Preface
I was reading the textbook Fundamentals of Mathematical Statistics by S.C. Gupta and V.K. Kapoor and I came across exceptional cases where mode is simply not the observation with the highest frequency. The book states for ungrouped data: (pg. 2.17)
But in any one (or more) of the following cases:
(i) If the maximum frequency is repeated,
(ii) if the maximum frequency occurs at the very beginning or at the end of the distribution, and
(iii) if there are irregularities in the distribution, the value of mode is determined by the method of grouping
Similarly for grouped data, it states: (pg. 2.21)
Remarks. 1.
In case of irregularities in the distribution or the maximum frequency being repeated or the maximum frequency occurring in the very beginning or at the end of the distribution, the modal class is determined by the method of grouping and the mode is obtained by using (2·7)2.7 is the formula for evaluating Mode for grouped data as follows: $$Mode = l + \frac{h(f_1 - f_0)}{2f_1 - f_0 - f_2} \qquad...(2.7)$$
The example given for the ungrouped data is as follows
| x | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| f | 3 | 8 | 15 | 23 | 35 | 40 | 32 | 28 | 20 | 45 | 14 | 6 |
After applying the method of grouping the mode is found to be 6 not 10.
Here are my doubts:
- Why does this case and the other exceptional cases exist?
- How does the method of Grouping solve such problems in these cases?
- When are these methods applicable and are there cases where these don't apply?
Also, considering this example
Let's say, I take the data of marks obtained by students out of 20 in our class of 70 students and it turns out that a score of 20 was obtained by 40 students, hence any other score can't have a higher frequency. In this case, can I say that the mode of the data is 20 or should I calculate the mode based on the method of grouping because 20 is at the extreme end of the distribution?
Images
Example question for ungrouped data from textbook pg 2.17
Solution for above question using the method of grouping pg 2.18
