Range as a Measure of Spread

Two archers both average 5. Archer A scores 4, 5, 5, 6, 5 — consistent, and you know roughly what you will get. Archer B scores 1, 2, 5, 8, 9 — sometimes brilliant, sometimes terrible. Identical means, completely different performances, and the mean alone cannot tell them apart. That is because an average tells you where the data sits, not how spread out it is. The simplest measure of spread is the range: the largest value minus the smallest. Archer A has 6 minus 4, which is 2, while Archer B has 9 minus 1, which is 8. A small range means consistent, a large range means variable. Always report an average and a spread together.

Finding the range takes three short steps: find the largest value, find the smallest, and subtract the smaller from the larger. For 12, 7, 19, 4 and 15 that is 19 minus 4, which is 15. Sorting the data first makes both ends obvious — written as 4, 7, 12, 15, 19, the smallest sits at one end and the largest at the other, so you cannot miss them. One check that catches errors instantly: the range is never negative. If your answer comes out below zero you subtracted the wrong way round, so swap them.

You can read the range straight off most displays. On a dot plot it is the leftmost dot to the rightmost, so 4 minus 0, which is 4. On a stem-and-leaf plot it is the first leaf to the last, so 51 minus 23, which is 28. But a grouped frequency table cannot give you an exact range. If the intervals run from 150 to 154 up to 165 to 169, the smallest value could be anywhere from 150 to 154 and the largest anywhere from 165 to 169, so all you can give is an estimate of about 19. That is the price of a tidy table — grouping loses the individual values.

The range has one real weakness. Take 12, 14, 15, 15, 16, 17 and 48. Six of the values are packed close together, but the 48 sits far away. With it, the range is 48 minus 12, which is 36, suggesting very spread out data. Without it the range is 17 minus 12, just 5, which describes six of the seven values honestly. The problem is that the range uses only the two end values and ignores everything in between, so a single outlier controls it completely. It is the same weakness the mean has — both get dragged by extremes. So report the range, but then say whether an outlier is inflating it. Senior maths fixes this with the interquartile range.

The main use of the range is comparing two data sets, and there is a way to write it that earns full marks. Suppose Class A and Class B both have a mean of 72, but Class A's range is 12 while Class B's is 46. A good comparison mentions both centre and spread: both classes averaged 72, so their typical performance was the same, but Class A's range of 12 is much smaller than Class B's 46, so Class A was far more consistent. What is not enough is 'Class A is better' — that says nothing measurable. Quote the actual numbers and interpret them in context. Centre, spread, then a sentence about what it means.