Download Section 3-3

Section 3-3
Measures of Variation
Measures of Variation
• Measures of variation (or dispersion
or spread) are values that measure the
amount that the data values vary
among themselves.
Range
• The range of a set of data is the
difference between the highest value
and the lowest value in the data set.
Range = Highest Value – Lowest Value
1
Standard Deviation
• The standard deviation of a set of
sample values is a measure of variation
of values about the mean.
Standard Deviation
• We use s to represent the sample
standard deviation.
• We use σ to represent the population
standard deviation.
• The standard deviation is sometimes
denoted SD.
Standard Deviation
• We will use Formula 3-4 on page 97 to
illustrate the calculation of the sample
standard deviation.
2
Sample Standard
Deviation Formula
S=
Σ (x - x)2
n-1
Formula 3-4
Population Standard
Deviation
σ =
Σ (x - µ)2
N
This formula is similar to Formula 3-4, but
instead the population mean and population
size are used
Standard Deviation
• The value of the standard deviation can
increase dramatically with the
inclusion of one or more outliers (data
values far away from all others).
• The units of the standard deviation are
the same as the units of the original
data values.
3
Standard Deviation
• The value of the standard deviation is
nonnegative. It is zero only when all of
the data values are the same number.
• Larger values of the standard deviation
indicate greater amounts of variation
among the data.
Comparing Variation in Two
Different Sets
When comparing variation in two
different data sets, use the standard
deviation only if the sets use the same
measurement scale and have
approximately the same mean.
An Example Involving the
Standard Deviation
• Brand A car batteries have a mean life
of 72 months with a standard deviation
of 6 months. Brand B car batteries
have a mean life of 72 months with a
standard deviation of 12 months.
Which is the better battery?
4
Variance
• The variance of a set of values is a
measure of variation equal to the
square of the standard deviation.
Variance - Notation
}
Notation
s2
σ
2
Sample variance
Population variance
Round-off Rule
for Measures of Variation
Carry one more decimal place than is
present in the original set of data.
5
Standard Deviation and Mean
from a Frequency Distribution
• We can estimate the value of the
sample mean and sample standard
deviation from a frequency table of the
data. To do this we assume that in each
class, all values are equal to the class
midpoint.
Range Rule of Thumb
• The range rule of thumb is rough
estimate for the value of the SD. It is
based upon the assumption that most
data values fall within 2 standard
deviations of the mean. Thus we have
s ≈ range / 4
The “Usual” Maximum &
Minimum
• The range rule of thumb can be used to
estimate the minimum and maximum
“usual” sample values:
• Minimum value ≈ mean – 2s
• Maximum value ≈ mean + 2s
6
Empirical Rule for Bell-Shaped
Data (or the 68-95-99.7 Rule)
• For data sets having a distribution that is
approximately bell shaped (normal), the
following properties apply:
• About 68% of all values fall within 1 SD of the mean
• About 95% of all values fall within 2 SD of the mean
• About 99.7% of all values fall within 3 SD of the
mean
The Empirical Rule
Example of Applying the
Empirical Rule
• The life times for a certain cell phone
battery come from a normal population with
a mean of 6 days and a standard deviation
of 1.5 days.
– 99.7% of these batteries have life times that fall
between what two values?
– What percent of these batteries have life times
that fall between 3 and 9 days?
7