Median definition
The median is a type of average that gives the middle value of a data set when the values are placed in order. To find the median, first arrange the numbers from smallest to largest. Then find the value in the middle. For example, in the ordered set 3, 5, 8, 10 and 12, the median is 8 because it is the middle value.
The data must be ordered before the median is found. This is one of the most important steps. If the numbers are 12, 3, 8, 5 and 10, the middle number in the written list is 8, but that is only correct because the ordered list would be 3, 5, 8, 10, 12. If the original order was different, choosing the middle item before ordering could give the wrong answer.
Median is used in statistics, data handling, probability, business, science, geography, finance and real-life comparisons. At GCSE level, students find the median from lists, tables, frequency tables and sometimes cumulative frequency diagrams. They may also compare the median with the mean and mode to decide which average best represents a set of data.
When there is an odd number of values, the median is the single middle value. For example, in 2, 4, 7, 9 and 11, there are five values, so the third value is the median. The median is 7. A useful position rule is that the median position is found by adding 1 to the number of values and dividing by 2. With 5 values, this gives the third position.
When there is an even number of values, there is no single middle value. In this case, find the two middle values and calculate their mean. For example, in 3, 6, 8 and 15, the two middle values are 6 and 8. The median is the mean of 6 and 8, which is 7. This keeps the median exactly halfway between the two central values.
The median is often useful when data contains outliers. An outlier is a value that is much larger or smaller than the rest of the data. For example, the numbers 5, 6, 7, 8 and 100 have a median of 7, which represents the central position better than the mean. The mean is pulled upwards by the outlier, but the median is more resistant to extreme values.
Median can also be found from frequency tables by using cumulative frequencies. Instead of writing every value out separately, students can count through the frequencies until they reach the middle position. This is especially useful when a value occurs many times. In grouped data, the exact median may not be known, but an estimate or median class can be found.
Related ideas include average, mean, mode, range, ordered data, outliers, frequency tables, cumulative frequency and box plots. A clear definition of median helps students understand that it is the central value by position, not by calculation from the total. In exams, always order the data first and check whether there is an odd or even number of values.