Median, quartiles and percentiles

The median is the middle value when the data are arranged in order. Half the values lie below it, half above. It is a positional average.

Arrange first, then find the position. The median is the (N + 1) ÷ 2 th item. For the weights of five people, 55, 62, 60, 59 and 70 kg, first arrange them: 55, 59, 60, 62, 70. The (5 + 1) ÷ 2 = 3rd item is 60 kg. With an even number of items, the median is the average of the two middle ones.

The median is not affected by extreme values: for 2, 3 and 900, the median is still 3, while the mean would be over 300. This makes it a good average for data with a few very large or very small values.

In a discrete series, find (N + 1) ÷ 2, and look down the cumulative frequency column for the first value whose cumulative frequency reaches it.

In a continuous series, find N ÷ 2 (not (N + 1) ÷ 2). The class whose cumulative frequency first reaches N ÷ 2 is the median class. Then:

Median = L + [(N/2 – c.f.) ÷ f] × h, where L is the lower limit of the median class, c.f. the cumulative frequency of the class before it, f the frequency of the median class, and h its width.

Quartiles divide ordered data into four equal parts. Q1 (lower quartile) is the (N + 1) ÷ 4 th item; Q2 is the median; Q3 (upper quartile) is the 3(N + 1) ÷ 4 th item.

Percentiles divide the data into 100 parts. If a candidate's score is at the 90th percentile, the score is better than that of 90% of all the candidates. Quartiles and percentiles, like the median, are positional values.