Question

For the following scores, find the (a) mean, (b) median, (C) sum of squared deviations, (d) variance, and (e) standard deviatiorn 28, 24, 23, 27, 34, 30, 29, 30, 26

Answer 1

Solution

[NOTE:

Answers are given below. Detailed Working and Back-up Theory follow at the end.]

Part (a)

Mean = 27.89 Answer

Part (b)

Median = 28 Answer

Part (c)

Sum of Squared Deviations (from the mean) = 90.89   Answer

Part (d)

Variance = 10.0987 Answer

Part (e)

Standard Deviation = 3.18 Answer

Back-up Theory and Details of Working

Mean (Average) = (1/n)Σ(i = 1, n)xi = 251/9 = 27.89 Answer (a)

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⁾

When n is odd, say n = 2k + 1, Median = (k + 1)th value in the ordered set; i.e., x^{(k + 1)}

Ordered set of the given observations:

23
24
26
27
28
29
30
30
34

Median = 5^th observation = 28. Answer (b)

Sum of squared Deviations (from the mean median)

= (1/n)Σ(i = 1, n)(xi – Mean)² = 90.89 Answer (c)

Variance (V) and Standard Deviation (SD)

V = (1/n)Σ(i = 1, n)(xi – Mean)² [Defining Formula] or equivalently[{ (1/n)Σ(i = 1, n)(xi)²} – Mean² [computation Formula] = 10.0987 Answer (d)

SD = sqrt(V) = 3.18 Answer (e)

DONE