Question

Sample of Size 5
108108	110110	9999	9595	109109

Sample of Size 12
108108	110110	9999	9595	109109	9292
118118	111111	106106	9999	9191	101101

Sample of Size 30
108108	110110	9999	9595	109109	9292
118118	111111	106106	9999	9191	101101
9797	9393	103103	9494	9696	117117
9494	9696	106106	107107	9191	106106
101101	119119	105105	118118	107107	107107

1. What is the median of the sample of size​ 30?

2.For each data set recalculate the mean and​ median, assuming that the individual whose IQ is 108, and108 is accidently recorded as 180. What is the median of the new sample of size​ 30?

Answer 1

Solution

Back-up Theory

Mean (Average)

= (1/n)Σ(i = 1, n)xi.............................................................................................................................................. (1)

Median

Median of a set of values is that value which divides the set into two equal halves in terms of the number

of observations, i.e., half the number of observations lie below the median and half above the median. ...... (2)

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)

Now, to work out the solution,

Q1

Since n = 30 is even, vide (2a),

Median = [x⁽¹⁵⁾ + x⁽¹⁶⁾]/2

= (103 + 105)/2

= 104 Answer 1

Ordered set 1 follows at the end.

Q2

Since n = 30 is even, vide (2a),

Median = [x⁽¹⁵⁾ + x⁽¹⁶⁾]/2

= (103 + 105)/2

= 104 Answer 2

Ordered set 2 follows at the end

DONE

Ordered set 1

i	x(i)
1	91
2	91
3	92
4	93
5	94
6	94
7	95
8	96
9	97
10	97
11	99
12	99
13	101
14	101
15	103
16	105
17	106
18	106
19	106
20	107
21	107
22	107
23	108
24	109
25	110
26	111
27	117
28	118
29	118
30	119

Ordered set 2

i	x(i)
1	91
2	91
3	92
4	93
5	94
6	94
7	95
8	96
9	97
10	97
11	99
12	99
13	101
14	101
15	103
16	105
17	106
18	106
19	106
20	107
21	107
22	107
23	109
24	110
25	111
26	117
27	118
28	118
29	119
30	180