to place our deep sense of thanks to our respected teacher Mr. Abid Awan, who guided us to complete this project in a true sense. His valuable experience and knowledge of the field removed the difficulty at all crucial junctures. In the end, we pay our compliments to all those people who had been cooperative to us during the completion of the project and those who directly or indirectly helped us in completing our project on time.
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Dedication We are dedicating our project on the “Fan Industry of Gujrat” to our respected teacher Mr. Abid Awan for his effort to transfer the essence of his experience and knowledge to his students.
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S. No 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39
Table of contents Executive Summary Objective Introduction Companies selected Profile of Al Jamil Fans Profile of Gull Fans Profile of Finex Fans Raw Data Explanation of Raw Data Data Analysis Excel Work of Al Jamil Fans Manual Work of Al Jamil Fans Mean + Interpretation Median + Interpretation Standard Deviation + Interpretation Coefficient of Variation + Interpretation Skewness + Interpretation Mean Deviation About Mean + Interpretation Correlation + Interpretation Regression + Interpretation Excel Work of Gull Fans Manual Work of Gull Fans Mean + Interpretation Median + Interpretation Standard Deviation + Interpretation Coefficient of Variation + Interpretation Skewness + Interpretation Mean Deviation About Mean + Interpretation Correlation + Interpretation Regression + Interpretation Excel Work of Finex Fans Manual Work of Finex Fans Mean + Interpretation Median + Interpretation Standard Deviation + Interpretation Coefficient of Variation + Interpretation Mean Deviation About Mean + Interpretation Skewness + Interpretation Correlation + Interpretation
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Page No 5 6 7 11 11 11 12 13 14 15 16 17 17 18 19 20 21 22 23 25 28 29 29 30 31 32 33 33 34 36 39 40 40 40 42 43 44 45 46
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40 41 42 43 44
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Regression + Interpretation Table of Comparison Comparison Explanation Conclusion Recommendation(s)
48 50 51 54 55
Executive summary This project report is basically related to the fan industry of Gujrat. Gujrat is the city having a large number of fan manufacturing units and among them we have selected three fan manufacturing companies for our project. First of all, we have selected two important variables of these companies for our project and these variables are: advertisements and sales. The data related to these variables has been collected for each of the three companies individually. Further various statistical tools including mean, median, and mode, mean deviation for mean; standard deviation and coefficient of variation have been applied on both of the variables of each of the three companies. After that the correlation between these variables has been found and the regression analysis of these variables is also done. Finally each of the three companies is compared with each
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other on the basis of the results obtained after applying these statistical tools. And in the end the conclusion of the whole report is given after analyzing the data completely.
Objective The main objective of performing this project is to enhance our knowledge through practically applying those statistical techniques that we have been learning throughout the whole semester. Although we have been studying these techniques in many books but still we have never applied them practically in the real life, so this project has greatly helped us to understand the real life situations and applications of the various statistical techniques and enhanced our knowledge related to this particular field of study.
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Introduction: Fan is a daily use item. Its utility increases, especially in the summer season. The Fan Industry is producing about 5 to 6 million fans per annum and meeting successfully the local as well as the export demand. Out of the total production, approximately 30 per cent fans consist of pedestals, 7 per cent brackets and the remaining 63 percent are ceiling fans. The industry belongs to the light engineering industry category, and is one of the industries that existed at the time of independence. In the early 1950s, it was declared as cottage industry and its more than 50 per cent units still fall in this category.
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The story of emergence of electric fan manufacturing industry in Pakistan is one of those heroic efforts made by a few enterprising individuals who. Starting with nothing in early 1940s struggled against all odds and turned this concern into a most efficient industry in Pakistan. Fan industry is mainly confined to Gujranwala and Gujrat cities of the Punjab province. The reason for its remaining a cottage industry is that majority of the units does not have full facilities of production under one roof. They usually give orders to the units having machines for different parts like fan guards, blade castings, core laminations etc. These units have lathes, shapers, milling machines, and power pressers, die casting machines and electroplating equipments. Therefore, most of the units are simply assembling units. Thus, they do not give brand names to their products. Besides small and medium units, a few units are quite large and have integrated system i.e. from motor winding to high-pressure dies casting. These companies have reputed brand names and the qualities of their products are of international level. These units are the main players in the export field. The industry is producing a variety of products in different sizes and designs. The major products are: ceiling, pedestal, table, table-cum-pedestal fans, circulator fans, wall bracket, exhaust fans and propellers. The industry supplies quality products to the local markets, whether branded or unbranded, at competitive prices. About 400 units have a production capacity of 5 to 6 million fans, on single shift basis. The production is equal to the demand, including a nominal quantity of exports. The actual production has remained about 2.5 million fans per annum, showing a 50 per cent idle capacity. The demand for fans is continuously increasing due to increase in population and speedy migration towards big cities, and for exports. The other factor is that during the last few years, local demand for quality products is increasing fast as compared to low price goods. This means people are becoming quality conscious. Innovators Page7
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Major fan producing countries are Japan, Korea, Taiwan, Hong Kong, India and China. Japan is covering high quality market segment of fan market. Korea and Hong Kong are in middle segment of market while Pakistan, India, Taiwan and China are supplying comparatively low quality products at cheaper prices. Exports: Although fans' exports have great potential, their export is negligible. In 1992-93, only two-lack fans were exported to only two countries, Iraq and Yemen. But now the industry is exporting fans to more than 25 countries. The figures given in table-I indicate that although the exports of ceiling fan increased from Rs.134 thousand to 209 million rupees, the trade is fluctuating a great deal. As against ceiling fans, the increase in pedestal fan export is quite stable i.e., continuously rising. Pakistan has also started exporting parts of fans. During the last four years, as shown in table-II, the amount earned from export rose from Rs4.2 million to Rs107 million in 1998-01, but decreased to about fifty per cent the very next year, 2001-02. However, in spite of big fluctuations, it is believed that if some bottlenecks are removed there is a big scope for enhancing the export of fans. Government Role: The situation is that from very the beginning, the government has not given any incentives to encourage fan industry i.e. whatever progress the industry made, it has made on its own resources, and is due to its dedicated and hard working manufacturers and laborers. In spite of the lack of proper training the industry has innovated and uplifted the standard to the level where its products can compete in the world markets. TABLE - I: Exports of fans
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Year
Amount in (000 Rs.)
1989-90
134 1,289
1992-93 1993-94
81,762 1,826 18,636 15,137
1998-99 1999-00 2000-01 2001-02
2,552 32,432 15,689 66,370 59,259 161,554 208,963 183,613
TABLE - II Parts of Fans Exported Year
(Million Rs.)
1998-99 1999-00 2000-01 2001-02
4.2 16.6 109.7 57.5
TABLE - III: Fan Exports by Type (Million Rs.)
Ceiling Fans Innovators Page9
1998-99
1999-00
2000-01
2001-02
Rs.2.6 Million
Rs.15.7 Million
Rs.59.3 Million
Rs.208.9 Million
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Pedestal Fans
Rs.32.4 Millions
Rs.66.4 Million
Rs.161.6 Million
Rs.183.6 Million
Table Fans
Rs.1.1 Million
-------------
------------------
Rs.0.9 Million
Exhaust Fans Other Fans
Rs.0.1 Million Rs.2.9 Million
Rs.0.1 Million Rs.6.9 Million
Rs.0.1 Million Rs.5.1 Million
---------------Rs.27.2 Million
Fan Blowers
Rs.24.9 Million
Rs.13.2 Million
Rs.0.7Million
----------------
Other Fans N.S
Rs.3.0 Million
Rs.9.7 Million
Rs.5.3 Million
Rs.2.0 Million
Total
Rs.67 Million
Rs.112 Million
Rs.232.1 Million
Rs.422.6 Million
Companies Selected: We have selected three fan manufacturing companies for our project and these companies are: •
Al Jamil Fans
•
Gull Fans
•
Finex Fans
The data related to these companies has been collected for the two specific variables which are advertisements and sales. Before going to the next section of the data analysis, we would first give a brief history or profile of each of the three companies. Profile of Al Jamil Fans: Al Jamil Fan industry was established in 1987 and the founder this industry was Mr. Khalid Javaid, who named the company after the name of his son Mr. Jamil Javaid. In the beginning this company had been operated on partnership basis and the name of Innovators Page10
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the partner was Mr. Fateh Muhammad but with the passage of time Mr. Khalid Javaid became a sole proprietor and run the company individually. The company has also got ISO 9001 quality certificate
Profile of Gull Fans: This company was established in 1974 and its founder is Sufi Ghulam Hussain who had also named his company after the name of his son Gull Pervaiz Butt. The company also produces export quality fans and also export there products to Dubai and Saudi Arabia. This company has also got the quality certificate ISO 9001
Profile of Finex Fans: The company was established in 1994 and its founders name is Chaudhary Imran Cheema. This company is not a big one and is comparatively on a lower level as compared to the above two companies. The present chairman of the company is the brother of Chaudhary Imran and his name is Chaudhary Nouman Cheema. This company does not have any quality certificate.
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Raw Data Al- Jamil Fan 2005 P.F=1700 C.F=1200 Sales Adver. Jan 475 900 1,382,000 6,250 Feb 450 925 1,353,500 6,000 March 500 875 1,378,500 7,000 April 500 900 1,343,250 7,500 May 470 900 1,324,250 5,500 Jun 400 1000 1,379,500 5,000 July 500 925 1,406,000 6,800 Aug 300 800 1,236,500 4,500
Fine X Fan 2005 P.F=1600 C.F=1150 Sales Adver. Jan 375 680 1,382,000 5,000 Feb 350 690 1,353,500 4,800 March 380 670 1,378,500 5,100 April 340 695 1,343,250 4,300 May 350 665 1,324,250 4,600 Jun 395 650 1,379,500 5,200 July 390 680 1,406,000 5,500 Aug 320 630 1,236,500 4,200
2006 P.F=1900 C.F=1350 Sales Adver. Jan 600 1300 1,927,000 9,000 Feb 550 1300 1,956,000 8,500 March 670 1250 1,994,500 9,100 April 500 1350 1,910,500 8,300 May 600 1300 1,921,250 8,900 Jun 700 1170 1,898,000 9,200 July 480 1100 1,915,000 7,000 Aug 450 1100 1,784,000 6,700
2006 P.F=1700 C.F=1250 Sales Adver. Jan 435 950 1,927,000 12,000 Feb 430 980 1,956,000 11,500 March 460 970 1,994,500 12,200 April 440 930 1,910,500 12,100 May 450 925 1,921,250 11,500 Jun 440 920 1,898,000 12,800 July 450 920 1,915,000 11,300 Aug 395 890 1,784,000 11,000
2007 P.F=1950 C.F=1400 Sales Adver. Jan 900 1400 3,015,000 9,500 Feb 900 1400 2,986,500 9,300 March 800 1450 2,978,250 9,000 April 600 1500 3,009,000 8,200 May 700 1500 3,021,000 9,500 Jun 700 1500 2,925,000 9,000 July 600 1400 2,955,000 7,500 Aug 550 1250 2,869,500 7,000
2007 P.F=1950 C.F=1350 Sales Adver. Jan 750 1150 3,015,000 15,500 Feb 770 1100 2,986,500 14,300 March 745 1130 2,978,250 15,000 April 740 1160 3,009,000 15,200 May 760 1140 3,021,000 14,700 Jun 735 1105 2,925,000 14,500 July 740 1120 2,955,000 14,800 Aug 710 1100 2,869,500 14,000
Gull Fans 2005 P.F=1600 C.F=1100 Sales Adver. Jan 650 1200 2,360,000 11,000 Feb 700 1100 2,330,000 9,500 March 600 1150 2,225,000 9,000 April 650 1250 1,479,000 8,500 May 550 1150 2,145,000 9,700 Jun 720 1300 2,582,000 12,000 July 500 1100 2,010,000 10,000 Aug 530 1200 2,168,000 9,800 2006 P.F=1700 C.F=1200 Sales Adver. Jan 1050 1500 3,585,000 16,000 Feb 1000 1450 3,440,000 14,500 March 1100 1500 3,670,000 16,100 April 1020 1470 3,498,000 15,000 May 1050 1400 3,465,000 14,300 Jun 1000 1500 3,500,000 14,500 July 1100 1500 3,670,000 14,900 Aug 1000 1300 3,260,000 15,500
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2007 P.F=1900 C.F=1400 Sales Adver. Jan 950 2000 4,605,000 13,500 Feb 1000 1800 4,420,000 13,000 March 1050 1900 4,655,000 14,000 April 900 1950 4,440,000 12,500 May 1000 2000 4,700,000 13,200 Jun 1100 1700 4,470,000 13,000 July 1000 1980 4,672,000 14,000 Aug 950 1780 4,297,000 12,000
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Explanation of Raw Data Given above is the data of three fan manufacturing companies that are Al Jamil Fans, Gull Fans and Finex Fans. The data has been collected for 24 months. We have selected two variables of each of these companies these are: •
Sales
•
Advertisements
We will calculate the sales of the fans and the advertising expenditures of these companies. For calculating the sales, we have chosen two types of fans that are produced by each of these companies, and these fans are: pedestal and ceiling fans. The price of a single unit of each of the two fans is multiplied by the number of units sold per month and it makes the total sales of the fans per month, and finally by adding the sales of each month we get the overall three year sales of the companies. Same is the case with the advertising expenditures, for which again we will calculate the overall advertising expenditure of three years for all these three companies. Finally this data will be analyzed by using various statistical tools an actual calculations and interpretation of this data is shown further.
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DATA ANALYSIS
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Excel Work of Al-Jamil Fans Sale Advertisement 1,888 6 1,875 6 1,900 7 1,930 8 1,879 6 1,880 5 1,960 7 1,470 5 2,895 9 2,800 9 2,961 9 2,773 8 2,895 9 2,910 9 2,397 7 2,340 7 3,715 10 3,715 9 3,590 9 3,270 8 3,465 10 3,465 9 3,130 8 Sale Advertisement 2,823 7 Average 2,663 8 Median 2,811 8 Mean deviation about 593.02430 1.3046875 mean 6 Standard deviation 689.07258 1.529763586 8 Skewness -0.0453332 -0.522564549 Correlation 0.8861954 9
Intercept X Variable 1 Innovators Page15
Coefficients -417.7015121 399.1813025
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Manual Work of Al-Jamil Fans
Mean In the very first phase we use the average method to analyze the data. Perhaps the most important measure of location is the mean, which is average value. Mean is taken by all small and large data values. It is denoted by x. As average value represent the whole data by taking the sum of all values and further divide by the total number of values. Mean of Sales: x = ∑x/ n = 63926/24 = 2,663.6 Interpretation: Here, we got the final value for sale that is 2,663.6. Which tells that average sale of the fans during last 24 months are 2,663.6 and this value is in thousands. Mean of advertisements: x = ∑x/ n = 187/24 = 7.8 Interpretation:
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The average value for advertisement is 7.8 that are also in thousands. It is representing the whole data. It represents the amount that the companies spent on their advertisements in 24 months.
Median Median is also used to measure the average, the important characteristic of median is that it s used when there is variation in the data. Median is more appropriate for the data that has got some outliers. Median is the middle value of the data which is arranged in ascending order. The median provides the better value than the mean. Median of Sales: 1,470, 1,875, 1,879, 1,880, 1,887, 1,900, 1,930, 1,960, 2,340, 2,397, 2,773, 2,800, 2,895, 2,895, 2,910, 2,961, 2,823, 3,130, 3,270, 3,465, 3,465, 3,590, 3,715, 3,715, = (n + 1)th/2 = (24 + 1)th/2 = 12.5th Value = 12th value + .5 (13th – 12th) = 2,800 + .5 (2,823 – 2,800) = 2811.5 Interpretation: The median in this case is 2811.5 which shows the better measure of the sales of the company and is a better method of calculating the average. Median of Advertisements: Innovators Page17
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5, 5, 6, 6, 6, 7, 7, 7, 7, 7, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 9, 10, 10 = (n + 1)th/2 = (24 + 1)th/2 =12.5th Value =12th value + .5 (13th – 12th) = 8 + .5 (8 – 8) =8 Interpretation: The median in this case is 8 which shows the better measure of the sales of the company and is a better method of calculating the average.
Standard Deviation Standard deviation is measured in the same units as the original data. It is the most appropriate and reliable method of calculating the dispersion. We use the standard deviation to bring the values to there actual position that have been made too large by using variance by squaring the values. So, for this reason we move towards standard deviation Standard Deviation of Sales: =√∑X2/n – (∑X/n)2 = √181193148/24 - (63926/24)2 = √7549714.5 – (2663.6)2 = √7549714.5 – 7,094,764.96 = √454,949.54
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= 674.5 Interpretation: This value basically shows the variation in the data. The fluctuation of data around the mean is easily measured through this technique Here the spread in the data is 674.5, if there is less variation and spread in the data then the data is better. So we can say that the value of sales can lie between the ranges of 1989.1-3338.1 that is in thousands. Standard Deviation of Advertisements: =√∑X2/n – (∑X/n)2 = √1507/24 - (187/24)2 = √62.8 – (7.79)2 = √62.8 – 60.68 = √2.2 = 1.48 Interpretation: Here the spread in the data is 1.48, if there is less variation and spread in the data then the data is better. So we can say that the value of advertisements can lie between the ranges of 6.32 – 9.28 that is in thousands.
Coefficient of Variation Coefficient of variation is used for the comparison between the two different variables. This is the relative measure of variation; it measures the standard deviation relative to mean. Coefficient of Variation of Sales: = S/ ҳ X 100 = 674.5/2663.6 X 100 Innovators Page19
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= 25.322 Interpretation: The coefficient of variation of sales according to the situation is 25.322 which shows the per unit risk associated with a given variable. The smaller the CV the less risk involved, and the greater the CV the high risk involved. Coefficient of Variation of Advertisements: = S/ ҳ X 100 = 1.48/7.8 X 100 = 18.97 Interpretation: The coefficient of variation of advertisements according to the situation is 18.97 which shows the per unit risk associated with a given variable. The smaller the CV the less risk involved, and the greater the CV the high risk involved.
Skewness Skewness is a statistical tool that tells us about whether the given data is normally distributed or not. If the data is not normally distributed, then it would either be positively skewed or negatively skewed. Skewness of Sales: = 3 (mean – median)/S.D = 3 (2663.6 – 2847.5)/ 674.5 = - 0.818 Interpretation:
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In this case the value of Skewnesss is -0.818 which clearly shows that the data is negatively skewed and in case of negative Skewnesss, the mean of the data is the smallest, median is greater and finally the mode is the greatest one.
Skewness of Advertisements: = 3 (mean – median)/S.D = 3 (7.8 – 8)/1.48 = -0.40 Interpretation: The value of Skewnesss of advertising expenditures is -0.40 which is again negatively skewed.
Mean Deviation about Mean Mean deviation tells us about the actual deviation of the data from its arithmetic mean. One of the drawbacks of this technique is that it ignores all the negative values due to modulus. So it is not very much reliable. M.D (mean) of sales: = ∑|x-x|/n = 14236/24 = 593.16 Interpretation:
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The value of mean deviation about mean of sales is 593.16 which shows the deviation of the data from the mean value which shows that the data deviates fro the mean by an amount of 593.16. M.D (mean) of Advertisements: = ∑|x-x|/n = 29/24 = 1.20
Interpretation: The value of mean deviation about mean of sales is 1.20 which shows the deviation of the data from the mean value which shows that the data deviates fro the mean by an amount of 1.20.
Correlation It is basically a descriptive measure of the strength of linear association between two variables one is independent and the other one is dependent. Values of the correlation coefficient are always between -1 and +1. A value of +1 indicates that two variables are perfectly positively related and similarly a value of -1 indicates that two variables are strongly negatively related. Here we have given a scale of measurement of correlation between two variables.
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|______________|________________|_______________|______________| -1
-.5
Perfect -ve
0
Moderate
No relation
+.5 Moderate
+1 Perfect +ve
Table X
Y
6
1,888
6
1,875
7
1,900
8
1,930
6
1,879
5
1,880
7
1,960
5
1,470
9
2,895
9
2,800
9
2,961
8
2,773
9
2,895
9
2,910
7
2,397
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Xy
x2
y2
11,328
36
3564544
11,250
36
3515625
13,300
49
3610000
15,440
64
3724900
11,274
36
3530641
9,400
25
3534400
13,720
49
3841600
7,350
25
2160900
26,055
81
8381025
25,200
81
7840000
26,649
81
8767521
22,184
64
7689529
26,055
81
8381025
26,190
81
8468100
16,779
49
5745609
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7
2,340
10
3,715
9
3,715
9
3,590
8
3,270
10
3,465
9
3,465
8
3,130
7
2,823
187
63926
16,380
49
5475600
37,150
100
13801225
33,435
81
13801225
32,310
81
12888100
26,160
64
10692900
34,650
100
12006225
31,185
81
12006225
25,040
64
9796900
19,761
49
7969329
518,245
1507
181193148
N = 24 ∑x = 187 ∑y = 63926 ∑xy = 518245 x2 = 1507 y2 = 181193148 r = n∑xy – (∑x)(∑y)/ √{n∑x2 – (∑x)2}{n∑y2 – (∑y)2} = 24(518245) – (187) (63926)/ √{24(1507) – (187)2}{24(181193148) – (63926)2} = 0.28 Interpretation: The value of correlation between sale and advertisements of this company is 0.28 which shows that there exists a weekly positive relation between these two variables. Innovators Page24
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Regression Analysis Regression analysis is basically used to forecast the future. In regression terminology, the variable being predicted is called the dependent variable. The variable or variables being used to predict the value of dependent variable called the independent variable. For example in analyzing the effect of advertising expenditures on sales, a marketing manager’s desire to predict sales would suggest making sales the dependent variable. Advertising expenditure would be the independent variable used to help predict sales. Y = a + bx b = n ∑xy – (∑x)(∑y)/ n ∑x2 – (∑x)2 = 403.43 a = ∑y – b∑x/n = 63926 – 403.43(187)/24 = -479.8 Y = -479.8 + 403.43x
Interpretation: The regression equation of Al Jamil Fans shows that -479.8 is a fixed value which stays the same even if the value of b is zero. By changing the number of units or the value of x, we get different values of y. Y = -479.8 + 403.43(7) = -479.8 + 2,824.01 = 2,344.21
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If in x we put 7 the value which we got is 2,344.21. Due to one unit increase in x the value of y-intercept increase and automatically there is increase in the final value of y which is the given value of x.
Bar Graph 4,000
3,500
3,000
Sales
2,500
Series1 Series2
2,000
1,500
1,000
500
0 1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
Advertising Expenditure
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16
17
18
19
20
21
22
23
24
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Line Graph 4,000
3,500
3,000
Sales
2,500
Series1 Series2
2,000
1,500
1,000
500
0 1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
Advertising Expenditure
Excel Work of Gull Fans
Gull Fans
Sales 2360 2330 2225 1479 2145 2582 2010 2168 3585 3440 3670 3498 3465 3500 3670 3260 4605 4420 4655 Innovators 4440 Page27 4700 4470 4672 4297
Advertisements 11 9.5 9 8.5 9.7 12 10 9.8 16 14.5 16.1 15 14.3 14.5 14.9 15.5 13.5 13 14 12.5 13.2 13 14 12
23
24
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Coefficients -319.4670732 292.3509321
Intercept X Variable 1
Manual Work of Gull Fans Mean Mean Sales:
of
= ∑x/ n = 81646/24 = 3,401.9
Average Median Mean About Mean Skewness
Coefficient Variation Standard Deviation Correlation
Interpretation: Innovators Page28
Sales 3,402 3,499 Deviation 838.1875 -0.27873390 1 of 38.56 1013.45186 8 0.66999274 4
Advertisement 13 13 1.940972222 -0.394445995 17.83 2.322569635
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Here, we got the final value for sale that is 3,401.9. Which tells that average sale of the fans during last 24 months are 3,401.9 and this value is in thousands. Mean of advertisements: = ∑x/ n = 305.5/24 = 12.73 Interpretation: The average value for advertisement is 12.73 that are also in thousands. It is representing the whole data. It represents the amount that the companies spent on their advertisements in 24 months.
Median Median of Sales: 1,479, 2,010, 2,145, 2,168, 2,225, 2,330, 2,360, 2,582, 3,260, 3,440, 3,465, 3,498, 3,500, 3,585, 3,670, 3,670, 4,297, 4,420, 4,440, 4,470, 4,605, 4,655, 4,672, 4,700, = (n + 1)th/2 = (24 + 1)th/2 = 12.5th Value = 12th value + .5 (13th – 12th) Innovators Page29
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= 3,498 + .5 (3,500 – 3,498) = 3,499 Interpretation: The median in this case is 3,499 which shows the better measure of the sales of the company and is a better method of calculating the average. Median of Advertisements: 8.5, 9, 9.5, 9.7, 9.8, 10, 11, 12, 12, 12.5, 13, 13, 13.2, 13.5, 14, 14, 14.3, 14.5, 14.5, 14.9, 15, 15.5, 16, 16.1 =(n + 1)th/2 =(24 + 1)th/2 =12.5th Value =12th value + .5 (13th – 12th) = 13 + .5 (13.2 – 13) = 13.1
Interpretation: The median in this case is 13.1 which shows the better measure of the advertisements of the company and is a better method of calculating the average.
Standard Deviation Standard Deviation of Sales: =√∑X2/n – (∑X/n)2 Innovators Page30
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= √319046621/24 – (81646/24)2 = √13,293,609.21 – (3,401.92)2 = √13,293,609.21 – 11,573,059.69 = √1,720,549.52 = 1,311.7 Interpretation: This value basically shows the variation in the data. The fluctuation of data around the mean is easily measured through this technique Here the spread in the data is 1311.7, if there is less variation and spread in the data then the data is better. So we can say that the value of sales can lie between the ranges of 2090.2-4713.6 that is in thousands. Standard Deviation of Advertisements: =√∑X2/n – (∑X/n)2 = √4012.83/24 – (305.5/24)2 = √167.20 – (12.73)2 = √167.20 – 162.05 = √5.15 = 2.27 Interpretation: This value basically shows the variation in the data. The fluctuation of data around the mean is easily measured through this technique Here the spread in the data is 2.27, if there is less variation and spread in the data then the data is better. So we can say that the value of sales can lie between the ranges of 10.46-15 that is in thousands.
Coefficient of Variation Innovators Page31
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Coefficient of Variation of Sales: = S/ ҳ X 100 = 1311.7/3401.9 X 100 = 38.56 Interpretation: The coefficient of variation of sales according to the situation is 38.56 which shows the per unit risk associated with a given variable. The smaller the CV the less risk involved, and the greater the CV the high risk involved. Coefficient of Variation of Advertisement = S/ ҳ X 100 = 2.27/12.73 X 100 = 17.83 Interpretation: The coefficient of variation of advertisements according to the situation is 17.83 which shows the per unit risk associated with a given variable. The smaller the CV the less risk involved, and the greater the CV the high risk involved.
Skewness Skewness of Sales: = 3 (mean – median)/S.D = 3 (3401.9 - 3499)/ 1311.7 = -0.22 Interpretation: Innovators Page32
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In this case the value of Skewnesss is -0.22 which clearly shows that the data is negatively skewed and in case of negative Skewnesss, the mean of the data is the smallest, median is greater and finally the mode is the greatest one. Skewness of Advertisements: = 3 (mean – median)/S.D = 3 (12.73 – 13.1)/2.27 = -0.49 Interpretation: In this case the value of Skewnesss is -0.49 which clearly shows that the data is negatively skewed and in case of negative Skewnesss, the mean of the data is the smallest, median is greater and finally the mode is the greatest one.
Mean Deviation about mean M.D (mean) of sales: = ∑|x-x|/n = 20116/24 = 838.17
Interpretation: The value of mean deviation about mean of sales is 838.17 which shows the deviation of the data from the mean value which shows that the data deviates fro the mean by an amount of 838.17. M.D (mean) of Advertisements: =∑|x-x|/n Innovators Page33
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=45/24 =1.87 Interpretation: The value of mean deviation about mean of sales is 1.87 which shows the deviation of the data from the mean value which shows that the data deviates fro the mean by an amount of 1.87.
Correlation X
Y
11
2360
9.5
2330
9
2225
8.5
1479
9.7
2145
12
2582
10
2010
9.8
2168
16
3585
14.5
3440
16.1
3670
15
3498
14.3
3465
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Xy
x2
y2
25960
121
5569600
22,135
90.25
5428900
20,025
81
4950625
12,571.5
72.25
2187441
20,806.5
94.09
4601025
30,984
144
6666724
20,100
100
4040100
21,246.4
96.04
4700224
57,360
256
12852225
49,880
210.25
11833600
59,087
259.21
13468900
52,470
225
12236004
49,549.5
204.49
12006225
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14.5
3500
14.9
3670
15.5
3260
13.5
4605
13
4420
14
4655
12.5
4440
13.2
4700
13
4470
14
4672
12
4297
305.5
81646
50,750
210.25
12250000
54,683
222.01
13468900
50,530
240.25
10627600
62,167.5
182.25
21206025
57,460
169
19536400
65,170
196
21669025
55,500
156.25
19713600
62,040
174.24
22090000
58,110
169
19980900
65,408
196
21827584
51,564
144
18464209
4012.83
319046621
1075557
N = 24 ∑x = 305.5 ∑y = 81646 ∑xy = 1075557 X2 = 4012.83 Y2 = 319046621 r = n∑xy – (∑x)(∑y)/ √{n∑x2 – (∑x)2}{n∑y2 – (∑y)2} = 24(1075557) – (305.5) (81646)/ √{24(4012.83) – (305.5)2}{24(319046621) – (81646)2} = 0.507 Interpretation: Innovators Page35
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The value of correlation between sales and advertisements of this company is 0.507 which shows that there exists a moderately positive relation between the sales and advertisements of Gull Fans.
Regression Analysis Y = a + bx b = n ∑xy – (∑x)(∑y)/ n ∑x2 – (∑x)2 = 292.35 a = ∑y – b∑x/n = 81646 – 292.35(305.5)/24 = -319.46 Y = -319.46 + 292.35x Interpretation: The regression equation of Gull Fans shows that -319.46 is a fixed value which stays the same even if the value of b is zero. By changing the number of units or the value of x, we get different values of y. Y = -319.46 + 292.35(9) = -319.46 + 2,631.15 = 2,311.69 If in x we put 9 the value which we got is 2,311.69. Due to one unit increase in x the value of y-intercept increase and automatically there is increase in the final value of y which is the given value of x.
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Bar Graph 5000 4500 4000 3500
Sales
3000 Series1 Series2
2500 2000 1500 1000 500 0 1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
Advertising Expenditure
Line Graph 5000 4500 4000 3500
Sales
3000 2500 2000 1500 Innovators Page37 1000 500 0
Series1 Series2
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Excel work of Finex Finex Fans Sales (1000s) 1236 1343 1324 1353 1382 1378 1379 1406 1784 1915 1921 1956 1927 1910 1994 1898 2869 2986 2925 3021 2951 2978 3009 3015
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Mean Median S.D Skewness Coefficient of Variation Mean deviation about mean Co-relation
Advertisements (1000s) 4.2 4.3 4.6 4.8 5 5.1 5.2 5.5 11 11.3 11.5 11.5 12 12.1 12.2 12.8 14 14.3 14.5 14.7 14.8 15 15.2 15.5
Sale 2,078 1,918 687.6110503 0.37217925 0.32392
Advertisement 10 12 4.272949546 -0.466272 0.44263
594.5
3.75
0.916563558
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Coefficients 534.3322469 147.4951257
Intercept X Variable 1
Manual Work of Finex Fans Mean Mean of sales: = Σx/n = 49864/24 = 2077.6 Interpretation: Here, we got the final value for sales that is 2,077.6 which tells that average sale of the fans during last 24 months are 2,077.6 and this value is in thousands. Mean of Advertisements: =Σx/n =251.1/24 =10.46 Interpretation: The average value for advertisements is 10.46 that are also in thousands. It is representing the whole data. It represents the amount that the companies spent on their advertisements in 24 months. Innovators Page39
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Median Median of sales: 1236,1343,1324,1353,1382,1378,1379,1406,1784,1915,1921,1956,1927,1910,1994,189 8,2869,2925,3021,2951,2978,3009,3015 = (n+1)th/2 = (24+1)th/2 = 12.5th value = 12th value+0.5(13th-12th) = 1915+0.5(1921- 1915) = 1918 Interpretation: The median in this case is 1918 which shows the better measure of the sales of the company and is a better method of calculating the average.
Median of Advertisements: 4.2,4.3,4.6,4.8,5,5.1,5.2,5.5,11,11.3,11.5,11.5,12,12.1,12.2,12.8,14,14.3,14.5,14.7,14.8,15 ,15.2,15.5 = (n+1)th/n = (24+1)th/2 = 12.5th value Innovators Page40
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=12th value+0.5(13th-12th) =11.5+0.5(12- 11.5) =11.75
Interpretation: The median in this case is 11.75 which shows the better measure of the advertisements of the company and is a better method of calculating the average.
Standard Deviation Standard deviation of sales: = √Σx2/n- (Σx/n)2 = √(114458756/24)-(2077.5)2 = √4769114.8-4316006.25 = √453108.5 = 673.13 Interpretation: This value basically shows the variation in the data. The fluctuation of data around the mean is easily measured through this technique Here the spread in the data is 673.13, if
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there is less variation and spread in the data then the data is better. So we can say that the value of sales can lie between the ranges of 1404.47-2750.73 that is in thousands. Standard deviation of Advertisements: = √Σx2/n- (Σx/n)2 = √(2915.52/24)-(100) = √121.48-100 = √21.48 = 4.63 Interpretation: Here the spread in the data is 4.63, if there is less variation and spread in the data then the data is better. So we can say that the value of advertisements can lie between the ranges of 5.83-15.09 that is in thousands.
Coefficient of Variation Coefficient of variation of sales: = S/x X 100 = 673.13/2078 = 0.32393 Interpretation: Innovators Page42
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The coefficient of variation of sales according to the situation is 0.32393 which shows the per unit risk associated with a given variable. The smaller the CV the less risk involved, and the greater the CV the high risk involved.
Coefficient of variation of Advertisements: = S/x X 100 = 4.63/10.46 = 0.44263 Interpretation: The coefficient of variation of advertisements according to the situation is 0.44263 which shows the per unit risk associated with a given variable. The smaller the CV the less risk involved, and the greater the CV the high risk involved.
Mean Deviation about Mean M.D (mean) of sales: = Σ|x-x|/n = 14271/24 = 594.6 Interpretation: The value of mean deviation about mean of sales is 594.6 which shows the deviation of the data from the mean value which shows that the data deviates fro the mean by an amount of 594.6.
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M.D (mean) of Advertisements: = Σ|x-x|/n = 93.7/24 = 3.9 Interpretation: The value of mean deviation about mean of sales is 594.6 which shows the deviation of the data from the mean value which shows that the data deviates fro the mean by an amount of 594.6.
Skewness Skewness of sales: = 3(x-median)/S.D = 3(2077.6 – 1918)/673.13 = 0.711 Interpretation: In this case the value of Skewnesss is 0.711 which clearly shows that the data is positively or rightly skewed and in case of positive Skewnesss, the mean of the data is the largest, median is smaller then the mean and finally the mode is the smallest.
Skewness of Advertisements: = 3(x-median)/S.D = 3(10.46 – 11.75)/4.63
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= -0.83 Interpretation: In this case the value of Skewnesss is -0.83 which clearly shows that the data is negatively or left skewed and in case of negative Skewnesss, the mean of the data is the smallest, median is greater then the mean and finally the mode is the larges
Correlation X
Y
x2
Xy
y2
5
1382
6910
25
1909924
4.8
1353
6494.4
23.04
1830609
5.1
1378
7027.8
26.01
1898884
4.3
1343
5774.9
18.49
1803649
4.6
1324
6090.4
21.16
1752976
5.2
1379
7170.8
27.04
1901641
5.5
1406
7733
30.25
1976836
4.2
1236
5191.2
17.64
1527696
12
1927
23124
144
3713329
11.5
1956
22494
132.25
3825936
12.2
1994
24326.8
148.84
3976036
12.1
1910
23111
146.41
3648100
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11.5
1921
22091.5
132.25
3690241
12.8
1898
24294.4
163.84
3602404
11.3
1915
21639.5
127.69
3667225
11
1784
19624
121
3182656
15.5
3015
46732.5
240.25
9090225
14.3
2986
42699.8
204.49
8916196
15
2978
44670
225
8868484
15.2
3009
45736.8
231.04
9054081
14.7
3021
44408.7
216.09
9126441
14.5
2925
42412.5
210.25
8555625
14.8
2955
43734
219.04
8732025
14
2869
40166
196
8231161
251.1
49860
583658
3047.07
114482380
N = 24 ∑x = 251.1 ∑y = 49860 ∑xy = 583658 X2 = 3047.07 Y2 = 114482380
r = n∑xy – (∑x)(∑y)/ √{n∑x2 – (∑x)2}{n∑y2 – (∑y)2} Innovators Page46
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= 24(583658)- (251.1)(49860)/ √{ 24(3047.07)-(251.1)2}{ 24(114482380)-(49860)2 =14007792-12519846/√ (73129.68-63051.21)(2747577120-2486019600) = 1487946/ √ (10078.47)(261557520) = 1487946/1623606.9 = 0.916
Interpretation: The value of correlation between sale and advertisements of this company is 0.916 which shows that there exists an almost perfectly positive relation between these two variables.
Regression Analysis Y = a + bx b = n ∑xy – (∑x)(∑y)/ n ∑x2 – (∑x)2 = 1487946/10078.47 = 147.64 a = ∑y – b∑x/n = 49860-147.64(251.1)/24 = 12787.6/24 = 532.82 Y = 532.82+147.64x Interpretation: The regression equation of Finex Fans shows that 532.82 is a fixed value which stays the same even if the value of b is zero. By changing the number of units or the value of x, we get different values of y. Innovators Page47
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Y = 532.82 + 147.64(11) = 532.82 + 1,624.04 = 2,156.86 If in x we put 11 the value which we got is 2156.86. Due to one unit increase in x the value of y-intercept increase and automatically there is increase in the final value of y which is the given value of x.
Bar Graph 3500
3000
2500
Sales
2000 Series1 Series2 1500
1000
500
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2
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4
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7
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13
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Advertising Expenditure
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16
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23
24
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Line Graph 3500
3000
2500
Sales
2000 Series1 Series2 1500
1000
500
0 1
2
3
4
5
6
7
8
9
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12
13
14
15
Advertising Expenditure
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16
17
18
19
20
21
22
23
24
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Explanation of comparison By comparing the data of the three companies selected by us, we have got the following results: Mean: The average sale of Al Jamil Fans for the given period of 24 months is 2663.6, whereas that of Gull Fans and Finex Fans is 3401.9 and 2077.6 respectively. By looking at this data we have found that the sales of Gull Fans are more as compared to the other two companies. Similarly, the average advertising expenditures of Al Jamil Fans is 7.8 and those of other two companies’ are 12.73 and 10.46 respectively. This data again shows that the advertising expenditures of Gull Fans are again more then the other two companies. The sales and advertising expenditures of Gull Fans is more because the company has more investment and production, so it sales more as compared to the other companies. Innovators Page50
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Median: The median sales of Al Jamil Fans, Gull Fans and Finex Fans are 2847.5, 3499 and 1918. Again Gull fan has more value as compare to Al Jamil and Finex. Now in the case of advertisements the median expenditures for Al Jamil fans is 8, for Gull Fan 13.1 and for Finex 11.75. So the median value of advertising expenditures of Gull fans is more than other two companies and the reason is again the same because of more production and investment. Standard Deviation: Further by calculating the standard deviation of sales we have got the values of S.D for these companies as 674.5 for Al Jamil, 1,311.7 for Gull fan and 673.13 for the Finex fans. By using standard deviation we come to know that how much deviation is there in the data or in values from there actual values. However for sales the value for Finex is more reliable because there is less deviation in this value. But this is also clear that the mean of this company is not more than others. So, for getting the more accurate value further we calculate the CV. The standard deviation of advertising expenditures for Al Jamil fans is 1.48, for Gull fans are 2.27 and for the Finex fan is 4.63. In this scenario Al Jamil fans have less standard deviation as compared to Gull and Finex. This shows less deviation respectively. But mean is more for Gull fans from this we didn’t know which is better. For getting the more accurate value now we calculate the CV. Coefficient of Variation: The coefficient of variation of sales of Al Jamil Fans for the 24 months is 25.322, whereas that of Gull Fans and Finex Fans is 38.56 and 0.323 respectively. By looking at this data we have found that the per unit risk associated with the companies. According to these values, Gull Fans is associated with more risk as compared to the other two companies. The company having less value of CV has got less risk associated so the Finex Fans Has got less risk and is more reliable among these companies. Innovators Page51
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Similarly, the coefficient of variation of advertising expenditures for Al Jamil Fans is 18.97 and those of other two companies’ are 17.83 and 0.44236 respectively. This data again shows that the advertising expenditures of Gull Fans are again more then the other two companies and again Finex Fans is more reliable. M.D (mean): The values of mean deviation about mean of sales for the three companies are 593.16, 838.17 and 594.6 and the values for the mean deviation about mean of the advertising expenditures for these companies are 1.20, 1.87 and 3.9 respectively. Correlation: The correlation values of these companies are 0.28 for Al Jamil Fans, 0.502 for Gull Fans and 0.916 for Finex Fans which shows that the relation between sales and advertisements of Al Jamil Fans is weakly positive which shows that the sales of this company depend on the advertisements but not too much. Similarly the value of 0.502 for Gull Fans shows that the sales and advertisements of this company have got a moderately strong relationship. And finally the value of 0.916 for Finex Fans shows that the relationship between its sales and the advertisements is a strongly positive, which means that the sales of this company directly depend on the advertisements. Regression: The regression equations of the three companies are: Al Jamil Fans
y = - 479.8 + 403.43x
Gull Fans
y = - 319.46 + 292.35x
Finex Fans y = 532.8 + 147.64x The regression equations of all these companies are positive.
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Conclusion After collecting the data from the three fan manufacturing companies and analyzing the data by applying the statistical tools, we have concluded that the average sales and advertising expenditure of Gull Fans is more then Al Jamil Fans. The reason is that the Gull Fans has got a greater market share and has more investment and production as compared to the other two companies. Further the correlation between the advertising expenditure and sales is strongly positive for Finex Fans which means that the sales of this company greatly depend on the advertising of the company. Gull Fans has got a moderate correlation and Al Jamil Fans has got a weakly positively positive correlation. So Al Jamil Fans and Finex Fans are a bit behind Gull fans in case of their sales and advertising expenditures.
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Recommendations •
As the sales are directly dependent on the advertising expenditure so all these three companies, especially Al Jamil Fans and Finex Fans have to work hard for their advertisements which would then increase their sales.
•
These companies should try to bring some innovations in order to attract more customers.
•
New designs should also be introduced in their products.
•
Display Centers of all these companies are not much attractive, so they should work to make their showrooms more attractive.
•
These companies have got a manual system of data handling which usually creates problems, so a computerized system should be introduced.
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•
[PROBABILITY & STATISTICS]
The management and staff of these companies is not much educated and qualified, so the staff should focus on their qualification.
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[PROBABILITY & STATISTICS]
Acknowledgement We at first bow our heads before Allah Almighty, who has bestowed His countless blessings upon ourselves, guided us towards the way of success, blessed us with the courage of facing the problems and obstacles and finally enabled us to accomplish this project work. We wish to place our deep sense of thanks to our respected teacher Mr. Abid Awan, who guided us to complete this project in a true sense. His valuable experience and knowledge of the field removed the difficulty at all crucial junctures. In the end, we pay our compliments to all those people who had been cooperative to us during the completion of the project and those who directly or indirectly helped us in completing our project on time.
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Dedication We are dedicating our project on the “Fan Industry of Gujrat” to our respected teacher Mr. Abid Awan for his effort to transfer the essence of his experience and knowledge to his students.
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S. No 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39
Table of contents Executive Summary Objective Introduction Companies selected Profile of Al Jamil Fans Profile of Gull Fans Profile of Finex Fans Raw Data Explanation of Raw Data Data Analysis Excel Work of Al Jamil Fans Manual Work of Al Jamil Fans Mean + Interpretation Median + Interpretation Standard Deviation + Interpretation Coefficient of Variation + Interpretation Skewness + Interpretation Mean Deviation About Mean + Interpretation Correlation + Interpretation Regression + Interpretation Excel Work of Gull Fans Manual Work of Gull Fans Mean + Interpretation Median + Interpretation Standard Deviation + Interpretation Coefficient of Variation + Interpretation Skewness + Interpretation Mean Deviation About Mean + Interpretation Correlation + Interpretation Regression + Interpretation Excel Work of Finex Fans Manual Work of Finex Fans Mean + Interpretation Median + Interpretation Standard Deviation + Interpretation Coefficient of Variation + Interpretation Mean Deviation About Mean + Interpretation Skewness + Interpretation Correlation + Interpretation
Innovators Page3
Page No 5 6 7 11 11 11 12 13 14 15 16 17 17 18 19 20 21 22 23 25 28 29 29 30 31 32 33 33 34 36 39 40 40 40 42 43 44 45 46
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Regression + Interpretation Table of Comparison Comparison Explanation Conclusion Recommendation(s)
48 50 51 54 55
Executive summary This project report is basically related to the fan industry of Gujrat. Gujrat is the city having a large number of fan manufacturing units and among them we have selected three fan manufacturing companies for our project. First of all, we have selected two important variables of these companies for our project and these variables are: advertisements and sales. The data related to these variables has been collected for each of the three companies individually. Further various statistical tools including mean, median, and mode, mean deviation for mean; standard deviation and coefficient of variation have been applied on both of the variables of each of the three companies. After that the correlation between these variables has been found and the regression analysis of these variables is also done. Finally each of the three companies is compared with each
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other on the basis of the results obtained after applying these statistical tools. And in the end the conclusion of the whole report is given after analyzing the data completely.
Objective The main objective of performing this project is to enhance our knowledge through practically applying those statistical techniques that we have been learning throughout the whole semester. Although we have been studying these techniques in many books but still we have never applied them practically in the real life, so this project has greatly helped us to understand the real life situations and applications of the various statistical techniques and enhanced our knowledge related to this particular field of study.
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Introduction: Fan is a daily use item. Its utility increases, especially in the summer season. The Fan Industry is producing about 5 to 6 million fans per annum and meeting successfully the local as well as the export demand. Out of the total production, approximately 30 per cent fans consist of pedestals, 7 per cent brackets and the remaining 63 percent are ceiling fans. The industry belongs to the light engineering industry category, and is one of the industries that existed at the time of independence. In the early 1950s, it was declared as cottage industry and its more than 50 per cent units still fall in this category.
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Final Project
[PROBABILITY & STATISTICS]
The story of emergence of electric fan manufacturing industry in Pakistan is one of those heroic efforts made by a few enterprising individuals who. Starting with nothing in early 1940s struggled against all odds and turned this concern into a most efficient industry in Pakistan. Fan industry is mainly confined to Gujranwala and Gujrat cities of the Punjab province. The reason for its remaining a cottage industry is that majority of the units does not have full facilities of production under one roof. They usually give orders to the units having machines for different parts like fan guards, blade castings, core laminations etc. These units have lathes, shapers, milling machines, and power pressers, die casting machines and electroplating equipments. Therefore, most of the units are simply assembling units. Thus, they do not give brand names to their products. Besides small and medium units, a few units are quite large and have integrated system i.e. from motor winding to high-pressure dies casting. These companies have reputed brand names and the qualities of their products are of international level. These units are the main players in the export field. The industry is producing a variety of products in different sizes and designs. The major products are: ceiling, pedestal, table, table-cum-pedestal fans, circulator fans, wall bracket, exhaust fans and propellers. The industry supplies quality products to the local markets, whether branded or unbranded, at competitive prices. About 400 units have a production capacity of 5 to 6 million fans, on single shift basis. The production is equal to the demand, including a nominal quantity of exports. The actual production has remained about 2.5 million fans per annum, showing a 50 per cent idle capacity. The demand for fans is continuously increasing due to increase in population and speedy migration towards big cities, and for exports. The other factor is that during the last few years, local demand for quality products is increasing fast as compared to low price goods. This means people are becoming quality conscious. Innovators Page7
Final Project
[PROBABILITY & STATISTICS]
Major fan producing countries are Japan, Korea, Taiwan, Hong Kong, India and China. Japan is covering high quality market segment of fan market. Korea and Hong Kong are in middle segment of market while Pakistan, India, Taiwan and China are supplying comparatively low quality products at cheaper prices. Exports: Although fans' exports have great potential, their export is negligible. In 1992-93, only two-lack fans were exported to only two countries, Iraq and Yemen. But now the industry is exporting fans to more than 25 countries. The figures given in table-I indicate that although the exports of ceiling fan increased from Rs.134 thousand to 209 million rupees, the trade is fluctuating a great deal. As against ceiling fans, the increase in pedestal fan export is quite stable i.e., continuously rising. Pakistan has also started exporting parts of fans. During the last four years, as shown in table-II, the amount earned from export rose from Rs4.2 million to Rs107 million in 1998-01, but decreased to about fifty per cent the very next year, 2001-02. However, in spite of big fluctuations, it is believed that if some bottlenecks are removed there is a big scope for enhancing the export of fans. Government Role: The situation is that from very the beginning, the government has not given any incentives to encourage fan industry i.e. whatever progress the industry made, it has made on its own resources, and is due to its dedicated and hard working manufacturers and laborers. In spite of the lack of proper training the industry has innovated and uplifted the standard to the level where its products can compete in the world markets. TABLE - I: Exports of fans
Innovators Page8
Final Project
[PROBABILITY & STATISTICS]
Year
Amount in (000 Rs.)
1989-90
134 1,289
1992-93 1993-94
81,762 1,826 18,636 15,137
1998-99 1999-00 2000-01 2001-02
2,552 32,432 15,689 66,370 59,259 161,554 208,963 183,613
TABLE - II Parts of Fans Exported Year
(Million Rs.)
1998-99 1999-00 2000-01 2001-02
4.2 16.6 109.7 57.5
TABLE - III: Fan Exports by Type (Million Rs.)
Ceiling Fans Innovators Page9
1998-99
1999-00
2000-01
2001-02
Rs.2.6 Million
Rs.15.7 Million
Rs.59.3 Million
Rs.208.9 Million
Final Project
[PROBABILITY & STATISTICS]
Pedestal Fans
Rs.32.4 Millions
Rs.66.4 Million
Rs.161.6 Million
Rs.183.6 Million
Table Fans
Rs.1.1 Million
-------------
------------------
Rs.0.9 Million
Exhaust Fans Other Fans
Rs.0.1 Million Rs.2.9 Million
Rs.0.1 Million Rs.6.9 Million
Rs.0.1 Million Rs.5.1 Million
---------------Rs.27.2 Million
Fan Blowers
Rs.24.9 Million
Rs.13.2 Million
Rs.0.7Million
----------------
Other Fans N.S
Rs.3.0 Million
Rs.9.7 Million
Rs.5.3 Million
Rs.2.0 Million
Total
Rs.67 Million
Rs.112 Million
Rs.232.1 Million
Rs.422.6 Million
Companies Selected: We have selected three fan manufacturing companies for our project and these companies are: •
Al Jamil Fans
•
Gull Fans
•
Finex Fans
The data related to these companies has been collected for the two specific variables which are advertisements and sales. Before going to the next section of the data analysis, we would first give a brief history or profile of each of the three companies. Profile of Al Jamil Fans: Al Jamil Fan industry was established in 1987 and the founder this industry was Mr. Khalid Javaid, who named the company after the name of his son Mr. Jamil Javaid. In the beginning this company had been operated on partnership basis and the name of Innovators Page10
Final Project
[PROBABILITY & STATISTICS]
the partner was Mr. Fateh Muhammad but with the passage of time Mr. Khalid Javaid became a sole proprietor and run the company individually. The company has also got ISO 9001 quality certificate
Profile of Gull Fans: This company was established in 1974 and its founder is Sufi Ghulam Hussain who had also named his company after the name of his son Gull Pervaiz Butt. The company also produces export quality fans and also export there products to Dubai and Saudi Arabia. This company has also got the quality certificate ISO 9001
Profile of Finex Fans: The company was established in 1994 and its founders name is Chaudhary Imran Cheema. This company is not a big one and is comparatively on a lower level as compared to the above two companies. The present chairman of the company is the brother of Chaudhary Imran and his name is Chaudhary Nouman Cheema. This company does not have any quality certificate.
Innovators Page11
Final Project
[PROBABILITY & STATISTICS]
Raw Data Al- Jamil Fan 2005 P.F=1700 C.F=1200 Sales Adver. Jan 475 900 1,382,000 6,250 Feb 450 925 1,353,500 6,000 March 500 875 1,378,500 7,000 April 500 900 1,343,250 7,500 May 470 900 1,324,250 5,500 Jun 400 1000 1,379,500 5,000 July 500 925 1,406,000 6,800 Aug 300 800 1,236,500 4,500
Fine X Fan 2005 P.F=1600 C.F=1150 Sales Adver. Jan 375 680 1,382,000 5,000 Feb 350 690 1,353,500 4,800 March 380 670 1,378,500 5,100 April 340 695 1,343,250 4,300 May 350 665 1,324,250 4,600 Jun 395 650 1,379,500 5,200 July 390 680 1,406,000 5,500 Aug 320 630 1,236,500 4,200
2006 P.F=1900 C.F=1350 Sales Adver. Jan 600 1300 1,927,000 9,000 Feb 550 1300 1,956,000 8,500 March 670 1250 1,994,500 9,100 April 500 1350 1,910,500 8,300 May 600 1300 1,921,250 8,900 Jun 700 1170 1,898,000 9,200 July 480 1100 1,915,000 7,000 Aug 450 1100 1,784,000 6,700
2006 P.F=1700 C.F=1250 Sales Adver. Jan 435 950 1,927,000 12,000 Feb 430 980 1,956,000 11,500 March 460 970 1,994,500 12,200 April 440 930 1,910,500 12,100 May 450 925 1,921,250 11,500 Jun 440 920 1,898,000 12,800 July 450 920 1,915,000 11,300 Aug 395 890 1,784,000 11,000
2007 P.F=1950 C.F=1400 Sales Adver. Jan 900 1400 3,015,000 9,500 Feb 900 1400 2,986,500 9,300 March 800 1450 2,978,250 9,000 April 600 1500 3,009,000 8,200 May 700 1500 3,021,000 9,500 Jun 700 1500 2,925,000 9,000 July 600 1400 2,955,000 7,500 Aug 550 1250 2,869,500 7,000
2007 P.F=1950 C.F=1350 Sales Adver. Jan 750 1150 3,015,000 15,500 Feb 770 1100 2,986,500 14,300 March 745 1130 2,978,250 15,000 April 740 1160 3,009,000 15,200 May 760 1140 3,021,000 14,700 Jun 735 1105 2,925,000 14,500 July 740 1120 2,955,000 14,800 Aug 710 1100 2,869,500 14,000
Gull Fans 2005 P.F=1600 C.F=1100 Sales Adver. Jan 650 1200 2,360,000 11,000 Feb 700 1100 2,330,000 9,500 March 600 1150 2,225,000 9,000 April 650 1250 1,479,000 8,500 May 550 1150 2,145,000 9,700 Jun 720 1300 2,582,000 12,000 July 500 1100 2,010,000 10,000 Aug 530 1200 2,168,000 9,800 2006 P.F=1700 C.F=1200 Sales Adver. Jan 1050 1500 3,585,000 16,000 Feb 1000 1450 3,440,000 14,500 March 1100 1500 3,670,000 16,100 April 1020 1470 3,498,000 15,000 May 1050 1400 3,465,000 14,300 Jun 1000 1500 3,500,000 14,500 July 1100 1500 3,670,000 14,900 Aug 1000 1300 3,260,000 15,500
Innovators Page12
2007 P.F=1900 C.F=1400 Sales Adver. Jan 950 2000 4,605,000 13,500 Feb 1000 1800 4,420,000 13,000 March 1050 1900 4,655,000 14,000 April 900 1950 4,440,000 12,500 May 1000 2000 4,700,000 13,200 Jun 1100 1700 4,470,000 13,000 July 1000 1980 4,672,000 14,000 Aug 950 1780 4,297,000 12,000
Final Project
[PROBABILITY & STATISTICS]
Explanation of Raw Data Given above is the data of three fan manufacturing companies that are Al Jamil Fans, Gull Fans and Finex Fans. The data has been collected for 24 months. We have selected two variables of each of these companies these are: •
Sales
•
Advertisements
We will calculate the sales of the fans and the advertising expenditures of these companies. For calculating the sales, we have chosen two types of fans that are produced by each of these companies, and these fans are: pedestal and ceiling fans. The price of a single unit of each of the two fans is multiplied by the number of units sold per month and it makes the total sales of the fans per month, and finally by adding the sales of each month we get the overall three year sales of the companies. Same is the case with the advertising expenditures, for which again we will calculate the overall advertising expenditure of three years for all these three companies. Finally this data will be analyzed by using various statistical tools an actual calculations and interpretation of this data is shown further.
Innovators Page13
Final Project
[PROBABILITY & STATISTICS]
DATA ANALYSIS
Innovators Page14
Final Project
[PROBABILITY & STATISTICS]
Excel Work of Al-Jamil Fans Sale Advertisement 1,888 6 1,875 6 1,900 7 1,930 8 1,879 6 1,880 5 1,960 7 1,470 5 2,895 9 2,800 9 2,961 9 2,773 8 2,895 9 2,910 9 2,397 7 2,340 7 3,715 10 3,715 9 3,590 9 3,270 8 3,465 10 3,465 9 3,130 8 Sale Advertisement 2,823 7 Average 2,663 8 Median 2,811 8 Mean deviation about 593.02430 1.3046875 mean 6 Standard deviation 689.07258 1.529763586 8 Skewness -0.0453332 -0.522564549 Correlation 0.8861954 9
Intercept X Variable 1 Innovators Page15
Coefficients -417.7015121 399.1813025
Final Project
[PROBABILITY & STATISTICS]
Manual Work of Al-Jamil Fans
Mean In the very first phase we use the average method to analyze the data. Perhaps the most important measure of location is the mean, which is average value. Mean is taken by all small and large data values. It is denoted by x. As average value represent the whole data by taking the sum of all values and further divide by the total number of values. Mean of Sales: x = ∑x/ n = 63926/24 = 2,663.6 Interpretation: Here, we got the final value for sale that is 2,663.6. Which tells that average sale of the fans during last 24 months are 2,663.6 and this value is in thousands. Mean of advertisements: x = ∑x/ n = 187/24 = 7.8 Interpretation:
Innovators Page16
Final Project
[PROBABILITY & STATISTICS]
The average value for advertisement is 7.8 that are also in thousands. It is representing the whole data. It represents the amount that the companies spent on their advertisements in 24 months.
Median Median is also used to measure the average, the important characteristic of median is that it s used when there is variation in the data. Median is more appropriate for the data that has got some outliers. Median is the middle value of the data which is arranged in ascending order. The median provides the better value than the mean. Median of Sales: 1,470, 1,875, 1,879, 1,880, 1,887, 1,900, 1,930, 1,960, 2,340, 2,397, 2,773, 2,800, 2,895, 2,895, 2,910, 2,961, 2,823, 3,130, 3,270, 3,465, 3,465, 3,590, 3,715, 3,715, = (n + 1)th/2 = (24 + 1)th/2 = 12.5th Value = 12th value + .5 (13th – 12th) = 2,800 + .5 (2,823 – 2,800) = 2811.5 Interpretation: The median in this case is 2811.5 which shows the better measure of the sales of the company and is a better method of calculating the average. Median of Advertisements: Innovators Page17
Final Project
[PROBABILITY & STATISTICS]
5, 5, 6, 6, 6, 7, 7, 7, 7, 7, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 9, 10, 10 = (n + 1)th/2 = (24 + 1)th/2 =12.5th Value =12th value + .5 (13th – 12th) = 8 + .5 (8 – 8) =8 Interpretation: The median in this case is 8 which shows the better measure of the sales of the company and is a better method of calculating the average.
Standard Deviation Standard deviation is measured in the same units as the original data. It is the most appropriate and reliable method of calculating the dispersion. We use the standard deviation to bring the values to there actual position that have been made too large by using variance by squaring the values. So, for this reason we move towards standard deviation Standard Deviation of Sales: =√∑X2/n – (∑X/n)2 = √181193148/24 - (63926/24)2 = √7549714.5 – (2663.6)2 = √7549714.5 – 7,094,764.96 = √454,949.54
Innovators Page18
Final Project
[PROBABILITY & STATISTICS]
= 674.5 Interpretation: This value basically shows the variation in the data. The fluctuation of data around the mean is easily measured through this technique Here the spread in the data is 674.5, if there is less variation and spread in the data then the data is better. So we can say that the value of sales can lie between the ranges of 1989.1-3338.1 that is in thousands. Standard Deviation of Advertisements: =√∑X2/n – (∑X/n)2 = √1507/24 - (187/24)2 = √62.8 – (7.79)2 = √62.8 – 60.68 = √2.2 = 1.48 Interpretation: Here the spread in the data is 1.48, if there is less variation and spread in the data then the data is better. So we can say that the value of advertisements can lie between the ranges of 6.32 – 9.28 that is in thousands.
Coefficient of Variation Coefficient of variation is used for the comparison between the two different variables. This is the relative measure of variation; it measures the standard deviation relative to mean. Coefficient of Variation of Sales: = S/ ҳ X 100 = 674.5/2663.6 X 100 Innovators Page19
Final Project
[PROBABILITY & STATISTICS]
= 25.322 Interpretation: The coefficient of variation of sales according to the situation is 25.322 which shows the per unit risk associated with a given variable. The smaller the CV the less risk involved, and the greater the CV the high risk involved. Coefficient of Variation of Advertisements: = S/ ҳ X 100 = 1.48/7.8 X 100 = 18.97 Interpretation: The coefficient of variation of advertisements according to the situation is 18.97 which shows the per unit risk associated with a given variable. The smaller the CV the less risk involved, and the greater the CV the high risk involved.
Skewness Skewness is a statistical tool that tells us about whether the given data is normally distributed or not. If the data is not normally distributed, then it would either be positively skewed or negatively skewed. Skewness of Sales: = 3 (mean – median)/S.D = 3 (2663.6 – 2847.5)/ 674.5 = - 0.818 Interpretation:
Innovators Page20
Final Project
[PROBABILITY & STATISTICS]
In this case the value of Skewnesss is -0.818 which clearly shows that the data is negatively skewed and in case of negative Skewnesss, the mean of the data is the smallest, median is greater and finally the mode is the greatest one.
Skewness of Advertisements: = 3 (mean – median)/S.D = 3 (7.8 – 8)/1.48 = -0.40 Interpretation: The value of Skewnesss of advertising expenditures is -0.40 which is again negatively skewed.
Mean Deviation about Mean Mean deviation tells us about the actual deviation of the data from its arithmetic mean. One of the drawbacks of this technique is that it ignores all the negative values due to modulus. So it is not very much reliable. M.D (mean) of sales: = ∑|x-x|/n = 14236/24 = 593.16 Interpretation:
Innovators Page21
Final Project
[PROBABILITY & STATISTICS]
The value of mean deviation about mean of sales is 593.16 which shows the deviation of the data from the mean value which shows that the data deviates fro the mean by an amount of 593.16. M.D (mean) of Advertisements: = ∑|x-x|/n = 29/24 = 1.20
Interpretation: The value of mean deviation about mean of sales is 1.20 which shows the deviation of the data from the mean value which shows that the data deviates fro the mean by an amount of 1.20.
Correlation It is basically a descriptive measure of the strength of linear association between two variables one is independent and the other one is dependent. Values of the correlation coefficient are always between -1 and +1. A value of +1 indicates that two variables are perfectly positively related and similarly a value of -1 indicates that two variables are strongly negatively related. Here we have given a scale of measurement of correlation between two variables.
Innovators Page22
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[PROBABILITY & STATISTICS]
|______________|________________|_______________|______________| -1
-.5
Perfect -ve
0
Moderate
No relation
+.5 Moderate
+1 Perfect +ve
Table X
Y
6
1,888
6
1,875
7
1,900
8
1,930
6
1,879
5
1,880
7
1,960
5
1,470
9
2,895
9
2,800
9
2,961
8
2,773
9
2,895
9
2,910
7
2,397
Innovators Page23
Xy
x2
y2
11,328
36
3564544
11,250
36
3515625
13,300
49
3610000
15,440
64
3724900
11,274
36
3530641
9,400
25
3534400
13,720
49
3841600
7,350
25
2160900
26,055
81
8381025
25,200
81
7840000
26,649
81
8767521
22,184
64
7689529
26,055
81
8381025
26,190
81
8468100
16,779
49
5745609
Final Project
[PROBABILITY & STATISTICS]
7
2,340
10
3,715
9
3,715
9
3,590
8
3,270
10
3,465
9
3,465
8
3,130
7
2,823
187
63926
16,380
49
5475600
37,150
100
13801225
33,435
81
13801225
32,310
81
12888100
26,160
64
10692900
34,650
100
12006225
31,185
81
12006225
25,040
64
9796900
19,761
49
7969329
518,245
1507
181193148
N = 24 ∑x = 187 ∑y = 63926 ∑xy = 518245 x2 = 1507 y2 = 181193148 r = n∑xy – (∑x)(∑y)/ √{n∑x2 – (∑x)2}{n∑y2 – (∑y)2} = 24(518245) – (187) (63926)/ √{24(1507) – (187)2}{24(181193148) – (63926)2} = 0.28 Interpretation: The value of correlation between sale and advertisements of this company is 0.28 which shows that there exists a weekly positive relation between these two variables. Innovators Page24
Final Project
[PROBABILITY & STATISTICS]
Regression Analysis Regression analysis is basically used to forecast the future. In regression terminology, the variable being predicted is called the dependent variable. The variable or variables being used to predict the value of dependent variable called the independent variable. For example in analyzing the effect of advertising expenditures on sales, a marketing manager’s desire to predict sales would suggest making sales the dependent variable. Advertising expenditure would be the independent variable used to help predict sales. Y = a + bx b = n ∑xy – (∑x)(∑y)/ n ∑x2 – (∑x)2 = 403.43 a = ∑y – b∑x/n = 63926 – 403.43(187)/24 = -479.8 Y = -479.8 + 403.43x
Interpretation: The regression equation of Al Jamil Fans shows that -479.8 is a fixed value which stays the same even if the value of b is zero. By changing the number of units or the value of x, we get different values of y. Y = -479.8 + 403.43(7) = -479.8 + 2,824.01 = 2,344.21
Innovators Page25
Final Project
[PROBABILITY & STATISTICS]
If in x we put 7 the value which we got is 2,344.21. Due to one unit increase in x the value of y-intercept increase and automatically there is increase in the final value of y which is the given value of x.
Bar Graph 4,000
3,500
3,000
Sales
2,500
Series1 Series2
2,000
1,500
1,000
500
0 1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
Advertising Expenditure
Innovators Page26
16
17
18
19
20
21
22
23
24
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[PROBABILITY & STATISTICS]
Line Graph 4,000
3,500
3,000
Sales
2,500
Series1 Series2
2,000
1,500
1,000
500
0 1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
Advertising Expenditure
Excel Work of Gull Fans
Gull Fans
Sales 2360 2330 2225 1479 2145 2582 2010 2168 3585 3440 3670 3498 3465 3500 3670 3260 4605 4420 4655 Innovators 4440 Page27 4700 4470 4672 4297
Advertisements 11 9.5 9 8.5 9.7 12 10 9.8 16 14.5 16.1 15 14.3 14.5 14.9 15.5 13.5 13 14 12.5 13.2 13 14 12
23
24
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[PROBABILITY & STATISTICS]
Coefficients -319.4670732 292.3509321
Intercept X Variable 1
Manual Work of Gull Fans Mean Mean Sales:
of
= ∑x/ n = 81646/24 = 3,401.9
Average Median Mean About Mean Skewness
Coefficient Variation Standard Deviation Correlation
Interpretation: Innovators Page28
Sales 3,402 3,499 Deviation 838.1875 -0.27873390 1 of 38.56 1013.45186 8 0.66999274 4
Advertisement 13 13 1.940972222 -0.394445995 17.83 2.322569635
Final Project
[PROBABILITY & STATISTICS]
Here, we got the final value for sale that is 3,401.9. Which tells that average sale of the fans during last 24 months are 3,401.9 and this value is in thousands. Mean of advertisements: = ∑x/ n = 305.5/24 = 12.73 Interpretation: The average value for advertisement is 12.73 that are also in thousands. It is representing the whole data. It represents the amount that the companies spent on their advertisements in 24 months.
Median Median of Sales: 1,479, 2,010, 2,145, 2,168, 2,225, 2,330, 2,360, 2,582, 3,260, 3,440, 3,465, 3,498, 3,500, 3,585, 3,670, 3,670, 4,297, 4,420, 4,440, 4,470, 4,605, 4,655, 4,672, 4,700, = (n + 1)th/2 = (24 + 1)th/2 = 12.5th Value = 12th value + .5 (13th – 12th) Innovators Page29
Final Project
[PROBABILITY & STATISTICS]
= 3,498 + .5 (3,500 – 3,498) = 3,499 Interpretation: The median in this case is 3,499 which shows the better measure of the sales of the company and is a better method of calculating the average. Median of Advertisements: 8.5, 9, 9.5, 9.7, 9.8, 10, 11, 12, 12, 12.5, 13, 13, 13.2, 13.5, 14, 14, 14.3, 14.5, 14.5, 14.9, 15, 15.5, 16, 16.1 =(n + 1)th/2 =(24 + 1)th/2 =12.5th Value =12th value + .5 (13th – 12th) = 13 + .5 (13.2 – 13) = 13.1
Interpretation: The median in this case is 13.1 which shows the better measure of the advertisements of the company and is a better method of calculating the average.
Standard Deviation Standard Deviation of Sales: =√∑X2/n – (∑X/n)2 Innovators Page30
Final Project
[PROBABILITY & STATISTICS]
= √319046621/24 – (81646/24)2 = √13,293,609.21 – (3,401.92)2 = √13,293,609.21 – 11,573,059.69 = √1,720,549.52 = 1,311.7 Interpretation: This value basically shows the variation in the data. The fluctuation of data around the mean is easily measured through this technique Here the spread in the data is 1311.7, if there is less variation and spread in the data then the data is better. So we can say that the value of sales can lie between the ranges of 2090.2-4713.6 that is in thousands. Standard Deviation of Advertisements: =√∑X2/n – (∑X/n)2 = √4012.83/24 – (305.5/24)2 = √167.20 – (12.73)2 = √167.20 – 162.05 = √5.15 = 2.27 Interpretation: This value basically shows the variation in the data. The fluctuation of data around the mean is easily measured through this technique Here the spread in the data is 2.27, if there is less variation and spread in the data then the data is better. So we can say that the value of sales can lie between the ranges of 10.46-15 that is in thousands.
Coefficient of Variation Innovators Page31
Final Project
[PROBABILITY & STATISTICS]
Coefficient of Variation of Sales: = S/ ҳ X 100 = 1311.7/3401.9 X 100 = 38.56 Interpretation: The coefficient of variation of sales according to the situation is 38.56 which shows the per unit risk associated with a given variable. The smaller the CV the less risk involved, and the greater the CV the high risk involved. Coefficient of Variation of Advertisement = S/ ҳ X 100 = 2.27/12.73 X 100 = 17.83 Interpretation: The coefficient of variation of advertisements according to the situation is 17.83 which shows the per unit risk associated with a given variable. The smaller the CV the less risk involved, and the greater the CV the high risk involved.
Skewness Skewness of Sales: = 3 (mean – median)/S.D = 3 (3401.9 - 3499)/ 1311.7 = -0.22 Interpretation: Innovators Page32
Final Project
[PROBABILITY & STATISTICS]
In this case the value of Skewnesss is -0.22 which clearly shows that the data is negatively skewed and in case of negative Skewnesss, the mean of the data is the smallest, median is greater and finally the mode is the greatest one. Skewness of Advertisements: = 3 (mean – median)/S.D = 3 (12.73 – 13.1)/2.27 = -0.49 Interpretation: In this case the value of Skewnesss is -0.49 which clearly shows that the data is negatively skewed and in case of negative Skewnesss, the mean of the data is the smallest, median is greater and finally the mode is the greatest one.
Mean Deviation about mean M.D (mean) of sales: = ∑|x-x|/n = 20116/24 = 838.17
Interpretation: The value of mean deviation about mean of sales is 838.17 which shows the deviation of the data from the mean value which shows that the data deviates fro the mean by an amount of 838.17. M.D (mean) of Advertisements: =∑|x-x|/n Innovators Page33
Final Project
[PROBABILITY & STATISTICS]
=45/24 =1.87 Interpretation: The value of mean deviation about mean of sales is 1.87 which shows the deviation of the data from the mean value which shows that the data deviates fro the mean by an amount of 1.87.
Correlation X
Y
11
2360
9.5
2330
9
2225
8.5
1479
9.7
2145
12
2582
10
2010
9.8
2168
16
3585
14.5
3440
16.1
3670
15
3498
14.3
3465
Innovators Page34
Xy
x2
y2
25960
121
5569600
22,135
90.25
5428900
20,025
81
4950625
12,571.5
72.25
2187441
20,806.5
94.09
4601025
30,984
144
6666724
20,100
100
4040100
21,246.4
96.04
4700224
57,360
256
12852225
49,880
210.25
11833600
59,087
259.21
13468900
52,470
225
12236004
49,549.5
204.49
12006225
Final Project
[PROBABILITY & STATISTICS]
14.5
3500
14.9
3670
15.5
3260
13.5
4605
13
4420
14
4655
12.5
4440
13.2
4700
13
4470
14
4672
12
4297
305.5
81646
50,750
210.25
12250000
54,683
222.01
13468900
50,530
240.25
10627600
62,167.5
182.25
21206025
57,460
169
19536400
65,170
196
21669025
55,500
156.25
19713600
62,040
174.24
22090000
58,110
169
19980900
65,408
196
21827584
51,564
144
18464209
4012.83
319046621
1075557
N = 24 ∑x = 305.5 ∑y = 81646 ∑xy = 1075557 X2 = 4012.83 Y2 = 319046621 r = n∑xy – (∑x)(∑y)/ √{n∑x2 – (∑x)2}{n∑y2 – (∑y)2} = 24(1075557) – (305.5) (81646)/ √{24(4012.83) – (305.5)2}{24(319046621) – (81646)2} = 0.507 Interpretation: Innovators Page35
Final Project
[PROBABILITY & STATISTICS]
The value of correlation between sales and advertisements of this company is 0.507 which shows that there exists a moderately positive relation between the sales and advertisements of Gull Fans.
Regression Analysis Y = a + bx b = n ∑xy – (∑x)(∑y)/ n ∑x2 – (∑x)2 = 292.35 a = ∑y – b∑x/n = 81646 – 292.35(305.5)/24 = -319.46 Y = -319.46 + 292.35x Interpretation: The regression equation of Gull Fans shows that -319.46 is a fixed value which stays the same even if the value of b is zero. By changing the number of units or the value of x, we get different values of y. Y = -319.46 + 292.35(9) = -319.46 + 2,631.15 = 2,311.69 If in x we put 9 the value which we got is 2,311.69. Due to one unit increase in x the value of y-intercept increase and automatically there is increase in the final value of y which is the given value of x.
Innovators Page36
Final Project
[PROBABILITY & STATISTICS]
Bar Graph 5000 4500 4000 3500
Sales
3000 Series1 Series2
2500 2000 1500 1000 500 0 1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
Advertising Expenditure
Line Graph 5000 4500 4000 3500
Sales
3000 2500 2000 1500 Innovators Page37 1000 500 0
Series1 Series2
Final Project
[PROBABILITY & STATISTICS]
Excel work of Finex Finex Fans Sales (1000s) 1236 1343 1324 1353 1382 1378 1379 1406 1784 1915 1921 1956 1927 1910 1994 1898 2869 2986 2925 3021 2951 2978 3009 3015
Innovators Page38
Mean Median S.D Skewness Coefficient of Variation Mean deviation about mean Co-relation
Advertisements (1000s) 4.2 4.3 4.6 4.8 5 5.1 5.2 5.5 11 11.3 11.5 11.5 12 12.1 12.2 12.8 14 14.3 14.5 14.7 14.8 15 15.2 15.5
Sale 2,078 1,918 687.6110503 0.37217925 0.32392
Advertisement 10 12 4.272949546 -0.466272 0.44263
594.5
3.75
0.916563558
Final Project
[PROBABILITY & STATISTICS]
Coefficients 534.3322469 147.4951257
Intercept X Variable 1
Manual Work of Finex Fans Mean Mean of sales: = Σx/n = 49864/24 = 2077.6 Interpretation: Here, we got the final value for sales that is 2,077.6 which tells that average sale of the fans during last 24 months are 2,077.6 and this value is in thousands. Mean of Advertisements: =Σx/n =251.1/24 =10.46 Interpretation: The average value for advertisements is 10.46 that are also in thousands. It is representing the whole data. It represents the amount that the companies spent on their advertisements in 24 months. Innovators Page39
Final Project
[PROBABILITY & STATISTICS]
Median Median of sales: 1236,1343,1324,1353,1382,1378,1379,1406,1784,1915,1921,1956,1927,1910,1994,189 8,2869,2925,3021,2951,2978,3009,3015 = (n+1)th/2 = (24+1)th/2 = 12.5th value = 12th value+0.5(13th-12th) = 1915+0.5(1921- 1915) = 1918 Interpretation: The median in this case is 1918 which shows the better measure of the sales of the company and is a better method of calculating the average.
Median of Advertisements: 4.2,4.3,4.6,4.8,5,5.1,5.2,5.5,11,11.3,11.5,11.5,12,12.1,12.2,12.8,14,14.3,14.5,14.7,14.8,15 ,15.2,15.5 = (n+1)th/n = (24+1)th/2 = 12.5th value Innovators Page40
Final Project
[PROBABILITY & STATISTICS]
=12th value+0.5(13th-12th) =11.5+0.5(12- 11.5) =11.75
Interpretation: The median in this case is 11.75 which shows the better measure of the advertisements of the company and is a better method of calculating the average.
Standard Deviation Standard deviation of sales: = √Σx2/n- (Σx/n)2 = √(114458756/24)-(2077.5)2 = √4769114.8-4316006.25 = √453108.5 = 673.13 Interpretation: This value basically shows the variation in the data. The fluctuation of data around the mean is easily measured through this technique Here the spread in the data is 673.13, if
Innovators Page41
Final Project
[PROBABILITY & STATISTICS]
there is less variation and spread in the data then the data is better. So we can say that the value of sales can lie between the ranges of 1404.47-2750.73 that is in thousands. Standard deviation of Advertisements: = √Σx2/n- (Σx/n)2 = √(2915.52/24)-(100) = √121.48-100 = √21.48 = 4.63 Interpretation: Here the spread in the data is 4.63, if there is less variation and spread in the data then the data is better. So we can say that the value of advertisements can lie between the ranges of 5.83-15.09 that is in thousands.
Coefficient of Variation Coefficient of variation of sales: = S/x X 100 = 673.13/2078 = 0.32393 Interpretation: Innovators Page42
Final Project
[PROBABILITY & STATISTICS]
The coefficient of variation of sales according to the situation is 0.32393 which shows the per unit risk associated with a given variable. The smaller the CV the less risk involved, and the greater the CV the high risk involved.
Coefficient of variation of Advertisements: = S/x X 100 = 4.63/10.46 = 0.44263 Interpretation: The coefficient of variation of advertisements according to the situation is 0.44263 which shows the per unit risk associated with a given variable. The smaller the CV the less risk involved, and the greater the CV the high risk involved.
Mean Deviation about Mean M.D (mean) of sales: = Σ|x-x|/n = 14271/24 = 594.6 Interpretation: The value of mean deviation about mean of sales is 594.6 which shows the deviation of the data from the mean value which shows that the data deviates fro the mean by an amount of 594.6.
Innovators Page43
Final Project
[PROBABILITY & STATISTICS]
M.D (mean) of Advertisements: = Σ|x-x|/n = 93.7/24 = 3.9 Interpretation: The value of mean deviation about mean of sales is 594.6 which shows the deviation of the data from the mean value which shows that the data deviates fro the mean by an amount of 594.6.
Skewness Skewness of sales: = 3(x-median)/S.D = 3(2077.6 – 1918)/673.13 = 0.711 Interpretation: In this case the value of Skewnesss is 0.711 which clearly shows that the data is positively or rightly skewed and in case of positive Skewnesss, the mean of the data is the largest, median is smaller then the mean and finally the mode is the smallest.
Skewness of Advertisements: = 3(x-median)/S.D = 3(10.46 – 11.75)/4.63
Innovators Page44
Final Project
[PROBABILITY & STATISTICS]
= -0.83 Interpretation: In this case the value of Skewnesss is -0.83 which clearly shows that the data is negatively or left skewed and in case of negative Skewnesss, the mean of the data is the smallest, median is greater then the mean and finally the mode is the larges
Correlation X
Y
x2
Xy
y2
5
1382
6910
25
1909924
4.8
1353
6494.4
23.04
1830609
5.1
1378
7027.8
26.01
1898884
4.3
1343
5774.9
18.49
1803649
4.6
1324
6090.4
21.16
1752976
5.2
1379
7170.8
27.04
1901641
5.5
1406
7733
30.25
1976836
4.2
1236
5191.2
17.64
1527696
12
1927
23124
144
3713329
11.5
1956
22494
132.25
3825936
12.2
1994
24326.8
148.84
3976036
12.1
1910
23111
146.41
3648100
Innovators Page45
Final Project
[PROBABILITY & STATISTICS]
11.5
1921
22091.5
132.25
3690241
12.8
1898
24294.4
163.84
3602404
11.3
1915
21639.5
127.69
3667225
11
1784
19624
121
3182656
15.5
3015
46732.5
240.25
9090225
14.3
2986
42699.8
204.49
8916196
15
2978
44670
225
8868484
15.2
3009
45736.8
231.04
9054081
14.7
3021
44408.7
216.09
9126441
14.5
2925
42412.5
210.25
8555625
14.8
2955
43734
219.04
8732025
14
2869
40166
196
8231161
251.1
49860
583658
3047.07
114482380
N = 24 ∑x = 251.1 ∑y = 49860 ∑xy = 583658 X2 = 3047.07 Y2 = 114482380
r = n∑xy – (∑x)(∑y)/ √{n∑x2 – (∑x)2}{n∑y2 – (∑y)2} Innovators Page46
Final Project
[PROBABILITY & STATISTICS]
= 24(583658)- (251.1)(49860)/ √{ 24(3047.07)-(251.1)2}{ 24(114482380)-(49860)2 =14007792-12519846/√ (73129.68-63051.21)(2747577120-2486019600) = 1487946/ √ (10078.47)(261557520) = 1487946/1623606.9 = 0.916
Interpretation: The value of correlation between sale and advertisements of this company is 0.916 which shows that there exists an almost perfectly positive relation between these two variables.
Regression Analysis Y = a + bx b = n ∑xy – (∑x)(∑y)/ n ∑x2 – (∑x)2 = 1487946/10078.47 = 147.64 a = ∑y – b∑x/n = 49860-147.64(251.1)/24 = 12787.6/24 = 532.82 Y = 532.82+147.64x Interpretation: The regression equation of Finex Fans shows that 532.82 is a fixed value which stays the same even if the value of b is zero. By changing the number of units or the value of x, we get different values of y. Innovators Page47
Final Project
[PROBABILITY & STATISTICS]
Y = 532.82 + 147.64(11) = 532.82 + 1,624.04 = 2,156.86 If in x we put 11 the value which we got is 2156.86. Due to one unit increase in x the value of y-intercept increase and automatically there is increase in the final value of y which is the given value of x.
Bar Graph 3500
3000
2500
Sales
2000 Series1 Series2 1500
1000
500
0 1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
Advertising Expenditure
Innovators Page48
16
17
18
19
20
21
22
23
24
Final Project
[PROBABILITY & STATISTICS]
Line Graph 3500
3000
2500
Sales
2000 Series1 Series2 1500
1000
500
0 1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
Advertising Expenditure
Innovators Page49
16
17
18
19
20
21
22
23
24
Final Project
[PROBABILITY & STATISTICS]
Explanation of comparison By comparing the data of the three companies selected by us, we have got the following results: Mean: The average sale of Al Jamil Fans for the given period of 24 months is 2663.6, whereas that of Gull Fans and Finex Fans is 3401.9 and 2077.6 respectively. By looking at this data we have found that the sales of Gull Fans are more as compared to the other two companies. Similarly, the average advertising expenditures of Al Jamil Fans is 7.8 and those of other two companies’ are 12.73 and 10.46 respectively. This data again shows that the advertising expenditures of Gull Fans are again more then the other two companies. The sales and advertising expenditures of Gull Fans is more because the company has more investment and production, so it sales more as compared to the other companies. Innovators Page50
Final Project
[PROBABILITY & STATISTICS]
Median: The median sales of Al Jamil Fans, Gull Fans and Finex Fans are 2847.5, 3499 and 1918. Again Gull fan has more value as compare to Al Jamil and Finex. Now in the case of advertisements the median expenditures for Al Jamil fans is 8, for Gull Fan 13.1 and for Finex 11.75. So the median value of advertising expenditures of Gull fans is more than other two companies and the reason is again the same because of more production and investment. Standard Deviation: Further by calculating the standard deviation of sales we have got the values of S.D for these companies as 674.5 for Al Jamil, 1,311.7 for Gull fan and 673.13 for the Finex fans. By using standard deviation we come to know that how much deviation is there in the data or in values from there actual values. However for sales the value for Finex is more reliable because there is less deviation in this value. But this is also clear that the mean of this company is not more than others. So, for getting the more accurate value further we calculate the CV. The standard deviation of advertising expenditures for Al Jamil fans is 1.48, for Gull fans are 2.27 and for the Finex fan is 4.63. In this scenario Al Jamil fans have less standard deviation as compared to Gull and Finex. This shows less deviation respectively. But mean is more for Gull fans from this we didn’t know which is better. For getting the more accurate value now we calculate the CV. Coefficient of Variation: The coefficient of variation of sales of Al Jamil Fans for the 24 months is 25.322, whereas that of Gull Fans and Finex Fans is 38.56 and 0.323 respectively. By looking at this data we have found that the per unit risk associated with the companies. According to these values, Gull Fans is associated with more risk as compared to the other two companies. The company having less value of CV has got less risk associated so the Finex Fans Has got less risk and is more reliable among these companies. Innovators Page51
Final Project
[PROBABILITY & STATISTICS]
Similarly, the coefficient of variation of advertising expenditures for Al Jamil Fans is 18.97 and those of other two companies’ are 17.83 and 0.44236 respectively. This data again shows that the advertising expenditures of Gull Fans are again more then the other two companies and again Finex Fans is more reliable. M.D (mean): The values of mean deviation about mean of sales for the three companies are 593.16, 838.17 and 594.6 and the values for the mean deviation about mean of the advertising expenditures for these companies are 1.20, 1.87 and 3.9 respectively. Correlation: The correlation values of these companies are 0.28 for Al Jamil Fans, 0.502 for Gull Fans and 0.916 for Finex Fans which shows that the relation between sales and advertisements of Al Jamil Fans is weakly positive which shows that the sales of this company depend on the advertisements but not too much. Similarly the value of 0.502 for Gull Fans shows that the sales and advertisements of this company have got a moderately strong relationship. And finally the value of 0.916 for Finex Fans shows that the relationship between its sales and the advertisements is a strongly positive, which means that the sales of this company directly depend on the advertisements. Regression: The regression equations of the three companies are: Al Jamil Fans
y = - 479.8 + 403.43x
Gull Fans
y = - 319.46 + 292.35x
Finex Fans y = 532.8 + 147.64x The regression equations of all these companies are positive.
Innovators Page52
Final Project
[PROBABILITY & STATISTICS]
Conclusion After collecting the data from the three fan manufacturing companies and analyzing the data by applying the statistical tools, we have concluded that the average sales and advertising expenditure of Gull Fans is more then Al Jamil Fans. The reason is that the Gull Fans has got a greater market share and has more investment and production as compared to the other two companies. Further the correlation between the advertising expenditure and sales is strongly positive for Finex Fans which means that the sales of this company greatly depend on the advertising of the company. Gull Fans has got a moderate correlation and Al Jamil Fans has got a weakly positively positive correlation. So Al Jamil Fans and Finex Fans are a bit behind Gull fans in case of their sales and advertising expenditures.
Innovators Page53
Final Project
[PROBABILITY & STATISTICS]
Recommendations •
As the sales are directly dependent on the advertising expenditure so all these three companies, especially Al Jamil Fans and Finex Fans have to work hard for their advertisements which would then increase their sales.
•
These companies should try to bring some innovations in order to attract more customers.
•
New designs should also be introduced in their products.
•
Display Centers of all these companies are not much attractive, so they should work to make their showrooms more attractive.
•
These companies have got a manual system of data handling which usually creates problems, so a computerized system should be introduced.
Innovators Page54
Final Project
•
[PROBABILITY & STATISTICS]
The management and staff of these companies is not much educated and qualified, so the staff should focus on their qualification.
Innovators Page55
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