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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?
| Population form: | Sample form: |
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. |
The
population form should be
used unless you know a random sample is being analyzed.
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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.
• 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:
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.
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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 |
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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 BACK |
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