Standard deviation

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This article/section deals with mathematical concepts appropriate for late high school or early college.

Standard deviation is a measure for how much a set of values varies. It allows for one to find how likely it is for a specific value to be obtained by doing a Z-test.

Standard Deviation

The standard deviation of a set of values is a measure of how widely the values differ from each other. Specifically, standard deviation follows the equation:

$\sigma (x)={\sqrt {\sum (x-{\bar {x}}) \over n-1}}$

This is the square root of the variance, which is:

$Var(x)={\sum (x-{\bar {x}}) \over n-1}$

Where:

• ${\bar {x}}$ is the arithmetic mean of all values of x
• $\sum$ is the summation function
• $n$ is the number of $x$ values

If the distribution of the values is normal then it follows the Empirical rule, which states that:

For a more detailed treatment, see Empirical rule.

• 68% of the values will fall within 1$\sigma$ of the mean.
• 95% of all values will fall within 2$\sigma$ of the mean.
• 99.7% of all values will fall within 3$\sigma$ of the mean.