Finding the Median and the Mode

1. The median of ungrouped data

The median is the middle value when the data are arranged in order. Half of the values are smaller than or equal to it, and half are larger than or equal to it. Like the median strip that divides a road into two equal halves, the median divides the data into two equal halves.

1. 1Arrange the data in order, from the smallest to the largest. This is called an array.
2. 2Count how many values there are, $n$.
3. 3If $n$ is odd, the median is the middle value, in position $\frac{n+1}{2}$.
4. 4If $n$ is even, there are two middle values, in positions $\frac{n}{2}$ and $\frac{n}{2}+1$. The median is the mean of those two.

Odd number of values. Find the median of 19, 29, 36, 15 and 20. In order: 15, 19, 20, 29, 36. There are 5 values, so the middle one is the 3rd: the median is 20.

Even number of values. Find the median of 8, 9, 7, 6, 8 and 10. In order: 6, 7, 8, 8, 9, 10. There are 6 values, so the middle two are the 3rd and 4th, both equal to 8. The median is (8 + 8) ÷ 2 = 8. If the middle two were 7 and 9, the median would be (7 + 9) ÷ 2 = 8, a value that is not even in the list.

▶ Always arrange the data in order before looking for the middle. The red boxes show the middle value(s).

2. The median from a frequency table

When data are in a frequency table, they are already in order by value, so we do not need to write all the values. We find the position of the middle value and use the running total (cumulative frequency) of the frequencies to see which score it falls in.

There are 20 values (an even number), so the median is the mean of the 10th and 11th values. The running total reaches 8 after score 2 and 12 after score 3, so the 9th to 12th values are all 3. The 10th and 11th are both 3, so the median is 3.

Now a table with an odd total. The ages of 39 children at a party are shown below.

With 39 values, the median is at position (39 + 1) ÷ 2 = 20th. The running total passes 13 and reaches 23 at age 6, so the 14th to 23rd values are all 6. The 20th value is 6, so the median age is 6 years.

▶ The tallest bar is age 6 — it is both the mode and the group containing the median.

3. The mode

The mode is the value that occurs most often. It is the only measure that can also be used for non-number data, such as the most popular network or the most common shoe size, which is why shop owners use it to decide what stock to buy.

• Shoe sizes 38, 40, 39, 40, 41, 40, 39: size 40 occurs 3 times, so the mode is 40.
• In a frequency table, the mode is the value with the highest frequency: age 6 has frequency 10, so the mode of the ages is 6.
• A data set can have two modes (bimodal), for example 2, 3, 3, 5, 5, 9 has modes 3 and 5.
• If every value occurs once, there is no mode.

4. Mean, median or mode? Choosing the best

Each measure has a job. The mean uses every value, so it is best when the data are fairly even. The median is not affected by very large or very small values, so it is best when there are extreme values (like the GH₵10,000 shop rent). The mode is best for the most popular choice, and is the only choice for categories.

For the data 2, 3, 3, 4 and 8 (goals scored in five matches): mean = 20 ÷ 5 = 4, median = 3 (the middle value), mode = 3. The big score of 8 pulls the mean up, but the median and mode show that a typical match had 3 goals.

Worked examples

Try each one yourself first, then open the solution and compare every step.