# Is it true that negative values of the standard deviation indicates that the set of values is even less dispersed than would be expected by chance alone?

## Expert Answers

It is not true that negative values indicate the set of values is less dispersed as the standard deviation must be nonnegative.

By definition, the standard deviation is the square root of the variance. The variance for a population is the mean of the squared differences of the data values and the mean of the data values.

`sigma=sqrt(sigma^2)=sqrt((sum(x_i-bar(x))^2/n)) `

A set of data that consists of only value (e.g. {5} or {5,5,5,5,5}) will have a standard deviation of 0. No set of data could be less spread out than that.

A related value is the z-score. A negative z-score indicates a value is below the mean (or the mean of a sample is below the population mean.) But a z-score, while taking into account the spread of the data, does not in itself indicate how spread out the data are.

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