How is standard deviation calculated from a set of data values?

Standard deviation is calculated by finding the squared differences from the mean, averaging those squared differences to get the variance, and then taking the square root of the variance. The formula is

\sigma = \sqrt{\frac{\sum (X - \mu)^2}{N}}

Conditions

The mean $\mu$ of the dataset has already been calculated.

Reasoning, step by step

Subtract the mean $\mu$ from each data value $X$ and square the result.
Sum all the squared differences.
Divide the sum by the total number of values $N$ to get the variance.
Take the square root of the variance to obtain the standard deviation $\sigma$ .

Example

For heights 64, 70, 72, 68, 76 with mean 70: $(64-70)^2 + (70-70)^2 + (72-70)^2 + (68-70)^2 + (76-70)^2 = 36 + 0 + 4 + 4 + 36 = 80$ . Variance is $80 / 5 = 16$ . Standard deviation is $\sqrt{16} = 4$ .

Common misconceptions

Forgetting to take the square root of the variance, mistakenly reporting the variance as the standard deviation.

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