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:

<math>\sigma(x) = \sqrt {\sum(x - \bar x) \over n - 1}</math>

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

<math>Var(x) = {\sum (x - \bar x) \over n - 1}</math>

Where:

• <math>\bar x</math> is the arithmetic mean of all values of x
• <math>\sum</math> is the summation function
• <math>n</math> is the number of <math>x</math> 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<math>\sigma</math> of the mean.
• 95% of all values will fall within 2<math>\sigma</math> of the mean.
• 99.7% of all values will fall within 3<math>\sigma</math> of the mean.