Question

The following weights in kilograms were recorded for a hockey team:
73 75 76 77 78 86 81 75 100 92 82 73 85 79
84 92 80 78 77 81 83 74 80 69 92 79 72 76
a) Find the mean, mode, and median weights.
b) Which of these three measures of central tendency is the least representative of the set of weights? Why?

Answer 1

Solution

Back-up Theory

Let x_i = ith value, i = 1, 2, ……., n. Then,

Mean (Average), µ, = (1/n)Σ(i = 1, n)(x_i) ………….........................................................……………………………………. (1)

Median

Let x⁽¹⁾ , x⁽²⁾ , x⁽³⁾ , ……….. , x^{(n - 1)} , x⁽ⁿ⁾ be the ordered set of the given values; i.e.,

x⁽¹⁾ < x⁽²⁾ < x⁽³⁾ < ……….. < x^{(n - 1)} < x⁽ⁿ⁾

Case 1: n is even, say n = 2k

Median = Average of two middle values in the ordered set

= (x^(k) + x^{(k + 1)})/2 ............................................................................................................................................... (2a)

Case 2: n is odd, say n = 2k + 1

Median = (k + 1)th value in the ordered set; i.e., x^{(k + 1)} .................................................................................... (2b)

Mode is the value occurring maximum number of times (i.e., the value with the highest frequency) ............ (3)

Now, to work out the solution,

Final answers are given below. Details of Calculations follow at the end.

Part (a)

Mean = 80.32 Answer 1

Median = 79 Answer 2

Mode = 92 [occurs 3 times] Answer 3

Part (b)

Of three measures, mode is the least representative since it does not take into account all the observations.

Answer 4

Details of Calculations

Given data in the ordered form is given below:

i	xi
1	69
2	72
3	73
4	73
5	74
6	75
7	75
8	76
9	76
10	77
11	77
12	78
13	78
14	79
15	79
16	80
17	80
18	81
19	81
20	82
21	83
22	84
23	85
24	86
25	92
26	92
27	92
28	100
n =	28
Sum	2249
Mean	80.32143
Median	79
Mode	92

Since n = 28 is even, vide (2a), median = average of x⁽¹⁴⁾ and x⁽¹⁵⁾.

DONE