Variance and Standard Deviation for Raw Data

Variance and Standard Deviation (raw data)

The variance is the mean of the squared deviations:

$\sigma^2 = \frac{\sum (x_i - \bar{x})^2}{n}$

The standard deviation $\sigma$ (read "sigma") is the positive square root of the variance:

$\sigma = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n}}$

Taking the square root returns the measure to the original units of the data (the variance is in squared units, e.g. rupees², which is why the standard deviation, not the variance, is the reported measure of spread).

Short-cut formula (avoids the mean inside the sum)

$\sigma^2 = \frac{\sum x_i^{2}}{n} - \left(\frac{\sum x_i}{n}\right)^{2} = \frac{\sum x_i^{2}}{n} - \bar{x}^{\,2}$

This form needs only $\sum x_i$ and $\sum x_i^2$, so it avoids computing each $(x_i - \bar{x})$ separately — much quicker, and it is the standard cross-check for a variance found the long way.

Key properties of the standard deviation

• $\sigma \ge 0$ always, and $\sigma = 0$ only when every value is identical (no spread at all). …