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What is the standard deviation of these numbers: (36/47),(26/40),(31/44),(38/50),(34/47),(31/46),(29/40),(15/25),(28/49), and (33/50).

Guest Mar 18, 2017
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Find the (sample) standard deviation of the list:
(36/47, 26/40, 31/44, 38/50, 34/47, 31/46, 29/40, 15/25, 28/49, 33/50) = (36/47, 13/20, 31/44, 19/25, 34/47, 31/46, 29/40, 3/5, 4/7, 33/50)

The standard deviation is given by:
sqrt((variance))

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 = {36/47, 26/40, 31/44, 38/50, 34/47, 31/46, 29/40, 15/25, 28/49, 33/50}:
(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 = {36/47, 26/40, 31/44, 38/50, 34/47, 31/46, 29/40, 15/25, 28/49, 33/50} are:
X_1 = 36/47
X_2 = 13/20
X_3 = 31/44
X_4 = 19/25
X_5 = 34/47
X_6 = 31/46
X_7 = 29/40
X_8 = 3/5
X_9 = 4/7
X_10 = 33/50
(abs(36/47 - μ)^2 + abs(13/20 - μ)^2 + abs(31/44 - μ)^2 + abs(19/25 - μ)^2 + abs(34/47 - μ)^2 + abs(31/46 - μ)^2 + abs(29/40 - μ)^2 + abs(3/5 - μ)^2 + abs(4/7 - μ)^2 + abs(33/50 - 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 = (36/47 + 13/20 + 31/44 + 19/25 + 34/47 + 31/46 + 29/40 + 3/5 + 4/7 + 33/50)/10 = 113772473/166474000:
(abs(36/47 - 113772473/166474000)^2 + abs(13/20 - 113772473/166474000)^2 + abs(31/44 - 113772473/166474000)^2 + abs(19/25 - 113772473/166474000)^2 + abs(34/47 - 113772473/166474000)^2 + abs(31/46 - 113772473/166474000)^2 + abs(29/40 - 113772473/166474000)^2 + abs(3/5 - 113772473/166474000)^2 + abs(4/7 - 113772473/166474000)^2 + abs(33/50 - 113772473/166474000)^2)/(10 - 1)

The values of the differences are:
36/47 - 113772473/166474000 = 13739527/166474000
13/20 - 113772473/166474000 = -5564373/166474000
31/44 - 113772473/166474000 = 3516027/166474000
19/25 - 113772473/166474000 = 12747767/166474000
34/47 - 113772473/166474000 = 6655527/166474000
31/46 - 113772473/166474000 = -1583473/166474000
29/40 - 113772473/166474000 = 6921177/166474000
3/5 - 113772473/166474000 = -13888073/166474000
4/7 - 113772473/166474000 = -18644473/166474000
33/50 - 113772473/166474000 = -3899633/166474000
10 - 1 = 9
(abs(13739527/166474000)^2 + abs(-5564373/166474000)^2 + abs(3516027/166474000)^2 + abs(12747767/166474000)^2 + abs(6655527/166474000)^2 + abs(-1583473/166474000)^2 + abs(6921177/166474000)^2 + abs(-13888073/166474000)^2 + abs(-18644473/166474000)^2 + abs(-3899633/166474000)^2)/(9)

The values of the terms in the numerator are:
abs(13739527/166474000)^2 = 188774602183729/27713592676000000
abs(-5564373/166474000)^2 = 30962246883129/27713592676000000
abs(3516027/166474000)^2 = 12362445864729/27713592676000000
abs(12747767/166474000)^2 = 162505563486289/27713592676000000
abs(6655527/166474000)^2 = 44296039647729/27713592676000000
abs(-1583473/166474000)^2 = 2507386741729/27713592676000000
abs(6921177/166474000)^2 = 47902691065329/27713592676000000
abs(-13888073/166474000)^2 = 192878571653329/27713592676000000
abs(-18644473/166474000)^2 = 347616373447729/27713592676000000
abs(-3899633/166474000)^2 = 15207137534689/27713592676000000
(188774602183729/27713592676000000 + 30962246883129/27713592676000000 + 12362445864729/27713592676000000 + 162505563486289/27713592676000000 + 44296039647729/27713592676000000 + 2507386741729/27713592676000000 + 47902691065329/27713592676000000 + 192878571653329/27713592676000000 + 347616373447729/27713592676000000 + 15207137534689/27713592676000000)/9

188774602183729/27713592676000000 + 30962246883129/27713592676000000 + 12362445864729/27713592676000000 + 162505563486289/27713592676000000 + 44296039647729/27713592676000000 + 2507386741729/27713592676000000 + 47902691065329/27713592676000000 + 192878571653329/27713592676000000 + 347616373447729/27713592676000000 + 15207137534689/27713592676000000 = 104501305850841/2771359267600000:
11611256205649/2771359267600000

The standard deviation is given by
sqrt((variance)) = sqrt(11611256205649/2771359267600000) = (sqrt(11611256205649/10))/(16647400):
Answer: |(sqrt(11611256205649/10))/(16647400) ≈0.064728.........

Guest Mar 18, 2017

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