Topic 5: Measures of Dispersion

Objectives

Find and calculate the different types of spreads namely Standard Deviation/Variance of ungrouped and grouped data, The Range and the Quartiles.

Use the different types of measure of spreads to compare different sets of data.

Explain the relationship between variance and standard deviation.

Range

The range is the difference between the largest and smallest values in the data set. i.e.

Range = Xmax - Xmin

Example 1,

Find the range of values 1, 4, 5, 6, 9, 10, 15

solution:

Range = Xmax - Xmin

           = 15 -1

           = 14

Quartiles and Interquartile Range

The lower quartile, Q1, is the value 25% of the way through the distribution and the upper quartile, Q3, is the value 75% of the way through the distribution.

Example 2,

Find the median and interquartile range of the following numbers:

2, 3, 3, 9, 6, 6, 12, 11, 8, 2, 3, 5, 7, 5, 4, 4, 5, 12

Solution: 

Example 3,

Find the median and interquartile range of the following distributions:

Q1 = 52  Q2 = 59  Q3 = 70  max = 84 and min = 46

Standard Deviation and Variance

Standard Deviation 

Perhaps the most commonly used measure of dispersion is the standard deviation which is the square root of the variance.

Standard deviation of a set of values can be obtained by using the formula below.

Variance 

Example 1,

Find the variance and standard deviation of values $2, $3, $5, $7 and $10.