measures of central tendency

Question

The following give the weights of 50 men in a sports camp

64 14 17 48 58 60 43 44 15 32 47 21 23 37 51 26 10 36 22 24 43 45 29 33 46 38 19 38 36 30 31 72 57 41 41 44 54 24 26 41 22 25 35 35 37 36 51 52 19 62

(a) Using a class interval 10 – 19, 20 – 29,…   calculate 

(i) median 

(ii) mode of the distribution

      (iii) What is the mean mark of the distribution?

Answer 1

First, arrange the data in ascending order. That will help answer (i) and (ii).

The intervals and their frequencies are as follows:

10-19 -- 6

20-29 -- 10

30-39 -- 13

40-49 -- 11

50-59 -- 6

60-69 -- 3

70-79 -- 1

(i) the median is the middle value in an ordered set of data. When N=50, the median is the average of values 25 and 26 (this is because N is even). Looking at the intervals and their frequencies, it is evident that the median is in the interval 30-39, because the sum of the frequencies of the first 3 intervals = 29, so values 25 and 26 are included. As it happens value 25=36 and value 26=37 and their average is 36.5. This is the median.

(ii) the mode is the value that occurs most frequently. It is not obvious from looking at the intervals where the mode is. If you look at the ordered data you'll see that the value 36 occurs 3 times as does the value 41. Thus there are 2 modal values, 36 and 41, because they happen most frequently and they are in different intervals.

(iii) the location of the mean is not necessarily obvious from the intervals. It is most likely to be in the intervals 30-39 or 40-49, but it very much depends on the individual data values. The mean is the sum of all values divided by the number of values contributing to the sum. Here x-bar works out to be 37.08, so it is in the same interval, 30-39, as the median.

I hope this helps!