# How to Find the Median From a Frequency Table (Mathematics Lesson)

# Finding the Median From a Frequency Table

We can find the median of a set of numbers that are presented in a frequency table.# How to Find the Median From a Frequency Table

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

The frequency table below shows the test scores for a class of students.What is the median test score?

Step 1

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

Step 2

**Cumulative Frequency**.

For each row of the table, add the entries in the

**Frequency**column up to that row.

Step 3

**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}.

Step 4

**Cumulative Frequency**column where this middle number (6) is first reached.

A cumulative frequency of 6 is first reached in the 3

^{rd}row.

Step 5

**Score**column of this row. This is the median.

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.

#### An Example Question

The frequency table below shows the test scores for a class of students.What is the median test score?

Step 1

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

Step 2

**Cumulative Frequency**.

For each row of the table, add the entries in the

**Frequency**column up to that row.

Step 3

**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.

Step 4

^{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.

Step 5

^{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.

Step 6

^{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

The median of the test scores is 6.5.# Another Real Example of How to Find the Median From a Frequency Table

The slider below gives another example of how to find the median from a frequency table.# How to Find Other Averages from a Frequency Table

We can use a frequency table to find the other types of average: the mean and the mode.Read more about finding the mean from a frequency table

Read more about finding the mode from a frequency table

##### Interactive Test

**show**

Here's a second test on finding the median from a frequency table.

Here's a third test on 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 a Frequency Table?

A frequency table shows how often (how*frequently*) each number appears in a list of numbers.

# 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.