Question

What is the variance of the values? 7,11,5,1,11

Answer 1

Find the (sample) variance of the list:
(7, 11, 5, 1, 11)
State the definition of variance.
The (sample) variance of a list of numbers X = {X_1, X_2, ..., X_n} with mean \(\mu = \frac{X_1+X_2+...+X_n}{n}\) is given by:
(abs(X_1-mu)^2+abs(X_2-mu)^2+...+abs(X_n-mu)^2)/(n-1)
Find the value of n (the number of elements in the list) and plug it into the definition.
There are n = 5 elements in the list X = {7, 11, 5, 1, 11}:
(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)/(5-1)
Plug the elements of the list into the definition.
The elements \(X_i\) of the list X = {7, 11, 5, 1, 11} are:
\[X_1 = 7 \]\[X_2 = 11 \]\[X_3 = 5 \]\[X_4 = 1 \]\[X_5 = 11 \](abs(7-mu)^2+abs(11-mu)^2+abs(5-mu)^2+abs(1-mu)^2+abs(11-mu)^2)/(5-1)
Next, find the mean \((\mu) \) of the elements and plug it into the definition.
The mean \((\mu) \) is given by

\[\mu = \frac{X_1+X_2+X_3+X_4+X_5}{5} = \frac{7+11+5+1+11}{5} = 7: \]\[\frac{|7-7|^2+|11-7|^2+|5-7|^2+|1-7|^2+|11-7^2}{5-1}\]Evaluate the differences within the absolute values and in the denominator.
The values of the differences are:
\[7-7 = 0 \]\[11-7 = 4 \]\[5-7 = -2 \]\[1-7 = -6 \]\[11-7 = 4 \]\[5-1 = 4 \]\[\frac{|0|^2+|4|^2+|-2|^2+|-6|^2+|4|^2}{4}\]Evaluate the squared absolute values in the numerator.
The values of the terms in the numerator are:
\[|0|^2 = 0\]\[|4|^2 = 16\]\[|-2|^2 = 4\]\[|-6|^2 = 36\]\[|4|^2 = 16 \]\[\frac{0+16+4+36+16}{4}\]Evaluate the sum in the numerator.
\[0+16+4+36+16 = 72: \]

Answer 2

Which leaves you with...?