Study this example to learn about "standard deviation"
Find the (sample) standard deviation of the list:
(2, 8, 15, 4, 14, 10, 15, 7, 4, 9)
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 mu = (X_1+X_2+...+X_n)/n is given by:
(abs(X_1-mu)^2+abs(X_2-mu)^2+...+abs(X_n-mu)^2)/(n-1)
There are n = 10 elements in the list X = {2, 8, 15, 4, 14, 10, 15, 7, 4, 9}:
(abs(X_1-mu)^2+abs(X_2-mu)^2+abs(X_3-mu)^2+abs(X_4-mu)^2+abs(X_5-mu)^2+abs(X_6-mu)^2+abs(X_7-mu)^2+abs(X_8-mu)^2+abs(X_9-mu)^2+abs(X_10-mu)^2)/(10-1)
The elements X_i of the list X = {2, 8, 15, 4, 14, 10, 15, 7, 4, 9} are:
X_1 = 2
X_2 = 8
X_3 = 15
X_4 = 4
X_5 = 14
X_6 = 10
X_7 = 15
X_8 = 7
X_9 = 4
X_10 = 9
(abs(2-mu)^2+abs(8-mu)^2+abs(15-mu)^2+abs(4-mu)^2+abs(14-mu)^2+abs(10-mu)^2+abs(15-mu)^2+abs(7-mu)^2+abs(4-mu)^2+abs(9-mu)^2)/(10-1)
The mean (mu) is given by
mu = (X_1+X_2+X_3+X_4+X_5+X_6+X_7+X_8+X_9+X_10)/10 = (2+8+15+4+14+10+15+7+4+9)/10 = 44/5:
(abs(2-44/5)^2+abs(8-44/5)^2+abs(15-44/5)^2+abs(4-44/5)^2+abs(14-44/5)^2+abs(10-44/5)^2+abs(15-44/5)^2+abs(7-44/5)^2+abs(4-44/5)^2+abs(9-44/5)^2)/(10-1)
The values of the differences are:
2-44/5 = -34/5
8-44/5 = -4/5
15-44/5 = 31/5
4-44/5 = -24/5
14-44/5 = 26/5
10-44/5 = 6/5
15-44/5 = 31/5
7-44/5 = -9/5
4-44/5 = -24/5
9-44/5 = 1/5
10-1 = 9
(abs(-34/5)^2+abs(-4/5)^2+abs(31/5)^2+abs(-24/5)^2+abs(26/5)^2+abs(6/5)^2+abs(31/5)^2+abs(-9/5)^2+abs(-24/5)^2+abs(1/5)^2)/(9)
The values of the terms in the numerator are:
abs(-34/5)^2 = 1156/25
abs(-4/5)^2 = 16/25
abs(31/5)^2 = 961/25
abs(-24/5)^2 = 576/25
abs(26/5)^2 = 676/25
abs(6/5)^2 = 36/25
abs(31/5)^2 = 961/25
abs(-9/5)^2 = 81/25
abs(-24/5)^2 = 576/25
abs(1/5)^2 = 1/25
(1156/25+16/25+961/25+576/25+676/25+36/25+961/25+81/25+576/25+1/25)/9
1156/25+16/25+961/25+576/25+676/25+36/25+961/25+81/25+576/25+1/25 = 1008/5:
112/5
The standard deviation is given by
sqrt((variance)) = sqrt(112/5) = 4 sqrt(7/5):
Answer: | 4 sqrt(7/5)
