MEASURES OF DISPERSION 1 Lecture 4

2 Objectives Explain the importance of measures of dispersion. Compute and interpret the range, the mean deviation, the variance and the standard deviation of ungrouped data and of grouped data. Discuss the characteristics and uses of each measure.

Importance of measures of Dispersions Measures of central tendency tell us only part of what we need to know about the characteristics of the data collected. They do not provide enough information to describe a set of data since variability or spread of the values in the data is ignored. 3 When the spread of data around the central item is high, the mean or median is less significant; low spread enhances the meaningfulness of the median or mean. Thus to increase our understanding of the pattern of the data, we must also measure its dispersion.

Types of Measures of Dispersion 1.Range 2.Variance 3.Standard deviation 4

Range The range is the difference between the largest and smallest values in the data set, i.e. Range = Xmaximum - Xminimum 5

Example Find the range of values 1,4,5,6,9,10,15 Solution Range = X max - X min = 15 – 1 = 14 6 We can see clearly that the range can be badly affected by extreme values

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. 7 Standard deviation for a sample

Example Find the standard deviation of values $2, $3, $5, $7 and $10. Solution 8

Variance Variance, = square of standard deviation 9

Uses of the variance and standard deviation Can be used to determine the spread of data ie the larger the value, the more dispersed the data. Can be used to determine the consistency of a variable. Can be used to determine the number of data values that fall within a specified interval in a distribution, 75% of the data values will fall within 2 standard deviations of the mean. They are used quite often in inferential statistics. 10

Recapitulate Importance of measures of dispersions Range Variance Standard Deviation 11 √ √ √ √

12 References : Lecture & Tutorial Notes from Department of Business & Management, Institute Technology Brunei, Brunei Darussalam.