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2022-10-05

What is the variance of {-4, 5, 8 ,-1, 0 ,4 ,-12, 4}?

Quinn Alvarez

First, let's assume that this is the entire population of values. Therefore we are looking for the population variance . If these numbers were a set of samples from a larger population, we would be looking for the sample variance which differs from the population variance by a factor of n/(n-1)
The formula for the population variance is
${\sigma }^{2}=\frac{1}{N}\sum _{i=1}^{N}\left({x}_{i}-\mu {\right)}^{2}$
where $\mu$ is the population mean, which can be calculated from
$\mu =\frac{1}{N}\sum _{i=1}^{N}{x}_{i}$
In our population the mean is
$\mu =\frac{-4+5+8-1+0+4-12+4}{8}=\frac{4}{8}=\frac{1}{2}$
Now we can proceed with the variance calculation:
${\sigma }^{2}$
$\frac{\left(-4-\frac{1}{2}{\right)}^{2}+\left(5-\frac{1}{2}{\right)}^{2}+\left(8-\frac{1}{2}{\right)}^{2}+\left(-1-\frac{1}{2}{\right)}^{2}+\left(0-\frac{1}{2}{\right)}^{2}+\left(4-\frac{1}{2}{\right)}^{2}+\left(-12-\frac{1}{2}{\right)}^{2}+\left(4-\frac{1}{2}{\right)}^{2}}{8}$
${\sigma }^{2}=35$

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