Question

find the variance of the data.
465, 460, 497, 462, 462, 474
\\( \overline { x } = 470 \\)
variance \\( ( \sigma ^ { 2 } ) = ? \\)
round to the nearest tenth

Explanation:

Step1: Calculate each data - mean difference squared

For \(465\): \((465 - 470)^2=(-5)^2 = 25\)
For \(460\): \((460 - 470)^2=(-10)^2=100\)
For \(497\): \((497 - 470)^2=(27)^2 = 729\)
For \(462\): \((462 - 470)^2=(-8)^2 = 64\) (do this twice since there are two \(462\)s)
For \(474\): \((474 - 470)^2=(4)^2=16\)

Step2: Sum up the squared differences

\(25+100 + 729+64+64+16\)
\(=25+100+729+(64\times2)+16\)
\(=25+100+729 + 128+16\)
\(=125+729+128+16\)
\(=854+128+16\)
\(=982+16\)
\(=998\)

Step3: Divide by the number of data points

There are \(n = 6\) data points.
The variance formula is \(\sigma^{2}=\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}}{n}\)
So \(\sigma^{2}=\frac{998}{6}\approx166.3\)

Answer:

\(166.3\)