Averages, spread and which to use
Maths · GCSE · The School
The three averages
The MEAN adds everything and divides by how many. The MEDIAN is the middle value once ordered, and for an even count it is halfway between the middle two. The MODE is the most common value, and it is the only average that works for categories like favourite colour. Ordering the list first is what people forget when finding a median.
Spread is half the answer
An average on its own tells you almost nothing. Two classes can both average 60% while one ranges from 55 to 65 and the other from 20 to 100. The RANGE is the largest minus the smallest; the INTERQUARTILE RANGE is the middle half, and it ignores outliers. Exam questions asking you to compare two sets want one comment on average and one on spread — that is where the second mark is.
Averages from a table
For a frequency table, the mean is not the mean of the values column. Multiply each value by its frequency, add those products, and divide by the total frequency. For grouped data you cannot know the exact values, so use the midpoint of each class and say your answer is an estimate — the word "estimate" is worth a mark.
Town A salaries, in thousands: 22, 24, 25, 26, 28. Town B: 18, 20, 25, 40, 95. Both have a MEDIAN of 25. But the MEANS are 25 and 39.6, because Town B has a 95 pulling it up, and the RANGES are 6 and 77. Which average describes a typical resident? The median, in both, and especially in B — most people there earn nowhere near 39.6. Now write the comparison an examiner wants, one sentence on each: "The medians are equal at 25, so a typical salary is similar. But B is far more spread, with a range of 77 against 6, so incomes there are much less consistent." That is the shape of every compare-two-sets answer.