Measures of Spread

The first quartile (Q₁) lies between the 25 th and 26 th student’s marks, the second quartile (Q₂) between the 50 th and 51 st student’s marks, and the third quartile (Q₃) between the 75 th and 76 th 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 (in other words, the 100 students rather than an odd amount such as 99). 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), then we would only need to take one score for each quartile (that is, the 25^th, 50^th and 75^th scores).

Whilst quartiles are a useful measure of spread, the interquartile range is slightly more useful because it is not affected by outliers; in other words, one or a small number of particularly low or high scores that do not represent well the general performance of our students. The interquartile range is more useful because it is 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

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.

Variation

Quartiles are useful but they are also somewhat limited because they do not take into account every score in our group of data. To get a more representative idea of the performance our 100 students (the example used throughout this guide) we need to use measures of spread that take into account all of the students’ scores. The absolute deviation, variance and standard deviation are such measures.

In order to calculate the variation in a frequency distribution, we start with the mean score because this represents the ‘central’ position within our group of data. We are then interested in how far each score deviates from this mean score. To find the total variability in our group of data, we simply add up the deviation of each score from the mean. The average deviation of a score can then be calculated by dividing this total by the number of scores. How we calculate the deviation of a score from the mean depends on our choice of statistic, whether we use absolute deviation, variance or standard deviation.

Absolute Deviation and Mean Absolute Deviation

Perhaps the simplest way of calculating the deviation of a score from the mean is to take each score and minus the mean score. For example, the mean score for our 100 students was 58.75 out of 100. Therefore, if we took a student that scored 60 out of 100, the deviation of a score from the mean is 60 – 58.75 = +1.25. It is important to note that scores above the mean take on a positive (+) sign and those below the score a minus (-) sign.

To find out the total variability in our group of data, we would perform this calculation for all of the 100 students’ scores. However, the problem is that because we have both positive and minus signs, when we add up all of these deviations, they cancel each other out, giving us a total of zero deviations. Since we are only interested in the deviations of the scores and not whether they are above or below the mean score, we can ignore the minus sign and take only the absolute value, giving us the absolute deviation. Adding up all of these absolute deviations and dividing them by the total number of scores then gives us the mean absolute deviation (see below). Therefore, for our 100 students the mean absolute deviation is 12.81.