How to Find the Median from a Frequency Table

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.

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 6th.

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 3rd 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 5.5th.

There is no 5.5th number. It is halfway between the 5th and 6th numbers. Find the 5th and 6th number.

4

Find the 5th 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 2nd row. This is a Score of 6. The 5th number is 6.

5

Find the 6th 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 3rd row. This is a Score of 7. The 6th number is 7.

6

The median is the 5.5th number, which is the mean of the 5th number (6) and the 6th 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.

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See Also

What is the median? What is a frequency table? What is an average? Learn more about finding the median from a frequency table ( interactive widget)