Statictics And Measures Of Central Tendency

STATICTICS AND MEASURES OF CENTRAL TENDENCY

STATICTICS The word statistics refers to quantitative information or to a method of dealing with quantitative information. The methods by which statistical data are analyzed are called STATISTICAL METHODS. STATISTICAL METHODS are applicable to a very large number of fieldseconomics,sociology,antropology,business,agricultur e,psychology,medicines,education.

FUNCTIONS OF STATISTICS It presents facts in a definite form It simplifies mass of figures It facilitates comparison It helps in formulating and testing hypothesis It helps in prediction It helps in the formulation of suitable policies

CHARTING DATA Most appropriate way to represent the data is

through charts. Pictorial representation helps in quick

understanding of the data. A chart can take the shape of either a

diagram or a graph.

A pie chart is a circular representation of data when a circle is divided into sectors with areas equal to the corresponding component. These sectors are called slices and represent the percentage breakdown of the corresponding components.

BAR-CHART A bar chart is a graphical device

used in depicting data that have been summarized as frequency, relative frequency, or percentage frequency

The statistical meaning of histogram is that it is graph that represent the class frequencies in frequency distribution by vertical adjacent rectangle Example: The following is a list of prices (in dollars) of birthday cards found in various drug stores: 1.45 0.78 1.75

2.20 0.75 1.23 1.25 1.25 3.09 1.99 2.00 1.32 2.25 3.15 3.85 0.52 0.99 1.38 1.22 1.75

we Make a frequency distribution table for this data. We omit the units (dollars) while calculating. The values go from 0.52 to 3.85 , which is roughly 0.50 to 4.00 . We can divide this into 7 intervals of equal length: 0.50 - 0.99 , 1.00 - 1.49 , 1.50 1.99 , 2.00 - 2.49 , 2.50 - 2.99 , 3.00 - 3.49 , and 3.50 - 3.99 . Then we can count the number of data points which fall into each interval--for example, 4 points fall into the first interval: 0.75, 0.78, 0.55, and 0.99--and make a frequency distribution table:

Intervals in (dollars) 0.50- 0.99 1.00- 1.49 1.50 -1.99

frequency 4 7 3

2.00 -2.49 2.50 -2.99 3.00 -3.49

3 0 2

3.50 -3.99 Total

1 20

Making a Histogram Using a Frequency Distribution Table A histogram is a bar graph which shows frequency distribution. To make a histogram, follow these steps: 1.On the vertical axis, place frequencies. Label this axis "Frequency". 2.On the horizontal axis, place the lower value of each interval. Label this axis with the type of data shown (price of birthday cards, etc.) 3.Draw a bar extending from the lower value of each interval to the lower value of the next interval. The height of each bar should be equal to the frequency of its corresponding interval.

Example: Make a histogram showing the frequency distribution of the price of birthday cards.

Histogram

A frequency polygon is a graph of frequency distribution. It is particularly effective in comparing two or more frequency distribution. There are two way in which a frequency polygon may be constructed: 1. we may draw histogram of the given data and then join by straight lines the mid-point of the upper horizontal side of each rectangle with the adjacent rectangle. 2. anther method of constructing frequency polygon is to take the mid-points of the various class-intervals and then plot the frequency corresponding to each point and to join all these points by straight lines.

# Draw an ogive and determine the no. of companies getting profits between Rs. 45 crore and Rs. 75 crore.

OGIVE

75

PARETO CHART  It is a special type of vertical bar chart in which the categorized responses are plotted in the descending rank order of their frequencies and combined with a cumulative polygon on the same graph.

 The main focus of the Pareto chart is to separate the “vital few” from the “trivial many.”

Case: A departmental store conducted a customer satisfaction survey of 455 frequent customers . The survey group prepared a questionnaire that was divided in two parts: satisfaction reasons and dissatisfaction reasons. Departmental store has decided to focus on the reasons of dissatisfaction to improve its quality of service. The following observations regarding categories of dissatisfaction were made: Sl. No. 1 2 3 4 5 6 Total

Dissatisfaction reasons Poor product range Lack of staffs Late Home delivery Parking Missing price tags No card payment service

No. of customers 125 40 65 30 75 120 455

Pareto chart of the case of department store

STEM AND LEAF PLOT 

Stem-and-leaf plot can be constructed by separating the digits of each number into two groups, one as a stem and the other as a leaf.



This plot is mainly used for examining the shape and spread of data.



After separating the data, the left-most digit is termed as the stem and is the higher valued digit. The right-most digit is termed as the leaf and is the lower valued digit.

State of auto manufacturing According to data released by the Automotive News Data Center, General MotorsCorporation is number one in the world in total vehicle sales of cars and light trucks. FordMotor Company is number two followed by Toyota Motor Corporation and Volkswagen,respectively. Between 1999 and 2000, while General Motors maintained its number oneposition, it sold nearly 200,000 fewer cars worldwide. During this same period, Ford Motorincreased sales by more than 200,000. The greatest percentage growth from 1999 to 2000was for PSA Peugeot-Citroen, which increased sales by 14.2%. The global sales figures Company 1999 2000 % change forthe top 10 auto manufacturers of cars and light trucks for both 1999 and 2000 follow.Suppose you are a businessGeneral analyst for motors 8786000 8591327 -2.2 one of these companies. Your manager asks you toprepare a brief report showing the state of car and light truck sales in the Fordcompare motorsyour company's 7148000 2.8 world. You areto position with7350495 other firms.

Toyota motors

5359000

5703446

6.4

volkswagen

4860203

5161188

6.2

daimlerchrysler 4864500

4749000

2.4

Psa peugeotcitroen

2519600

2877900

14.2

fiat

2521000

2646500

5.0

Hyundai motor

2600862

2634560

1.3

Nissan motor

2567878

2629044

2.4

Honda motor

2395000

2540000

6.1

Q.:Suppose DaimlerChrysler randomly samples 40 dealerships and discovers that the following data tell how many car and light trucks were sold at these dealerships last month. How can you summarize these data in a report? 34 58 40 49 49 57 44 57 69 45 64 31 47 30 44 44 51 65 60 65 61 62 68 43 66 63 44 34 57 44 67 61 47 67 52 34 58 59 33 SOLUTION: A stem and leaf plot of these data would appear as follows. FREQUENCY Stem & Leaf 6.00 3. 0 1 3 4 4 4 15.00 4. 0 3 4 4 4 4 4 5 5 7 7 9 9 10.00 5. 1 2 7 7 7 8 8 9 13.00 6. 0 I I 2 3 4 55 6 7 7 8 9  

45

SCATTER PLOT  The scatter plot is a graphical presentation of the relationship between two numerical variables.

 It generally shows the nature of the relationship between two variables.

 The application of a scatter plot is very common in regression, multiple regressions, correlation, etc.

Q. How might you graphically depict the 1999 data against the 2000 data?

Company

1999

2000

% change

General motors 8786000

8591327

-2.2

Ford motors

7148000

7350495

2.8

Toyota motors

5359000

5703446

6.4

volkswagen

4860203

5161188

6.2

daimlerchrysler 4864500

4749000

2.4

Psa peugeotcitroen

2519600

2877900

14.2

fiat

2521000

2646500

5.0

Hyundai motor

2600862

2634560

1.3

Nissan motor

2567878

2629044

2.4

Honda motor

2395000

2540000

6.1

Representation through scatter plot SALESVALUESFORYEAR1999AND2000 10000000 8000000 6000000 XAXIS-1999, Y AXIS2000

4000000 2000000 0 0

5000000

10000000

ARITHMETIC MEAN  The arithmetic mean (AM) of a set of observations is their sum, 

divided by the number of observations. It is generally denoted by x or AM. Population mean is denoted by μ.

Arithmetic mean is of two types:

 Simple arithmetic mean  Weighted arithmetic mean

Weighted Arithmetic Mean

 The weighted mean enables us to calculate an average

that takes into account the importance of each value to the overall total.

GEOMETRIC MEAN  Geometric mean (GM) is the nth root of the product of n items of a series.



Commonly used in the calculation of average rate of growth.

Chemical,industrial & pharmaceutical laboratories(cipla) Cipla was registered as a public ltd co. with authorized capital of rs.60000 million in 1935.operations officially started in sept 1937.its products & services are categorized as prescription,animal health care products etc.It now exports to countries in europe,america,africa,asia,mid-east,asia.it has won “EXPRESS PHARMA PULSE AWARD” for “sustained growth” for 205-06.It overtook ranbaxy and glaxosmithkline to become the largest pharmaceutical co. in the domestic market in 2007.

Mean= 8631.138889

HARMONIC MEAN  Based on the reciprocal of the numbers averaged.  Defined as the reciprocal of the arithmetic mean of the

reciprocal of the individual observation.  It can be written as

Applications of Harmonic Mean Useful for computing average rates

e.g. Average rate of increase of profits or average speed at which any journey has been performed.

Chemical,industrial & pharmaceutical laboratories(cipla)

Sales turnover for year 1989-2006 SALES (In million Rs.) (x)

YEAR

1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006

971.2 928.9 1236.4 1514 1990.3 2454.7 2987.1 3623.6 4525.8 5170.8 6255.4 7721.4 10643.1 14008.1 15730.2 20554.2 24008.9 31036.2

1/X

0.0010297 0.0010765 0.0008088 0.0006605 0.0005024 0.0004074 0.0003348 0.000276 0.000221 0.0001934 0.0001599 0.000129 0.000093 0.000071 0.000063 0.000048 0.000041 0.000032 ∑1/x = 0.0061473

n=18

∑ 1/x=0.0061473 Harmonic Mean =

= =

n ∑ 1/x 18/0.0061473 2928.1148

Merit and limitations It is useful in special cases of averaging rates. It can not be used when there are both

positive and negative observations or one or more observations have zero values. It is rarely used in business problems

The median may be defined as the middle or central value of

the variable when values are arranged in the order of magnitude. In other words, median is defined as that value of the variable that divides the group into two equal parts, one part comprising all values greater and the other all values lesser than the median.

To measure the qualitative characteristics of data, other measures of central tendency, namely median and mode are used.

 Positional averages, as the name indicates, mainly focus on the position of the value of an observation in the data set.

UNGROUPED DATA: 1) If n(no. of observation) is odd, middle term of a series is size of ((n+1)/2)th term & is the value of a median. 2) If n(no. of observation) is even, middle term of a series are (n/2)th and (n/2+1)th terms . So, arithmetic mean of both the observations is our median. GROUPED DATA: MEDIAN=L+ [N/2-p.c.f]/F* I Where L=lower limit of class N=sum of all the frequencies p.c.f=preceding cumulative frequency to median class F=frequency of median class i=class interval of median class

Chemical,industrial & pharmaceutical laboratories(cipla) Cipla was registered as a public ltd co. with authorized capital of rs.60000 million in 1935.operations officially started in sept 1937.its products & services are categorized as prescription,animal health care products etc.It now exports to countries in europe,america,africa,asia,mid-east,asia.it has won “EXPRESS PHARMA PULSE AWARD” for “sustained growth” for 205-06.It overtook ranbaxy and glaxosmithkline to become the largest pharmaceutical co. in the domestic market in 2007. SALES TURNOVER FROM 1989-2006 MEDIAN : total observations(n)=18 n/2th term=9th term=4525.8 (n/2+1)th term=10th term=5170.8 median=(4525.8+5170.8)/2 =4848.3

MODE

UNGROUPED DATA: Mode is that value which occurs the maximum no. of times GROUPED DATA: Mode=L+(f-f1)/(2f-f1-f2) *I Where, L=lower limit of modal class f = frequency of modal class f 1=frequency of class preceding the modal class f 2=frequency of class succeeding the modal class i=size of modal class

Chemical,industrial & pharmaceutical laboratories(cipla)  Cipla was registered as a public ltd co. with authorized capital of rs.60000 million in 1935.operations

officially started in sept 1937.its products & services are categorized as prescription,animal health care products etc.It now exports to countries in europe,america,africa,asia,mid-east,asia.it has won “EXPRESS PHARMA PULSE AWARD” for “sustained growth” for 205-06.It overtook ranbaxy and glaxosmithkline to become the largest pharmaceutical co. in the domestic market in 2007.  SALES TURNOVER FROM 1989-2006

mode=0

RELATIONSHIP BETWEEN MEAN,MEDIAN & MODE *distribution in which values of mean,median,mode coincide are symmetrical distribution. *distribution in which values of mean,median,mode are not equal are asymmetrical or skewed. The distance between mean & median is approximately one-third of the distance between the mean and mode. Acc. To Karl Pearson : Mean-median=1/3(mean-mode) Mode=3median-2mean Median=(2mean+mode)/3

QUARTILES  Partition values are measures that divide the data into several

equal parts. Quartiles divide data into 4 equal parts, deciles divide data into 10 equal parts, and percentiles divide data into 100 equal parts.

 For an individual series, the first and third quartiles can be computed using the following formula:  Values of quartiles can be measured as : Q(for k=1 to n)=k(n+1)/4

 Example : Q1=first quartile = (n+1)/4

DECILES  In a data series, when the observations are arranged

in an ordered sequence, deciles divide the data into 10 equal parts. In the case of individual series and discrete frequency distribution, the generalized formula for computing deciles is given as: Values of decile can be measured as : D(for k=1 to n)=k(n+1)/10 Example: D1=(N+1)/10

PERCENTILES  In a data series, when observations are arranged in an ordered

sequence, percentiles divide the data into 100 equal parts. For an individual series and a discrete frequency distribution, the generalized formula for computing percentiles is given as: VALUES OF PERCENTILE CAN BE MEASURED AS : P(for k=1 to n)=k(n+1)/100 Example : P1=(n+1)/100

 Example: From the following data, find the first and third

quartiles.

 The first and third quartiles can be computed by applying the

formula discussed above. The data is already arranged in an ordered manner:

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