Mode

Mode

Mode is defined as that value which occurs the maximum number of times i.e. having maximum frequency. For example, if the data is: Size of shoes

5

6

7

8

9

10

11

Number of persons

10

20

25

40

22

15

16

The modal size is 8 since it appears maximum number of times in the data.

or Wages per month 2000 3000 Rs.

4000

5000

7000

9000

10000

Number of persons 100

250

400

220

150

160

200

The modal wage of a group of workers is 5000 since the largest number of workers receive this wage and as such this wage may be considered as the representative wage of the group.

Why mode is used?

There are many situations in which arithmetic mean and median fail to reveal the true characteristics of data. For example:- when we talk of most common wage, most common income, most common height, most common size of shoes or ready made garments, we have in mind mode and not the arithmetic mean or median discussed earlier. The modal wage of a group of workers is that wage which the largest number of worker receive and as such this wage may be considered as the representative wage of the group.

The mean does not always provide an accurate reflection of the data due to the presence of extreme values. Median may also prove to be quite representative of the data due to uneven distribution of the series. For example:- the values in the lower half of a distribution ranges from, say, Rs. 10 to 100,while the same number of items in the upper half of the series ranges from Rs.100 to 6000 with most of them near the higher limit. In such a distribution the median value Rs.100 will provide little indication of the true nature of the data. Mode is used in business, because it is most likely to occur

Calculation of Mode

From a simple frequency distribution, mode can be determined by inspection only. It is the value of the variable which corresponds to the largest frequency. For example, Wages per month 2000 3000 Rs.

4000

5000

7000

9000

10000

Number of persons 100

250

400

220

150

160

200

The modal wage of a group of workers is 5000 which can be determined by inspection only.

Mode from grouped frequency distribution

From a grouped frequency distribution, it is difficult to find the mode accurately. However, if all classes are of equal width, mode is usually calculated by the following formula,

d1 Mode = I1 + -------------- x c d1 + d2 or f0 – f -1 Mode = I1 + ----------------------- x c 2f0 - f -1 - f 1 Where, I1 = Lower boundary of the modal class. d1 = f0 – f -1 = The difference between the frequency of the modal class and the frequency of the class just preceding the modal class. d2 = f0 – f 1 = The difference between the frequency of the modal class and the frequency of the class just succeeding (following) the modal class. c = Common width of the classes. If d1 = d2 then mode will lie exactly mid-way between upper and lower boundary of the modal class.

The formula is based on the assumption that the distances of mode from the two boundaries of modal class are proportional to the differences in frequencies of the modal class and its two adjoining classes i.e. d1 and d2. M0 – I1 d1 ----------------------- = -------------I2 -M0 d2

Where I1 = Lower class boundary of the modal class. I2 = Upper class boundary of the modal class.

A

distribution having only one mode is called unimodal.If it

contains more than one mode, it is called bimodal or multimodal. If however the frequency distribution has classes of unequal width or it contains more than one mode, the above formula can not be applied. In this case, an approximate value of mode is usually obtained by the following approximate formula based upon the relationship between mean, median and mode when the value of mean and median are known or mode is obviously eliminated as a measure of central tendency.

Mean – Mode = 3 (Mean – Median) or Mode = 3 Median - 2 Mean

Illustration 1:The following data relate to the sales of 100 companies Sales (Rs.lakhs)

Number of companies

Below 60

12

60 - 62

18

62 – 64

25

64 – 66

30

66 – 68

10

68 – 70

3

70 - 72

2

d1 Mode = I1 + -------------- x c d1 + d2 Here, I1 = 64 d1 = f0 – f -1 = 30 – 25 = 5 d2 = f0 – f 1 = 30 – 10 = 20 c=2 5 Mode = 64 + -------------- x 2 5 + 20 = 64 + (10/25) = 64.4 (Ans)

Locating Mode graphically The monthly profits in rupees of 100 shops are distributed as follows,

Profits per shop0 - 100

100 - 200

200 - 300 300 - 400

400 - 500

500 - 600

Number of shops

18

27

17

6

12

20

Draw the histogram to the data and hence find the modal value graphically. Check this value by direct calculation.

In a frequency distribution the value of mode can also be determined graphically. The steps are:1) Draw a histogram of the given data. 2) In the histogram ,the top right corner of the highest rectangle is joined by a straight line to the top right corner of the preceding rectangale.Similarly, the top left corner of the highest rectangle is joined by a straight line to the top left corner of the following rectangle. 3) From the point of intersection of these two lines a perpendicular is drawn on the horizontal axis. The foot of the perpendicular indicate the Mode. From the above diagram , the modal value is found to be Rs.256 approximately. Mode can also be determined from frequency polygon in which case perpendicular is brawn on the base from the apex of the polygon and the point where it meets the base gives the modal value.

or We may calculate the mode directly. by simply applying the formula i.e. d1 Mode = I1 + -------------- x c d1 + d2 Here, I1 = 200 d1 = f0 – f -1 = 27 – 18 = 9 d2 = f0 – f 1 = 27 – 20 = 7 c = 100 9 Mode = 200 + -------------- x 100 = 256.25 (Ans)

9 +7

Advantages 1)

2) 3) 4)

From a simple frequency distribution, mode can be obtained only by inspection. Also, for a simple series with a small umber of observations, mode can often be determined without any calculation. Mode is unaffected by the presence of extreme value. Unlike A.M. ,it can be calculated from frequency distribution, with open end classes. Mode can be easily used to describe qualitative phenomenon. For example:-when we want to compare the consumer preference for different types of products, say, soap. toothpastes,etc.we should compare the modal preferences. In such distributions where there is an outstanding large frequency, mode happens to be meaningful as an average.

Disadvantages 1) 2)

c) d) e)

Mode has no significance unless a large number of observations is available. It is particular measure of central tendency. For any given set of observations, it is always possible to find the values of A.M.,G.M.,H.M., or Median. But mode may not exist. When all values occur with equal frequency, there is no mode. On the other hand if two or more values have the same maximum frequency, there is more than one mode. For example:Mode of observations 2,5,8,4,3,4,5,2,4 is 4. For the observations 5,3,6,3,5,6, there is no mode. For the observations 5,3,6,3,5,10,7,2, there are two modes i.e. 3 and 5.

3) From the grouped frequency distribution, it is difficult to locate the mode accurately. An approximate value of the mode is obtained by the formula .But if the frequency distribution has classes of unequal width or it contains more than one mode, the above formula can not be applied. 4) Mode can not be treated algebraically.