2. Chapter 2 Exploring Data with Graphs and Numerical Summaries
Section 2.4
Measuring the Variability of Quantitative Data

3. Range One way to measure the spread is to calculate the range.
The range is the difference between the largest and smallest values in the data set:
Range = max  min
The range is simple to compute and easy to understand, but it uses only the extreme values and ignores the other values. Therefore, it’s affected severely by outliers.

4. Standard Deviation The deviation of an observation from the mean is
, the difference between the observation and the sample mean.
Each data value has an associated deviation from the
mean.
A deviation is positive if the value falls above the mean
and negative if the value falls below the mean.
The sum of the deviations for all the values in a data
set is always zero.

5. Standard Deviation
For the cereal sodium values, the mean is = 167. The observation of 210 for Honeycomb has a deviation of = 43. The observation of 50 for Honey Smacks has a deviation of = Figure 2.11 shows these deviations
Figure 2.9 Dot Plot for Cereal Sodium Data, Showing Deviations for Two Observations. Question: When is a deviation positive and when is it negative?

6. The Standard Deviation s of n Observations
Gives a measure of variation by summarizing the deviations of each observation from the mean and calculating an adjusted average of these deviations.

7. Standard Deviation Find the mean.
Find the deviation of each value from the mean.
Square the deviations.
Sum the squared deviations.
Divide the sum by n-1 and take the square root of that value.

8. Standard Deviation
Metabolic rates of 7 men (cal./24hr):

9. Standard Deviation

10. Properties of the Standard Deviation
The most basic property of the standard deviation is this:
The larger the standard deviation , the greater the variability of the data.
measures the spread of the data.
only when all observations have the same value, otherwise
. As the spread of the data increases, gets larger.
has the same units of measurement as the original
observations. The variance = has units that are squared.
is not resistant. Strong skewness or a few outliers can greatly
increase .

11. Magnitude of s: The Empirical Rule
If a distribution of data is bell shaped, then approximately:
68% of the observations fall within 1 standard deviation
of the mean, that is, between the values of and
(denoted ).
95% of the observations fall within 2 standard deviations
of the mean
All or nearly all observations fall within 3 standard
deviations of the mean

12. Magnitude of s: The Empirical Rule
Figure 2.12 The Empirical Rule. For bell-shaped distributions, this tells us approximately how much of the data fall within 1, 2, and 3 standard deviations of the mean. Question: About what percentage would fall more than 2 standard deviations from the mean?