Median and Mode

Finding the median always starts the same way: put the data in order from smallest to largest. This relies on the same skill you use when comparing and ordering numbers. Once the data is ordered, there are two cases.

Case 1: An odd number of values

When there is an odd number of data points, the median is simply the value in the very middle of the ordered list.

For the data set 3, 5, 7, 9, 11 there are five values, so the middle one, 7, is the median. There are exactly two values below it and two values above it.

Case 2: An even number of values

When there is an even number of data points, there is no single middle number. Instead, take the two middle values and find their average by adding them together and dividing by 2.

For example, in the ordered set 4, 6, 8, 10, the two middle values are 6 and 8. The median is \( \frac{6 + 8}{2} = 7 \).

The mode is the value that occurs most often. To find it, count how many times each value appears in the data set and pick the one with the highest count.

In the data set 2, 2, 3, 5, 7, the value 2 appears twice while every other value appears only once, so the mode is 2.

Special cases for the mode

• No mode: if every value in the data set appears exactly the same number of times (for example, 1, 2, 3, 4), there is no mode.
• One mode (unimodal): most data sets have exactly one value that repeats more than the rest.
• Two modes (bimodal): if two different values are tied for the highest frequency, the data set has two modes. For example, in 3, 4, 4, 6, 6, 9 both 4 and 6 appear twice, so the modes are 4 and 6.
• More than two modes (multimodal): it is possible for three or more values to be tied for the most frequent, especially in larger data sets.

Find the median and mode of this data set: 8, 3, 8, 5, 10, 3, 8.

Step 1: Order the data. 3, 3, 5, 8, 8, 8, 10

Step 2: Find the median. There are 7 values (an odd count), so the median is the 4th value in the ordered list, which is 8.

Step 3: Find the mode. The value 8 appears three times, more than any other value, so the mode is also 8.

In this case the median and mode are the same number, but that will not always happen — they measure different things and can give very different answers.

The median is useful when a data set has an extreme value, or outlier, that would pull the mean too far in one direction; the median stays close to the "typical" middle value regardless of extreme numbers. You can explore how outliers affect a data set further on the range and outliers page.

The mode is especially useful for categorical data, such as favorite colors or shoe sizes, where finding an average by adding and dividing does not make sense, but identifying the most common answer does.

Related lessons

• Statistics Help