Question

find the variance of the data.
156, 90, 146, 132, 122, 134
\\( \overline { x } = 130 \\)
variance \\( ( \sigma ^ { 2 } ) = \\)

Explanation:

Step1: Calculate the squared differences

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

• For \(x_1 = 156\): \((156 - 130)^2=26^2 = 676\)
• For \(x_2 = 90\): \((90 - 130)^2=(- 40)^2 = 1600\)
• For \(x_3 = 146\): \((146 - 130)^2=16^2 = 256\)
• For \(x_4 = 132\): \((132 - 130)^2=2^2 = 4\)
• For \(x_5 = 122\): \((122 - 130)^2=(-8)^2 = 64\)
• For \(x_6 = 134\): \((134 - 130)^2=4^2 = 16\)

Step2: Sum the squared differences

$$676+1600 + 256+4+64+16=\sum_{i = 1}^{6}(x_i-\bar{x})^2$$

$$676+1600=2276$$

$$2276+256 = 2532$$

$$2532+4=2536$$

$$2536+64=2600$$

$$2600+16=2616$$

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

The formula for variance \(\sigma^{2}=\frac{1}{n}\sum_{i = 1}^{n}(x_i-\bar{x})^2\), where \(n = 6\)
\(\sigma^{2}=\frac{2616}{6}\)

Answer:

\(436\)