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.
Reading a box plot
A box plot shows five numbers in one picture: the lowest and highest values at the whisker ends, the median as the line inside the box, and the lower and upper quartiles as the box edges. The IQR is the width of the box — upper quartile minus lower quartile — read straight off without any calculation. Comparing two box plots side by side is faster than comparing two lists of raw data, which is exactly why exams draw them.
Estimating the median from grouped data
When data is grouped into classes you cannot pick out the exact middle value, so find which class it falls in instead. Add a running total (the cumulative frequency) down the table until it passes half the total frequency — that class contains the median. Reading the matching point off a cumulative frequency graph gives an estimate of the actual value. Like the grouped mean, this is an estimate, because the exact values inside each class are unknown.