In this explainer, we will learn how to find and interpret the median of a data set.

The median is an example of a measure of center (or a measure of central tendency). Often, we would like to find a single number that can represent a whole data set or at least give us some information about typical values in the data set. There are a number of ways to describe typical values. For example, one way to describe a typical value is to see what value is the most common; this is the mode. The mean, which is the average of all the values in a data set, is another example. We could also describe a typical value by looking at the number in the middle; this is the median.

The mode, mean, and median are all different examples of measures of center. We will only discuss the median here.

### Definition: The Median

The median of a set of data represents the middle value.

Half of the data values are above the median, and half of the data values are below the median.

Let us start by looking at an example to see how to calculate the median when we have an odd number of data values.

### Example 1: Finding the Median of a Data Set with an Odd Number of Data Values

Find the median of the values 6, 8, 16, 6, and 19.

### Answer

The median is the middle value. To find the middle value, we first put the data values in order from least to greatest:

Now, to find the middle value, we can count in from each side until we reach the middle.

The middle value is 8. Hence, we have found that the median of these five values is 8.

As we saw in the above example, when there are an odd number of data values, finding the median is easy because there is exactly one middle value.

We have to think about how the situation is different when there are an even number of data values. Let us look at an example that shows what to do in this case.

### Example 2: Finding the Median of a Data Set with an Even Number of Data Values

Find the median of the values 13, 5, 9, 10, 2, and 15.

### Answer

Let us remember that the median of a data set is the value in the middle. We will write the numbers in order from least to greatest so we can find this value:

There are 6 data values, which is an even number, so there are two values in the middle.

To find the median, we find the number that is halfway between these middle values. The number halfway between 9 and 10 is

Hence, the median is 9.5 because half of the data values are above it and half are below it.

Next, we look at what happens when there are an even number of data values with a repeated value in the middle.

### Example 3: Finding the Median of a Data Set with a Repeated Middle Value

What is the median of the numbers 11, 11, 8, 8, 9, and 9?

### Answer

Recall that the median represents the middle of the data.

We start by writing the numbers in order and finding the middle value (or values).

Normally, when there are two middle values, we have to calculate the number halfway between them. However, since both middle numbers are the same here, this value is the median.

Hence, the median is 9.

Now that we have seen some examples of how to find the median of a data set, let us summarize what we have learned.

### How To: Finding the Median of a Data Set

- Write the data in order from least to greatest.
- Find the middle value (or values).

If there are an odd number of data values, there will be exactly one value in the middle.

If there are an even number of data values, there will be two values in the middle. - Find the median.

If there are an odd number of data values, the median is the middle value.

If there are an even number of data values, the median is halfway between the two middle values.

In the above examples, it was relatively simple to order the data values from least to greatest and then work out the middle value (or values). When we have larger sets of data, however, we need to make sure we count the number of values given in the question so that we do not omit any by accident when reordering them. Also, if the data values include decimal numbers, we need to make sure that we copy them down accurately, reorder them correctly, and carry out any calculations carefully. Here is an example of this type.

### Example 4: Finding the Median of a Larger Data Set with an Even Number of Values

Calculate the median of the values 2.9, 5.4, 5.1, 3.7, 3.4, 1.9, 6.1, 8.1, 12.4, 2.6, 8.8, and 5.5.

### Answer

Recall that the median of a data set is the middle value, which we can find by ordering the values and finding the single value (or pair of values) in the center.

In this question, we have 12 data values. Since 12 is an even number, when we order the values from least to greatest, there will be two values in the middle. The median will be the value halfway between these two.

Writing the data values from least to greatest, we have two middle values, as shown below.

As the two middle values are 5.1 and 5.4, then the number halfway between them is

Therefore, 5.25 is the median of this data set.

The fact that the median is the middle value can help us solve problems where there is a missing data value, as shown in the next example.

### Example 5: Finding an Unknown Value in a Data Set with a Known Median

Farida has the following data: 10, 8, 7, 9, .

If the median is 8, what number could be?

- 8.5
- 7
- 9
- 10
- 9.5

### Answer

Recall that the median represents the middle of the data. Let us start by ordering the data values from least to greatest and seeing what clues we can figure out about the number .

We are told that the median is 8, and we know that there must be the same number of data values above the median as below it. As there are already two values above the median, the number can be at most the median value.

This means that can be any number that is 8 or below. However, out of the available choices, we see that none of 8.5, 9, 9.5, or 10 will work.

Therefore, we must have that .

Note that, in the above example, the fact that there were already two values above
the median number of 8 did not mean that we could immediately assume the missing value
must be *less than* 8. We could only assume it was
*at most* 8. This is because a data set such as
would still have a median of 8, even though only one of the data values is
strictly less than the median value. In our example, the missing value turned
out to be less than 8 because 7 was the only one of the five available
answer options that satisfied the condition of being at most 8.

When working out the median for data sets involving a mixture of positive and negative numbers, it is especially important to make sure the numbers are ordered correctly, without dropping any minus signs. Let us look at an example.

### Example 6: Finding the Median of a Data Set Involving Negative Numbers

Nader’s game scores were , 2, 2, , , , and 8. Determine the median.

### Answer

We have been asked to find the median of some data, which we can recall is the value that lies in the middle of the other values when they are ordered.

Here, the data values are game scores. The number of data values is 7, which is an odd number. Therefore, when we order the values from least to greatest, there will be a single value in the middle; this is the median.

Writing the data values from least to greatest, not forgetting any minus signs, we get one middle value, as shown below.

We conclude that the median of Nader’s game scores was .

In our final example, we can apply the ideas explored here to calculate the medians of two data sets listed in a table and then find the difference between the two medians.

### Example 7: Calculating the Median for Two Data Sets and Finding the Difference between the Two Values

The table records the heights, in inches, of a group of fifth graders and a group of sixth graders. What is the difference between the medians of the heights of both groups?

Fifth Grade | |
---|---|

Sixth Grade |

### Answer

Recall that the median represents the middle of the data.

In this question, we have two data sets made up of the heights, in inches, of two groups of students. Each data set has 7 values, which is an odd number. This means that in both cases, the median will be the single value in the middle that we obtain when we order the data values from least to greatest.

Writing the fifth grade heights from least to greatest, we get the following list.

Therefore, the median height of the fifth graders is 61 inches.

Similarly, writing the sixth graders’ heights from least to greatest, we get another list.

From this, we see that the median height of the sixth graders is 60 inches.

Finally, we work out the difference between the medians by subtracting the smaller number from the larger one, to get . Thus, we have found that the difference between the medians of the heights of the groups is 1 inch.

Let us finish by recapping some key concepts from this explainer.

### Key Points

- The median of a set of data represents the middle value.
- If there are an odd number of data values, there will be exactly one value in the middle; this is the median.
- If there are an even number of data values, there will be two values in the middle; the value halfway between the two is the median.
- Always make sure that you count the number of data values given in the question so that none are omitted accidentally when we reorder them from least to greatest.
- If the data values include decimal numbers or negative numbers, make sure you copy them down accurately and reorder them carefully.
- When given the median of a data set, we can sometimes work backward to find an unknown data value.