Statistical Inference: A Short Course

3.6 The Standard Deviation

We previously denoted the ith deviation from the (sample) mean as . Since the sum of all the deviations from the mean is zero, it is evident that one way to circumvent the issue of the sign of the difference is to square it. Then all of these squared deviations from the mean are nonnegative.

Suppose we have a population of X values or X: X₁, X₂, . . . , X_N. Then we can define the variance of X as follows:

(3.4)

that is, the populationvariance of X is the average of the squared deviations from the mean of X or from μ. Since the variance of any variable is expressed in terms of (units of X)² (e.g., if X is measured in “inches,” then is expressed in terms of “inches²”), a more convenient way of assessing variability about μ is to work with the standard deviation of X, which is defined as the positive square root of the variance of X or

(3.5)

Here σ is expressed in the original units of X.

If we have a sample of observations on X or X: X₁, X₂, . . . , X_n, then ...