12.3 – Measures of DispersionDispersion is another analytical method to study data.

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

Range

For any set of data, the range of the set is given by the following formula:

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

A common technique in inferential statistics is to draw comparisons between populations by analyzing samples that come from those populations.

A main use of dispersion is to compare the amounts of spread in two (or more) data sets.

Example:

Set A 1 2 7 12 13

Set B 5 6 7 8 9

The two sets below have the same mean and median (7). Find the range of each set.

Range of Set A:

12.3 – Measures of DispersionRange

13 – 1 = 12

Range of Set A: 9 – 5 = 4

One of the most useful measures of dispersion is the standard deviation.

12.3 – Measures of DispersionStandard Deviation

It is based on deviations from the mean of the data.

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

The mean is 7. To find each deviation, subtract the mean from each data value.

Data Value 1 2 8 11 13

Deviation – 6 – 5 1 4 6

The sum of the deviations is always equal to zero.

The sample standard deviation is found by calculating the square root of the variance.

12.3 – Measures of DispersionStandard Deviation

Calculating the Sample Standard Deviation

2( ).

1

x xs

n

The sample standard deviation is denoted by the letter s.

The variance is found by summing the squares of the deviations and dividing that sum by n – 1 (since it is a sample instead of a population).

The standard deviation of a population is denoted by .

6. Take the square root of the quotient in Step 5.

12.3 – Measures of DispersionStandard Deviation

Calculating the Sample Standard Deviation

1. Calculate the mean of the numbers.

2. Find the deviations from the mean.

3. Square each deviation.

4. Sum the squared deviations.

5. Divide the sum in Step 4 by n – 1.

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

Data Value 1 2 8 11 13

Deviation

Sum of the (Deviations)2 =

12.3 – Measures of DispersionStandard Deviation

Calculating the Sample Standard Deviation

Example:

= 7

– 6 – 5 1 4 6

(Deviation)2 16 1 25 36 36

36 + 25 + 1 + 16 + 36 = 114

Sum of the (Deviations)2 =

12.3 – Measures of DispersionStandard Deviation

Calculating the Sample Standard Deviation

36 + 25 + 1 + 16 + 36 = 114

Divide 114 by n – 1 with n = 5:

114

5 – 1= 28.5

Take the square root of 28.5:

5.34

The sample standard deviation of the data is 5.34.

Example: Interpreting Measures

Two companies, A and B, sell small packs of sugar for coffee. The mean and standard deviation for samples from each company are given below. Which company consistently provides more sugar in their packs? Which company fills its packs more consistently?

Company A Company B

1.013 tspAx 1.007 tspBx

.0021As .0018Bs

12.3 – Measures of DispersionStandard Deviation

Example: Interpreting Measures

The sample mean for Company A is greater than the sample mean of Company B.

Company A Company B

1.013 tspAx 1.007 tspBx

.0021As .0018Bs

12.3 – Measures of DispersionStandard Deviation

Which company consistently provides more sugar in their packs?

The inference can be made that Company A provides more sugar in their packs.

Example: Interpreting Measures

Which company fills its packs more consistently?

Company A Company B

1.013 tspAx 1.007 tspBx

.0021As .0018Bs

12.3 – Measures of DispersionStandard Deviation

The standard deviation for Company B is less than the standard deviation for Company A.

The inference can be made that Company B fills their packs more closer to their mean than Company A.

Chebyshev’s Theorem

For any set of numbers, regardless of how they are distributed, the fraction of them that lie within k standard deviations of their mean (where k > 1) is at least

2

11

.k

12.3 – Measures of Dispersion

What is the minimum percentage of the items in a data set which lie within 2, and 3 standard deviations of the mean?

75% 88.9%

Coefficient of Variation

The coefficient of variation expresses the standard deviation as a percentage of the mean.

12.3 – Measures of Dispersion

It is not strictly a measure of dispersion as it combines central tendency and dispersion.

For any set of data, the coefficient of variation is given by

for a sample or

for a population.

100s

Vx

100V

Example: Comparing Samples

Compare the dispersions in the two samples A and B. A: 12, 13, 16, 18, 18, 20B: 125, 131, 144, 158, 168, 193

Coefficient of Variation

12.3 – Measures of Dispersion

Sample A Sample B16.167Ax 153.167Bx 3.125As 25.294Bs 19.3AV 16.5BV

Sample B has a larger dispersion than sample A, but sample A has the larger relative dispersion (coefficient of variation).