Mean, median and mode

Three summaries answer different questions

For 2, 3, 3, 4, 18, the total is 30 and the mean is 6. The median is the middle ordered value, 3. The mode is the most frequent value, also 3. The large final value pulls the mean upward; the median better describes the middle of this particular set.

For a categorical variable such as favourite fruit, a mode can be meaningful but an arithmetic mean of category names is not.

Sort before locating the middle

For 9, 2, 4, 6, order them as 2, 4, 6, 9. The median is (4 + 6)/2 = 5, even though no observation equals 5. Averaging the two middle entries of an unsorted list would not find the median.

For a frequency table, count observations, not distinct labels. Values 1, 2, 3 with frequencies 2, 5, 3 have total count 10 and total 1(2) + 2(5) + 3(3) = 21, so mean 2.1. The fifth and sixth ordered observations are both 2, so median 2; the modal value is 2, whose frequency is greatest.

Combine groups through totals

Four observations with mean 7 have total 28. Six with mean 12 have total 72. Combined mean is 100/10 = 10. Averaging 7 and 12 directly would give equal weight to unequal group sizes.

Check your understanding 1 Choose what changes

The data 2, 4, 4, 5, 10 are changed by replacing 10 with 30. Find the new mean, median and mode.

Your answer or drawing on paper

Show answer

The new total is 45, so mean 9. Median 4 and mode 4 stay unchanged. One changed extreme can affect the mean without changing the middle or most common value.

State what a summary leaves unknown

Two classes have the same mean score. That alone does not tell you which has more pupils above a chosen threshold or which has less variation. Use the actual distribution or an appropriate spread summary rather than inferring every learner’s result from the mean.