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Measures of Spread

Introduction

A measure of spread, sometimes also called a measure of dispersion, is used to describe the variability in a sample or population. It is usually used in conjunction with a measure of central tendency, such as the mean or median, to provide an overall description of a set of data.

Why is it important to measure the spread of data?

There are many reasons why the measure of the spread of data values is important, but one of the main reasons regards its relationship with measures of central tendency. A measure of spread gives us an idea of how well the mean, for example, represents the data. If the spread of values in the data set is large, the mean is not as representative of the data as if the spread of data is small. This is because a large spread indicates that there are probably large differences between individual scores. Additionally, in research, it is often seen as positive if there is little variation in each data group as it indicates that the similar.

We will be looking at the range, quartiles, variance, absolute deviation and standard deviation.

Range

The range is the difference between the highest and lowest scores in a data set and is the simplest measure of spread. So we calculate range as:

Range = maximum value - minimum value

For example, let us consider the following data set:

23 56 45 65 59 55 62 54 85 25

The maximum value is 85 and the minimum value is 23. This results in a range of 62, which is 85 minus 23. Whilst using the range as a measure of spread is limited, it does set the boundaries of the scores. This can be useful if you are measuring a variable that has either a critical low or high threshold (or both) that should not be crossed. The range will instantly inform you whether at least one value broke these critical thresholds. In addition, the range can be used to detect any errors when entering data. For example, if you have recorded the age of school children in your study and your range is 7 to 123 years old you know you have made a mistake!

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Quartiles and Interquartile Range

Quartiles tell us about the spread of a data set by breaking the data set into quarters, just like the median breaks it in half. For example, consider the marks of the 100 students below, which have been ordered from the lowest to the highest scores, and the quartiles highlighted in red.

Order Score Order Score Order Score Order Score Order Score
1st 35 21st 42 41st 53 61st 64 81st 74
2nd 37 22nd 42 42nd 53 62nd 64 82nd 74
3rd 37 23rd 44 43rd 54 63rd 65 83rd 74
4th 38 24th 44 44th 55 64th 66 84th 75
5th 39 25th 45 45th 55 65th 67 85th 75
6th 39 26th 45 46th 56 66th 67 86th 76
7th 39 27th 45 47th 57 67th 67 87th 77
8th 39 28th 45 48th 57 68th 67 88th 77
9th 39 29th 47 49th 58 69th 68 89th 79
10th 40 30th 48 50th 58 70th 69 90th 80
11th 40 31st 49 51st 59 71st 69 91st 81
12th 40 32nd 49 52nd 60 72nd 69 92nd 81
13th 40 33rd 49 53rd 61 73rd 70 93rd 81
14th 40 34th 49 54th 62 74th 70 94th 81
15th 40 35th 51 55th 62 75th 71 95th 81
16th 41 36th 51 56th 62 76th 71 96th 81
17th 41 37th 51 57th 63 77th 71 97th 83
18th 42 38th 51 58th 63 78th 72 98th 84
19th 42 39th 52 59th 64 79th 74 99th 84
20th 42 40th 52 60th 64 80th 74 100th 85


The first quartile (Q1) lies between the 25th and 26th student's marks, the second quartile (Q2) between the 50th and 51st student's marks, and the third quartile (Q3) between the 75th and 76th student's marks. Hence:

First quartile (Q1) = 45 + 45 ÷ 2 = 45
Second quartile (Q2) = 58 + 59 ÷ 2 = 58.5
Third quartile (Q3) = 71 + 71 ÷ 2 = 71

In the above example, we have an even number of scores (100 students, rather than an odd number, such as 99 students). This means that when we calculate the quartiles, we take the sum of the two scores around each quartile and then half them (hence Q1= 45 + 45 ÷ 2 = 45) . However, if we had an odd number of scores (say, 99 students), we would only need to take one score for each quartile (that is, the 25th, 50th and 75th scores). You should recognize that the second quartile is also the median.

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Quartiles are a useful measure of spread because they are much less affected by outliers or a skewed data set than the equivalent measures of mean and standard deviation. For this reason, quartiles are often reported along with the median as the best choice of measure of spread and central tendency, respectively, when dealing with skewed and/or data with outliers. A common way of expressing quartiles is as an interquartile range. The interquartile range describes the difference between the third quartile (Q3) and the first quartile (Q1), telling us about the range of the middle half of the scores in the distribution. Hence, for our 100 students:

Interquartile range = Q3 - Q1
= 71 - 45
= 26

However, it should be noted that in journals and other publications you will usually see the interquartile range reported as 45 to 71, rather than the calculated range.

A slight variation on this is the semi-interquartile range, which is half the interquartile range = ½ (Q3 - Q1). Hence, for our 100 students, this would be 26 ÷ 2 = 13.

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