Maths Is Ohsem. : MEASURES OF DISPERSION

MEASURES OF DISPERSION

DEFINITION

A measures of dispersion is a method of measuring the degree by which numerical data or value tend to spread from or cluster about control point of average.

The common measures of dispersion are the following:

RANGE
MEAN ABSOLUTE DEVIATION
VIARANCE
STANDARD DEVIATION

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:

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.

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:

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.

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

Population varianceis 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 deviationis 2.3
For more detailed information on using the graphing calculator, follow the links provided above.