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Sample/Population Variance Calculator

Calculate sample/population variance step by step

For the given set of values, the calculator will find their variance (either sample or population), with steps shown.

Comma-separated.

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Your Input

Find the sample variance of 22, 11, 99, −3-3, 52\frac{5}{2}.

Solution

The sample variance of data is given by the formula s2=∑i=1n(xi−μ)2n−1s^{2} = \frac{\sum_{i=1}^{n} \left(x_{i} - \mu\right)^{2}}{n - 1}, where nn is the number of values, xi,i=1..n‾x_i, i=\overline{1..n} are the values themselves, and μ\mu is the mean of the values.

Actually, it is the square of standard deviation.

The mean of the data is μ=2310\mu = \frac{23}{10} (for calculating it, see mean calculator).

Since we have nn points, n=5n = 5.

The sum of (xi−μ)2\left(x_{i} - \mu\right)^{2} is (2−2310)2+(1−2310)2+(9−2310)2+(−3−2310)2+(52−2310)2=3745.\left(2 - \frac{23}{10}\right)^{2} + \left(1 - \frac{23}{10}\right)^{2} + \left(9 - \frac{23}{10}\right)^{2} + \left(-3 - \frac{23}{10}\right)^{2} + \left(\frac{5}{2} - \frac{23}{10}\right)^{2} = \frac{374}{5}.

Thus, s2=∑i=1n(xi−μ)2n−1=37454=18710s^{2} = \frac{\sum_{i=1}^{n} \left(x_{i} - \mu\right)^{2}}{n - 1} = \frac{\frac{374}{5}}{4} = \frac{187}{10}.

Answer

The sample variance is s2=18710=18.7s^{2} = \frac{187}{10} = 18.7A.