Question

find the variance of the data.
16, 13, 20, 16, 12, 17, 18
\\( \overline { x } = 16 \\)
variance \\( \left( \sigma ^ { 2 } \
ight) = ? \\)
round to the nearest tenth

Explanation:

Step1: Calculate the squared differences

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

• For \(x_1 = 16\): \((16 - 16)^2=0\)
• For \(x_2 = 13\): \((13 - 16)^2 = 9\)
• For \(x_3 = 20\): \((20 - 16)^2=16\)
• For \(x_4 = 16\): \((16 - 16)^2 = 0\)
• For \(x_5 = 12\): \((12 - 16)^2=16\)
• For \(x_6 = 17\): \((17 - 16)^2 = 1\)
• For \(x_7 = 18\): \((18 - 16)^2=4\)

Step2: Find the sum of squared differences

\(\sum_{i = 1}^{7}(x_i-\bar{x})^2=0 + 9+16 + 0+16+1+4=46\)

Step3: Calculate the variance

The formula for variance \(\sigma^{2}=\frac{1}{n}\sum_{i = 1}^{n}(x_i-\bar{x})^2\), where \(n = 7\).
\(\sigma^{2}=\frac{46}{7}\approx6.6\)

Answer:

\(6.6\)