## How to Find the Median from a Frequency Table

Finding the median from a frequency table is easy.## Question

The frequency table below shows the test scores for a class of students. What is the median test score?## Step-by-Step:

## 1

Make sure the entries in the

**Score**column are in numerical order. In our example, the entries are in numerical order.## 2

Add another column onto the table, labelled

**Cumulative Frequency**. For each row of the table, add the entries in the**Frequency**column up to that row.## 3

Find the entry in the bottom row of the

**Cumulative Frequency**column. In this example, it is 11. (**Check:**You should get the same result if you add up the numbers in the**Frequency**column). Find the middle number of this number. (See**Finding the Middle Number**in the**Top Tip**). In this example, there are 11 numbers, so the middle number is the 6^{th}.## 4

Find the first entry in the

**Cumulative Frequency**column where this middle number (6) is first reached. A cumulative frequency of 6 is first reached in the 3^{rd}row.## 5

Find the entry in the

**Score**column of this row. This is the median.## Answer:

The median of the test scores is 7.## How to Find the Median from a Frequency Table (with an Even Numbered Set)

In the example above, there were 11 numbers, an odd number. (The total number of numbers is the sum of the**Frequency**column or the last entry in the

**Cumulative Frequency**). The procedure is slightly different if there is an even number of numbers, which will be the case in the example below.

## Question

The frequency table below shows the test scores for a class of students. What is the median test score?## Step-by-Step:

## 1

Make sure the entries in the

**Scores**column are in numerical order. In our example, the entries are in numerical order.## 2

Add another column onto the table, labelled

**Cumulative Frequency**. For each row of the table, add the entries in the**Frequency**column up to that row.## 3

Find the entry in the bottom row of the

**Cumulative Frequency**column. In this example, it is 10. (**Check:**You should get the same result if you add up the numbers in the**Frequency**column). Find the middle number of this number. (See**Finding the Middle Number**in the**Top Tip**). In this example, there are 10 numbers, so the middle number is the 5.5^{th}. There is no 5.5^{th}number. It is halfway between the 5^{th}and 6^{th}numbers. Find the 5^{th}and 6^{th}number.## 4

Find the 5

A cumulative frequency of 5 is first reached in the 2

^{th}number. Find the first entry in the**Cumulative Frequency**column where 5 is first reached and read off the**Score**.A cumulative frequency of 5 is first reached in the 2

^{nd}row. This is a**Score**of 6. The 5^{th}number is 6.## 5

Find the 6

A cumulative frequency of 6 is first reached in the 3

^{th}number. Find the first entry in the**Cumulative Frequency**column where 6 is first reached and read off the**Score**.A cumulative frequency of 6 is first reached in the 3

^{rd}row. This is a**Score**of 7. The 6^{th}number is 7.## 6

The median is the 5.5

^{th}number, which is the mean of the 5^{th}number (6) and the 6^{th}number (7). Find the mean by adding the two numbers and dividing by 2.
Median = (6 + 7) ÷ 2 = 13 ÷ 2 = 6.5

## Answer:

The median of the test scores is 6.5.## Interactive Widget

Here is an interactive widget to help you learn about finding the median from a frequency table.## Beware

## Make Sure the Numbers Are in Order

Make sure the numbers in the left most column are in order. The median is the middle number in a set of numbers that have been**arranged in order**. A frequency table is just a way of presenting these numbers. In the frequency table below, the numbers are not in order:

The first step is to rearrange the rows so the number in the left most column are in order:

**Don't forget:**The whole row, not just that in the first column, needs to be rearranged. Each number in the

**Frequency**column corresponds to the

**Number**to its left.

## Note

## What Is the Median?

The median is an average of a set of numbers. The median is the middle number in a set of numbers that has been arranged in order. (If there are an even number of numbers in a set, the median is the mean of the middle two numbers).## Top Tip

## A Formula to Find the Middle Number

The median is the middle number in an ordered set. If you know how many numbers there are in a set, which is the middle number? The formula for finding the middle number is:In this formula,

**n**is how many numbers there are in the set. In the examples on this page, we asserted that the 6

^{th}number is the middle of 11 and that the 5.5

^{th}is the middle of 10. We can now use the formula to show this:

**n**= 11(n + 1) ÷ 2 = (11 + 1) ÷ 2 = 6**n**= 10(n + 1) ÷ 2 = (10 + 1) ÷ 2 = 5.5Remember that the 5.5^{th}number is halfway between (or the mean of) the 5^{th}and 5^{th}numbers.

**n**is odd, the middle number will always be a whole number. When

**n**is even, the middle number will always have a .5 on the end, indicating it is halfway between two numbers.

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