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3.2 Measures of Variation
Measures of variation or measures of spread: is a descriptive measure
that describes how much variation or spread there is in a data set.
Range of a Data Set
The range of a data set is given by the formula.
Range  Max  Min,
Where Max and Min denote the maximum and minimum observations,
respectively
Sample Standard Deviation
For a variable x, the standard deviation of the observations for a sample is
called a sample standard deviation. It is denoted by s x or, when no
confusion will arise, simply
s . We have
xi  x 2

s
n 1
Procedure for finding the Standard Deviation
1. Calculate the sample mean,
x
2. Construct a table to obtain the sum of squared deviations,  xi  x 2
3. Apply the definition to determine the sample standard deviation
Rounding Rule
Do not perform any rounding (five decimal places to the right of the decimal
point is fine) until the computation is complete; otherwise, substantial round
off error can result.
Example 1
Determine the range and sample standard deviation for each of the data
sets. For the sample standard deviation, round each answer to one more
decimal place than that used for the observations
Variation and the Standard Deviation
The more variation that there is in a data set, the larger is its standard
deviation.
Computing Formula for a Sample Standard Deviation
A sample standard deviation can be computed using the formula
s

s.
2


x

i
x2 
i
n 1
n
where n is the sample size.
Example 2
Use the computing formula to find the standard deviation for the hurricane
example
Empirical Rule
For any data set having roughly a bell-shaped distribution.
 Approximately 68% of the observations lie within one standard
deviation to either side of the mean.
 Approximately 95% of the observations lie within two standard
deviations to either side of the mean.
 Approximately 99.7% of the observations lie within three standard
deviations to either side of the mean.
Example 3
1. Find the one standard deviation limits from the mean, the two standard
deviation limits from the mean, and the three standard deviation limits
from the mean for the hurricane data.
2. Does the data meet the appropriate percentages according to the
Empirical Rule?
Outliers: Are extreme data points that do not appear to be part of a
distribution. Observations that fall well outside of the overall pattern of the
data. These data points are important and should not be ignored.
Chebychev’s Rule
For any data set and any real number k>1, at least 100(1-1/k2)% of the
observations lie within k standard deviations to either side of the mean.