Question

find the variance of the data.
95, 80, 191, 86, 122, 86
\\( \overline { x } = 110 \\)
variance \\( ( \sigma ^ { 2 } ) = \\)

Explanation:

Step1: Calculate the squared differences

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

• For \(x_1 = 95\): \((95 - 110)^2=(- 15)^2 = 225\)
• For \(x_2 = 80\): \((80 - 110)^2=(-30)^2 = 900\)
• For \(x_3 = 191\): \((191 - 110)^2=(81)^2 = 6561\)
• For \(x_4 = 86\): \((86 - 110)^2=(-24)^2 = 576\)
• For \(x_5 = 122\): \((122 - 110)^2=(12)^2 = 144\)
• For \(x_6 = 86\): \((86 - 110)^2=(-24)^2 = 576\)

Step2: Sum the squared differences

\(\sum_{i = 1}^{6}(x_i-\bar{x})^2=225 + 900+6561+576+144+576\)
\(=225+900 = 1125\); \(1125+6561=7686\); \(7686 + 576=8262\); \(8262+144 = 8406\); \(8406+576=8982\)

Step3: Divide by the number of data points \(n\)

The formula for variance \(\sigma^{2}=\frac{\sum_{i = 1}^{n}(x_i-\bar{x})^2}{n}\), here \(n = 6\)
\(\sigma^{2}=\frac{8982}{6}=1497\)

Answer:

\(1497\)