The mode is another measure of central tendency. It is the value at the point around which the items are most heavily concentrated. As an example, consider the following series: 8,9, 11, 15, 16, 12, 15,3, 7, 15

There are ten observations in the series wherein the figure 15 occurs maximum number of times three. The mode is therefore 15. The series given above is a discrete series; as such, the variable cannot be in fraction. If the series were continuous, we could say that the mode is approximately 15, without further computation.

In the case of grouped data, mode is determined by the following formula: While applying the above formula, we should ensure that the class-intervals are uniform throughout. If the class-intervals are not uniform, then they should be made uniform on the assumption that the frequencies are evenly distributed throughout the class. In the case of inequal class-intervals, the application of the above formula will give misleading results.

Example 2.11:           Let us take the following frequency distribution:

 Class intervals (1) Frequency (2) 30-40 4 40-50 6 50-60 8 60-70 12 70-80 9 80-90 7 90-100 4

We have to calculate the mode in respect of this series.

Solution: We can see from Column (2) of the table that the maximum frequency of 12 lies in the class-interval of 60-70. This suggests that the mode lies in this class- interval. Applying the formula given earlier, we get: In several cases, just by inspection one can identify the class-interval in which the mode lies. One should see which the highest frequency is and then identify to which class-interval this frequency belongs. Having done this, the formula given for calculating the mode in a grouped frequency distribution can be applied.

At times, it is not possible to identify by inspection the class where the mode lies. In such cases, it becomes necessary to use the method of grouping. This method consists of two parts:

1. Preparation of a grouping table: A grouping table has six columns, the first column showing the frequencies as given in the problem. Column 2 shows frequencies grouped in two's, starting from the Leaving the first frequency, column 3 shows frequencies grouped in two's. Column 4 shows the frequencies of the first three items, then second to fourth item and so on. Column 5 leaves the first frequency and groups the remaining items in three's. Column 6 leaves the first two frequencies and then groups the remaining in three's. Now, the maximum total in each column is marked and shown either in a circle or in a bold type.
2. Preparation of an analysis table: After having prepared a grouping table, an analysis table is prepared. On the left-hand side, provide the first column for column numbers and on the right-hand side the different possible values of mode. The highest values marked in the grouping table are shown here by a bar or by simply entering 1 in the relevant cell corresponding to the values they represent. The last row of this table will show the number of times a particular value has occurred in the grouping table. The highest value in the analysis table will indicate the class-interval in which the mode lies. The procedure of preparing both the grouping and analysis tables to locate the modal class will be clear by taking an example.

Example 2.12: The following table gives some frequency data:

 Size of Item Frequency 10-20 10 20-30 18 30-40 25 40-50 26 50-60 17 60-70 4

Solution: 6 1 1 1 Total 1 3 5 5 2

This is a bi-modal series as is evident from the analysis table, which shows that the two classes 30-40 and 40-50 have occurred five times each in the grouping. In such a situation, we may have to determine mode indirectly by applying the following formula:

Mode = 3 median - 2 mean

Median = Size of (n + l)/2th item, that is, 101/2 = 50.5th item. This lies in the class 30-40. Applying the formula for the median, as given earlier, we get

=30 + (40 - 30)/25 (50.5 - 28)

= 30+9 =39

Now, arithmetic mean is to be calculated. This is shown in the following table.

 Class- interval Frequency Mid- points d d' = d/10 fd' 10-20 10 15 -20 -2 -20 20-30 18 25 -10 -I -18 30-40 25 35 0 0 0 40-50 26 45 10 1 26 50-60 17 55 20 2 34 60-70 4 65 30 3 12 Total 100 34

Deviation is taken from arbitrary mean = 35 Mean =  A + å fd ' / n*i

= 35+34/100*10

=38.4

Mode =          3 median - 2 mean

=   (3 x 39) - (2 x 38.4)

=   117 -76.8

=     40.2

This formula, Mode = 3 Median-2 Mean, is an empirical formula only. And it can give only approximate results. As such, its frequent use should be avoided. However, when mode is ill defined or the series is bimodal (as is the case in the present example) it may be used.