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 the squared differences

For each data point \(x_i\), calculate \((x_i - \bar{x})^2\).

• For \(x_1 = 465\): \((465 - 470)^2=(- 5)^2 = 25\)
• For \(x_2 = 460\): \((460 - 470)^2=(-10)^2 = 100\)
• For \(x_3 = 497\): \((497 - 470)^2=(27)^2 = 729\)
• For \(x_4 = 462\): \((462 - 470)^2=(-8)^2 = 64\)
• For \(x_5 = 462\): \((462 - 470)^2=(-8)^2 = 64\)
• For \(x_6 = 474\): \((474 - 470)^2=(4)^2 = 16\)

Step2: Find the sum of squared differences

\(\sum_{i = 1}^{6}(x_i-\bar{x})^2=25 + 100+729+64+64+16\)
\(=998\)

Step3: Calculate the variance

The formula for variance \(\sigma^{2}=\frac{\sum_{i = 1}^{n}(x_i - \bar{x})^2}{n}\) (assuming population variance, \(n = 6\))
\(\sigma^{2}=\frac{998}{6}\approx166.3\)

Answer:

\(166.3\)