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Sample Variance

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How do i find sample variance

Guest Apr 5, 2017
#1
+1

THIS IS HOW YOU CALCULATE "SAMPLE VARIANCE." STUDY IT CAREFULLY:

Find the (sample) variance of the list:
(11, 23, 18, 8, 33, 29, 7, 21, 19, 27)

The (sample) variance of a list of numbers X = {X_1, X_2, ..., X_n} with mean μ = (X_1 + X_2 + ... + X_n)/n is given by:
(abs(X_1 - μ)^2 + abs(X_2 - μ)^2 + ... + abs(X_n - μ)^2)/(n - 1)

There are n = 10 elements in the list X = {11, 23, 18, 8, 33, 29, 7, 21, 19, 27}:
(abs(X_1 - μ)^2 + abs(X_2 - μ)^2 + abs(X_3 - μ)^2 + abs(X_4 - μ)^2 + abs(X_5 - μ)^2 + abs(X_6 - μ)^2 + abs(X_7 - μ)^2 + abs(X_8 - μ)^2 + abs(X_9 - μ)^2 + abs(X_10 - μ)^2)/(10 - 1)

The elements X_i of the list X = {11, 23, 18, 8, 33, 29, 7, 21, 19, 27} are:
X_1 = 11
X_2 = 23
X_3 = 18
X_4 = 8
X_5 = 33
X_6 = 29
X_7 = 7
X_8 = 21
X_9 = 19
X_10 = 27
(abs(11 - μ)^2 + abs(23 - μ)^2 + abs(18 - μ)^2 + abs(8 - μ)^2 + abs(33 - μ)^2 + abs(29 - μ)^2 + abs(7 - μ)^2 + abs(21 - μ)^2 + abs(19 - μ)^2 + abs(27 - μ)^2)/(10 - 1)

The mean (μ) is given by

μ = (X_1 + X_2 + X_3 + X_4 + X_5 + X_6 + X_7 + X_8 + X_9 + X_10)/10 = (11 + 23 + 18 + 8 + 33 + 29 + 7 + 21 + 19 + 27)/10 = 98/5:
(abs(11 - 98/5)^2 + abs(23 - 98/5)^2 + abs(18 - 98/5)^2 + abs(8 - 98/5)^2 + abs(33 - 98/5)^2 + abs(29 - 98/5)^2 + abs(7 - 98/5)^2 + abs(21 - 98/5)^2 + abs(19 - 98/5)^2 + abs(27 - 98/5)^2)/(10 - 1)

The values of the differences are:
11 - 98/5 = -43/5
23 - 98/5 = 17/5
18 - 98/5 = -8/5
8 - 98/5 = -58/5
33 - 98/5 = 67/5
29 - 98/5 = 47/5
7 - 98/5 = -63/5
21 - 98/5 = 7/5
19 - 98/5 = -3/5
27 - 98/5 = 37/5
10 - 1 = 9
(abs(-43/5)^2 + abs(17/5)^2 + abs(-8/5)^2 + abs(-58/5)^2 + abs(67/5)^2 + abs(47/5)^2 + abs(-63/5)^2 + abs(7/5)^2 + abs(-3/5)^2 + abs(37/5)^2)/(9)

The values of the terms in the numerator are:
abs(-43/5)^2 = 1849/25
abs(17/5)^2 = 289/25
abs(-8/5)^2 = 64/25
abs(-58/5)^2 = 3364/25
abs(67/5)^2 = 4489/25
abs(47/5)^2 = 2209/25
abs(-63/5)^2 = 3969/25
abs(7/5)^2 = 49/25
abs(-3/5)^2 = 9/25
abs(37/5)^2 = 1369/25
(1849/25 + 289/25 + 64/25 + 3364/25 + 4489/25 + 2209/25 + 3969/25 + 49/25 + 9/25 + 1369/25)/9

1849/25 + 289/25 + 64/25 + 3364/25 + 4489/25 + 2209/25 + 3969/25 + 49/25 + 9/25 + 1369/25 = 3532/5:
Answer: | 3532/45 =78 + 22/45=~78.49

Guest Apr 5, 2017
#2
+93595
+1
Melody  Apr 5, 2017