Explainer: The Median of a Data Set

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 measure of central tendency). Often we would like to find a single number which 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 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 is above the median, and half of the data is 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 8, 6, 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: 6,6,8,16,19.

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 the median: median=8.

As we saw in the 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 this is different when there are an even number of data values. Let us do an example 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

We will write the numbers in order from least to greatest so we can find the middle value: 2,5,9,10,13,15.

There are 6 data values, which is even, 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 9+102=9.5.

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

We will end with a few more examples to practice finding the median.

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

What is the median of the following numbers: 11, 11, 8, 8, 9, 9?

Answer

Start by writing the numbers in order and finding the middle value(s).

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

Hence, the median is 9.

Example 4: Finding Missing Data Values given the Median

Isabella has the following data: 10, 8, 7, 9, 𝑚.

If the median is 8, which number could 𝑚 be?

  1. 10
  2. 9.5
  3. 7
  4. 8.5
  5. 9

Answer

The median represents the middle of the data; half of the data values are above it and half are below it.

Let us start by putting the data in order and thinking about what clues we can figure out about the number 𝑚.

The median is 8; we know that there has to be the same number of data values above and below the median.

Since there are already 2 values above the median, the number 𝑚 must be below the median.

This means that 𝑚 can be any number that is 8 or lower. Out of our choices, we see that none of 8.5, 9, 9.5, or 10 will work. Hence, 𝑚=7.

Now you have seen how to calculate the median of a set of data values, we will summarize what we have learned.

How To Find the Median of a Set of Data

  1. Write the data in order from least to greatest.
  2. Find the middle value(s).
    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.
  3. Find the median.
    If there are an odd number of data values, the median is the middle number.
    If there are an even number of data values, the median is halfway between the two.

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