## How to Find the Median

The median is found by ordering the numbers and finding the middle number. The middle number is slightly different depending on whether there are an odd or even number of numbers in the set.- If there are an odd number of numbers, the median is simply the middle number in the set.
- If there are an even number of numbers, the median is halfway between the middle two numbers. (It is the mean of the middle two numbers).

### Odd Numbered Set

## Question

What is the median of the test scores below?## Step-by-Step:

## 1

List the numbers in numerical order (going from the smallest to the largest number).

7 10 8 6 4 → 4 6 7 8 10

## 2

Find the middle number. In our example, the middle number is the 3

^{rd}number of 5.4 6 **7** 8 10

Median = 7

## Answer:

The median of the test scores is 7.### Even Numbered Set

This time, 6 students take a mathematics test.## Question

What is the median of the test scores below?## Step-by-Step:

## 1

List the numbers in numerical order (going from the smallest to the largest number).

8 9 5 6 5 10 → 5 5 6 8 9 10

## 2

Find the middle two numbers. In our example, the middle two numbers are the 3

^{rd}4^{th}numbers of 6.5 5 **6 8** 9 10

Middle two numbers = 6 8

## 3

## Answer:

The median of the test scores is 7.## A Formula to Find the Middle Number

In the two examples above, we have quoted that the 3^{rd}number is the middle of 5 and that the 3

^{rd}and 4

^{th}are the middle two of 6. It is relatively easy to see by eye where the middle number is when there are relatively few of them and they are written out in front of us. It may not always be this easy. Luckily, there is a formula for finding which is the middle number.

In this formula,

**n**is how many numbers there are in the set. Let's apply this formula to the two examples above.

- In the first example, we are asked to find the median of 5 numbers (7, 10, 8, 6 and 4). This means that
**n**= 5. Using the formula to find the middle number gives:Middle number = (n + 1) ÷ 2 = (5 + 1) ÷ 2 = 6 ÷ 2 = 3The median of**7 10 8 6 4**is the**3**number when they are ordered:^{rd}7 10 8 6 4 → 4 6

**7**8 10Median = 7

- In the second example, we are asked to find the median of 6 numbers (8, 9, 5, 6, 5 and 10). This means that
**n**= 6. Using the formula to find the middle number gives:Middle number = (n + 1) ÷ 2 = (6 + 1) ÷ 2 = 7 ÷ 2 = 3.5The median of**8 9 5 6 5 10**is the**3.5**number when they are ordered, which means halfway between the 3^{th}^{rd}and 4^{th}numbers:8 9 5 6 5 10 → 5 5

**6 8**9 10Halfway between 3

^{rd}and 4^{th}numbers = (6 + 8) ÷ 2 = 7Median = 7

## Top Tip

## The Median is the Middle

The median gives the number exactly in the middle of a set of numbers. Half of the numbers in the set are less than the median. Half of the numbers in the set are greater than the median.## Note

## Outliers

Sometimes in a group of numbers, a few of them are much larger...
3, 5, 2, 7,

...or much smaller than the rest:
**150****1**, 502, 847, 564, 980

**outliers**. The median is not affected by outliers, unlike the mean. For example, consider the numbers:

1, 2, 3

The mean is 2 and the median is 2.
If instead the numbers have a large outlier:
1, 2,

The median is still 2 but the mean is now 101.
If you read about the average salary in a country, ask if it is the mean or median average?
The mean average may dragged up by some very rich people, while the median won't be.
**300**## How to Find the Median from a Frequency Table

Sometimes data is presented in frequency tables. A frequency table representing the test scores is shown below:It is possible to find that the median test score is 7.

finding the median from a frequency table

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