Measures of Variation

Measure of variation is an important characteristicof a data set. It’s a fundamental concept in statistics.

Measure of variation is a measure that describeshow the values in a set of data are spread out around the measure of centraltendency.

It is also known as measures of dispersion ormeasures of spread.

The three most commonly used measures of variationare:

The range, the variance, and the standard deviation

RANGE

The range is thedifference between the extreme valuesof a distribution.

In other words, it is thedifference between the greatest and the least numbers in a set of data.

Range = Greatest – Least   

The range is the simplest measureof variation and is the easiest to compute. Though range is not the optimal measureof variation, it is still widely used in simple descriptions of data.

Example 1

The following are Brian’s Math test scores. Findthe range of his test scores.

79, 63, 74, 81, 77, 67, 84

Solution:

Step 1: First arrange the test scoresin order of their size.

63, 67, 74, 77, 79, 81, 84

Step 2: The lowest score is 63 andthe highest score is 84.

So, the range of these test scores is 84 – 63 = 21 points.

Example 2

Theheights in cm of ten students are:

157, 152, ___,151, 160, 156, 155, 162, 158, 163

Themaximum height is missing in the data set. Find the maximum height if the rangeof the data set is 14 cm.

Solution:

Step 1: First arrange the heights in order from least togreatest.

151, 152, 155, 156, 157, 158, 160, 162, 163,___ 

Step 2: The maximum height is missing in the data set. We’ll call it ‘h’.

Step 3: Minimum height= 151 cm

Step 4: Range = Maximumheight – Minimum height

Step 5: 14 = h – 151                     [Substitute the values.]

Step 6: h = 14 + 151                     [Solve the equation for ‘h’.]

Step 7: h = 165

Step 8: So, the maximum height is 165 cm.

VARIANCE

Variance tells us how measured data vary from theaverage value of the data set. Variance is the mean of the squares of thedeviations from the arithmetic mean of a data set.

Click the link below or copy/paste the link belowfor the formula for variance and a solved example on variance.

http://www.icoachmath.com/SiteMap/Variance.html

The variance is a squared value. It is difficult tointerpret the variance. Hence, statisticians often compute the square root ofthe variance which is the ‘standard deviation’. The variance is always thesquare of the standard deviation.

STANDARD DEVIATION

The standard deviation is the most popular measureof variation. It measures the variation around the mean. It takes into accountall values in the data set. The standard deviation is the ideal measure ofvariation when the mean is used as a measure of central tendency.

The values of variance and standarddeviation are never negative.

Click the link below or copy/paste the link belowfor the formula for standard deviation and a solved example on standarddeviation.

http://www.icoachmath.com/SiteMap/StandardDeviation.html