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How Do You Find Median in a Frequency Table?

The median is a measure of central tendency. Now, you’re going to learn how to determine the median given values in a frequency table.

The first step towards finding the median in a frequency table is to find the middle value in the data set. The sum of all the frequencies in a frequency table is called $n$ . The number $n$ is very important when you want to find the median in a frequency table.

Rule

The Middle Number in a Frequency Table

There are two cases:

1.

n

is an odd number:

Middle Number = \frac{n + 1}{2}

2.

n

is an even number, there are two middle numbers:

\begin{array}{l} {Middle Number}_{1} & = \frac{n}{2} \\ {Middle Number}_{2} & = \frac{n + 2}{2} \end{array}

Finding the median is easy once you have the middle number.

Rule

Median in a Frequency Table

The median is simply the value of the middle number. If $n$ is an even number, where there are two middle numbers, then the median is the mean of these two numbers.

This will be easier to understand after looking at a couple of examples.

Example 1

This frequency table shows how many PCs a group of youths have in their homes.

Number of PCs Frequency
0 2
1 7
2 6
3 3
4 0
5 1

Find the median number of PCs per house.

In order to find the middle number, you need to first know how many homes there are. This is the same as the sum of the frequencies—how many youths were asked this question. This sum is

2 + 7 + 6 + 3 + 0 + 1 = 19

Since 19 is an odd number there is only one number in the middle and we can use the first rule above:

\frac{19 + 1}{2} = 10

The tenth person is the median. So who was the tenth person? There were two youth with no PCs. There were seven with one, so that’s $2 + 7 = 9$ with one or less. So the tenth person must be the first with 2 PCs. So the median is 2.

You could have seen this more easily if the table were expanded to include the cumulative frequency. Let’s try that!


Number of PCs	Frequency	Cumulative Frequency

$0$	$2$	$2$

$1$	$7$	$9$

$2$	$6$	$15$

$3$	$3$	$18$

$4$	$0$	$18$

$5$	$1$	$19$

You see that the first row in cumulative frequency to cross 10 is the row with 2 PCs. This is another way of determining that the median is 2.

What if the sum $n$ is an even number? You will see that case in the next example.

Example 2

You take a walk along a street in Hollywood and count the number of cars parked in front of each home. You organize the information you find in the frequency table below. Since you remember that having the cumulative frequency can be helpful, you add that right away.

Number of Cars

Frequency

Cumulative Frequency

$0$

$1$

$1$

$1$

$4$

$5$

$2$

$7$

$12$

$3$

$6$

$18$

$4$

$2$

$20$

$5$

$4$

$24$

Find the median number of cars.

Just like in Example 1, you need to find the middle number by adding the number of homes together. This is the sum of the frequencies:

1 + 4 + 7 + 6 + 2 + 4 = 24

You could have also seen this from the last row in the cumulative frequency. Since 24 is an even number, there are now two middle numbers:

\frac{24}{2} = 12

and

\frac{24 + 2}{2} = 13

Both the 12^th and 13^th houses are the middle numbers here. You can see from the cumulative frequency that the 12^th house had two cars and the 13^th house was the first to have three cars. The median is the mean of these two:

\frac{2 + 3}{2} = \frac{5}{2} = 2.5

So the median number of cars per house on this street is $2.5$ .

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Number of Cars	Frequency	Cumulative Frequency

$0$	$1$	$1$

$1$	$4$	$5$

$2$	$7$	$12$

$3$	$6$	$18$

$4$	$2$	$20$

$5$	$4$	$24$