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Find the variance and standard deviation of the numbers: 4, 6, 8, 10, 12.

July 16, 2026Mathematics

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Answer

\text{8}

2\sqrt{2}

4 steps

Calculate the mean ( $\bar{x}$ ) of the numbers.
The given numbers are $4, 6, 8, 10, 12$ . There are $n=5$ numbers.
$\bar{x} = \frac{\sum x}{n} = \frac{4 + 6 + 8 + 10 + 12}{5} = \frac{40}{5} = 8$
Calculate the sum of the squared differences from the mean, $\sum (x - \bar{x})^2$ .
$\begin{align*} (4 - 8)^2 &= (-4)^2 = 16 \\ (6 - 8)^2 &= (-2)^2 = 4 \\ (8 - 8)^2 &= (0)^2 = 0 \\ (10 - 8)^2 &= (2)^2 = 4 \\ (12 - 8)^2 &= (4)^2 = 16 \end{align*}$ $\sum (x - \bar{x})^2 = 16 + 4 + 0 + 4 + 16 = 40$
Calculate the variance ( $\sigma^2$ ).
$\sigma^2 = \frac{\sum (x - \bar{x})^2}{n} = \frac{40}{5} = 8$
The variance is $\boxed{\text{8}}$ .
Calculate the standard deviation ( $\sigma$ ).
The standard deviation is the square root of the variance.
$\sigma = \sqrt{\sigma^2} = \sqrt{8} = \sqrt{4 \times 2} = 2\sqrt{2} \approx 2.83$
The standard deviation is $\boxed{2\sqrt{2}}$ .

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