Download Standard Deviation

AP Statistics
Variance and Standard Deviation
Introduction
 The 5-number summary is the most useful numerical
descriptor of a distribution.
 The combination of the mean and standard deviation
is most commonly used.
 The standard deviation is a measure of the average
spread of each data value from the mean of the data.
 Standard deviation is the square root of another
measure of spread (distance) called the variance.
Example
Example Continued - Variance

1
s 
xi  x

n 1
2

2
1
  214,870   35,811.67
6
Finally…Standard Deviation
s  35811.67  189.24
Notes on Variance
 The variance is large if the observations are widely
spread about the mean.
 It is small if the observations are all close to the mean.
 Variance has a different metric than the original
observations. Standard deviation is in the same
metric.
 *Read problems carefully to decide which measure to
use!
Why use n – 1 instead of n?
 “n – 1” are the degrees of freedom and is used when
dealing with samples. It corrects for error in sampling.
 “n” is used when doing calculations for populations,
where sampling error doesn’t exist.
Limitations
 Standard deviation should only be used with the
mean, as it measures spread about the mean.
 s = 0 when there is no spread, which occurs when all
the observations have the same value. Otherwise, s>0.
 Standard deviation gets larger as the spread gets larger.
 Standard deviation, like the mean, is strongly
influenced by extreme observations. Thus it is a nonresistant measure.
More Limitations
 The five number summary is better when describing
skewed distributions!
 Spreads from left and right of the mean are different.
When to Use
 The mean and standard deviation are better for
symmetrical distributions.
 Always plot your data. A picture is a better descriptor
for data than a numerical summary.
Homework
 Do the Variance and Standard Deviation Worksheet.
(#1 calculate by hand, the rest answer using the 1–var
stats function).
 Chapter 1 Review.