AP Statistics 2017

Variance and Standard Deviation

One measure of spread based on the mean is the variance . By definition, the variance is the average
squared deviation from the mean. That is, it is a measure of spread because the more distant a value is
from the mean, the larger will be the square of the difference between it and the mean.
Symbolically, the variance is defined by

Note that we average by dividing by n – 1 rather than n as you might expect. This is because there are
only n – 1 independent datapoints, not n , if you know . That is, if you know n – 1 of the values and you
also know , then the n th datapoint is determined.
One problem using the variance as a measure of spread is that the units for the variance won’t match
the units of the original data because each difference is squared. For example, if you find the variance of a
set of measurements made in inches, the variance will be in square inches. To correct this, we often take
the square root of the variance as our measure of spread.
The square root of the variance is known as the standard deviation . Symbolically,

As  discussed   earlier,    it  is  common  to  leave   off the indices and write:

In practice, you will rarely have to do this calculation by hand because it is one of the values returned
when you use you calculator to do 1-Var Stats on a list (it’s the Sx near the bottom of the first screen).

Calculator  Tip: When   you use 1-Var Stats ,   the calculator  will,   in  addition    to  Sx  ,   return  σ x ,   which

is  the standard    deviation   of  a   distribution.   Its formal  definition  is      .   Note    that    this

assumes you know    μ   ,   the population  mean,   which   you rarely  do  in  practice    unless  you are dealing

with    a   probability distribution    (see    Chapter 9   ).  Most    of  the time    in  statistics, you are dealing with

sample  data    and not a   distribution.   Thus,   with    the exception   of  the type    of  probability material    found   in

Chapters    9 and    10     ,   you should  use only    s and   not σ.

The definition of standard deviation has three useful qualities when it comes to describing the spread
of a distribution:

• It is independent of the mean . Because it depends on how far datapoints are from the mean, it doesn’t
matter where the mean is.