Question

find the variance of the data.
84, 98, 70, 76, 88, 76
\\(overline{x} = 82\\)
variance (\\(sigma^2\\)) = ?
round to the nearest tenth.

Explanation:

Step1: Recall variance formula

The formula for population variance is $\sigma^2=\frac{\sum (x_i - \overline{x})^2}{n}$, where $x_i$ are the data points, $\overline{x}$ is the mean, and $n$ is the number of data points. Here, $n = 6$, $\overline{x}=82$, and the data points are 84, 98, 70, 76, 88, 76.

Step2: Calculate each $(x_i - \overline{x})^2$

• For $x_1 = 84$: $(84 - 82)^2=(2)^2 = 4$
• For $x_2 = 98$: $(98 - 82)^2=(16)^2 = 256$
• For $x_3 = 70$: $(70 - 82)^2=(-12)^2 = 144$
• For $x_4 = 76$: $(76 - 82)^2=(-6)^2 = 36$
• For $x_5 = 88$: $(88 - 82)^2=(6)^2 = 36$
• For $x_6 = 76$: $(76 - 82)^2=(-6)^2 = 36$

Step3: Sum the squared differences

Sum $= 4 + 256 + 144 + 36 + 36 + 36$
$= 4 + 256 = 260$; $260 + 144 = 404$; $404 + 36 = 440$; $440 + 36 = 476$; $476 + 36 = 512$

Step4: Divide by $n$ to get variance

$\sigma^2=\frac{512}{6}\approx 85.333\cdots$

Step5: Round to nearest tenth

Rounding $85.333\cdots$ to the nearest tenth gives $85.3$.

Answer:

85.3