9.3 – measures of dispersion objective: tsw calculate the range and standard deviation from a set...

9.3 – Measures of Dispersion

Objective: TSW calculate the range and standard deviation from a set of

data.

© 2008 Pearson Addison-Wesley. All rights reserved

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Chapter 1


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Measures of Dispersion

Sometimes we want to look at a measure of dispersion, or spread, of data.

Two of the most common measures of dispersion are the range and the standard deviation.


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Range

For any set of data, the range of the set is given by

Range = (greatest value in set) – (least value in set).


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Example: Find the median and mean for each set of data below.

Set A 1 2 7 12 13

Set B 5 6 7 8 9

Now, let’s find the range.


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Standard Deviation

One of the most useful measures of dispersion, the standard deviation, is based on deviations from the mean of the data.

To find the deviations of each number:

1.Find the mean of the set of data

2.Subtract the number from the mean


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Example: Deviations from the Mean

Solution

Data Value 1 2 8 11 13

Deviation

Find the deviations from the mean for all data values of the sample 1, 2, 8, 11, 13.

The mean is 7. Subtract to find deviation.

The sum of the deviations for a set is always 0.


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Standard Deviation

Variance, to find:

1.Square each deviation

2.add the squares up

3.divide by (n-1) where n = number of items in the set.

standard deviation – the square root of the variance. Gives an average of the deviations from the mean. Which is denoted by the letter s. (The standard deviation of a population is denoted the lowercase Greek letter sigma.)

,


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Calculation of Standard Deviation

Let a sample of n numbers x1, x2,…xn have mean Then the sample standard deviation, s, of the numbers is given by

2( ).

1

x xs

n

.x

The smaller the number for standard deviation – the closer your data is together.

The higher the number for standard deviation, you data is further apart.


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Two standard deviations, or two sigmas, away from the mean (the red and green areas) account for roughly 95 percent of the data points. Three (3) standard deviations (the red, green and blue areas) account for about 99 percent of the data points.

If this curve were flatter and more spread out, the standard deviation would have to be larger in order to account for those 68 percent or so of the points. That's why the standard deviation can tell you how spread out the examples in a set are from the mean.


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Calculation of Standard Deviation

The individual steps involved in this calculation are as follows

Step 1 Calculate the mean of the numbers.Step 2 Find the deviations from the mean.Step 3 Square each deviation.Step 4 Sum the squared deviations.Step 5 Divide the sum in Step 4 by n – 1. Step 6 Take the square root of the quotient in

Step 5.


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Example

Find the standard deviation of the sample 1, 2, 8, 11, 13.

Data Value 1 2 8 11 13

Deviation

(Deviation)2

The mean =_____.


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Example: Find the Standard Deviation

12, 13, 16, 18, 18, 20 mean = _______

Data Value 12 13 16 18 18 20

Deviation

(Deviation)2

Example: Find the Standard Deviation

125, 131, 144, 158, 168, 193


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Data Value 125 131 144 158 168 193

Deviation

(Deviation)2

Homework

Worksheet


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9.3 – measures of dispersion objective: tsw calculate the range and standard deviation from a set...

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