Statistics · Mathematics

Measures of average

An average describes the centre of a set of data. The mean is the sum divided by the number of values. The mode is the most frequent value. The median is the middle value when the data are arranged in order. If there is an even number of values, the median is the mean of the two middle ones.

Σx is the sum of all the values and n is how many there are.

Worked exampleThe median

Find the median of 7, 3, 9, 4, 12, 6.

Arrange in order.
Six values: the median is the mean of the 3rd and 4th.

Answer: The median is 6.5.

Median of grouped data

For grouped data in a frequency table, find the median class, the class that holds the middle value. Use the cumulative frequency to find it, then estimate the median inside it.

L = lower boundary of the median class, n = total frequency, F = cumulative frequency before the class, f = frequency of the class, c = class width.

The tallest bar is the mode. The median is where half of the pupils are on each side.

Marks	Frequency	Cumulative frequency
0–9	2	2
10–19	4	6
20–29	8	14
30–39	4	18
40–49	2	20

Here n = 20, so the median is the 10th value, which is in the class 20–29 (cumulative frequency passes 10 there). The median is about 24.5.

Try itMean, median and mode of three scores

Choose three scores. The median is the middle one.

Check yourself Find the median of 5, 8, 2, 9, 4.

Check yourself The median of 4, 6, 8, 10 is...

Check yourself The median is best when the data...

Key terms

Mean: Sum of values divided by their number.
Median: The middle value in order.
Mode: The most frequent value.
Cumulative frequency: Running total of the frequencies.

Summary

The mean, median and mode are measures of average.
To find the median, order the data and find the middle.
For grouped data use the cumulative frequency.
The median is not affected by extreme values.