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While knowing the mean value for a set of data may give us some
information about the set itself, many varying sets can have the
same mean value. To determine how the sets are different,
we need more information.
Another way of examining single variable data is to look at
how the data is spread out, or
dispersed about the mean.
We will discuss 4
ways of examining the dispersion of data.
The smaller the values from these methods, the more consistent
the data.
1. Range: The simplest of
our methods for measuring dispersion is range. Range is the
difference between the largest value and the smallest value in the
data set. While being simple to compute, the range is often
unreliable as a measure of dispersion since it is based on only two
values in the set.
A range of 50 tells us very little
about how the values are dispersed.
Are the values all clustered to one end with the low value (12) or the
high value (62) being an outlier?
Or are the values more evenly dispersed among the range?
Before discussing our next methods, let's establish some vocabulary:
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Population form: |
Sample form: |
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The
population form is used
when the data being analyzed includes the entire set of
possible data. When using this form, divide by
n, the number of values in the data set.
All people living in
the US. |
The sample form is used
when the data is a random sample taken from the entire set of data.
When using this form, divide by n - 1.
(It can be shown that dividing by n - 1
makes S2 for the sample, a better estimate of
 for the
population from which the sample was taken.)
Sam, Pete and Claire who live in the US. |
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The
population form should be
used unless you know a random sample is being analyzed. |
2. Mean Absolute Deviation (MAD):
The mean absolute deviation is the
mean (average) of the absolute value of the difference between the
individual values in the data set and the mean. The method
tries to measure the average distances between the values in the
data set and the mean.
3. Variance:
To find the variance:
• subtract the mean,
 ,
from each of the values in the data set,
 .
• square the result
• add all of these squares
• and divide by the number of values in the data set.
4. Standard Deviation:
Standard deviation is the square root of the variance. The
formulas are:
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Mean absolute deviation, variance and standard
deviation are ways to describe the difference between
the mean and the values in the data set without
worrying about the signs of these differences.
These values are usually computed using a
calculator. |
Warning!!! Be sure you know
where to find "population" forms versus "sample" forms on
the calculator. If you are unsure, check out the
information at these links.
 |
See how to use
your
TI-83+/TI-84+
graphing
calculator with
measures of
dispersion.
Click
calculator. |
|
|
|
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See how to use
your
TI-83+/TI-84+
graphing
calculator with
measures of
dispersion on
grouped data.
Click
calculator. |
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Examples:
1.
Find, to the nearest tenth, the
standard deviation and
variance of the distribution:
| Score |
100 |
200 |
300 |
400 |
500 |
| Frequency |
15 |
21 |
19 |
24 |
17 |
Solution:
For more detailed information on using the graphing
calculator, follow the links provided above.
Grab
your graphing calculator.
Enter the data and frequencies
in lists. |
Choose 1-Var Stats and
enter as grouped data. |
Population standard
deviation
is 134.0 |
| |
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Population variance
is
17069.7 |
2. Find, to the nearest
tenth, the mean absolute deviation for
the set
{2, 5, 7, 9, 1, 3, 4, 2, 6, 7, 11, 5, 8, 2, 4}.
Enter the data in list. |
Be sure to have the calculator
first determine the mean. |
Mean absolute deviation
is 2.3 |
| For more
detailed information on using the graphing calculator, follow
the links provided above. |
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