## 11.2.1 Mean, Standard Deviation, and Variance

 mean( x)
Arithmetic mean (sample mean) of x:

 (11.1)

 variance( x)
Estimated (sample) variance of x:

 (11.2)

This function computes the mean via a call to mean. If you have already computed the mean then you can pass it directly to variance_m.

 variance_m( x, mean)
Estimated (sample) variance of x relative to mean:

 (11.3)

 sd( x)
 sd_m( x, mean)
The standard deviation is defined as the square root of the variance of x. These functions returns the square root of the respective variance-functions above.

 variance_with_fixed_mean( x, mean)
Compute an unbiased estimate of the variance of x when the population mean mean of the underlying distribution is known a priori. In this case the estimator for the variance uses the factor and the sample mean is replaced by the known population mean :

 (11.4)