Let
denote the data values.
The **average** or **arithmetic mean**
(denoted
in formulas) is computed as

(3.1) |

The average, which necessarily falls somewhere in the middle of the data points, is a commonly used statistic to indicate the

or in formulas,

The SD is the second formula with the denominator

Why do we use the squareroot of squares instead of just the average of
absolute values? Why do we replace *n* by *n*-1 in the denominator?
The long answers are mathematically complicated;
a short answer is ``because both adjustments give the statistic
better mathematical properties''.

Here are the monthly Pharmacia stock prices , sorted from smallest to largest.

39.60 40.52 40.56 42.65 44.40 44.62 45.95 48.56 49.93 50.37 51.68 51.70 51.93 52.26 54.75 55.00 56.02 58.56 60.18 61.00 61.00

Now, compute the deviations from average *X*_{i}-50.54.

-10.94 -10.02 -9.98 -7.89 -6.14 -5.92 -4.59 -1.98 -0.61 -0.17 1.14 1.16 1.39 1.72 4.21 4.46 5.48 8.02 9.64 10.46 10.46

The average size of the deviations (ignoring negative signs) is

The squareroot of the average of squares is

The standard deviation is