How to Find The Mean, Median, Mode and Range of a Series Of Numbers

Many calculators have buttons that will do these calculations for you.
Many calculators have functions that will do these calculations for you. Kick Images, Getty Images

When processing data, terms such as mean, median, mode, and range often come up. What do these terms mean and how do you calculate them? This worked example problem defines each term and demonstrates how to calculate the value of each term from a series of numbers.

Mean, Median, Mode, and Range Definitions

First, let's review what these terms mean:

Mean: The mean is the average of a set of numbers, determined by adding them together and dividing by how many numbers are present.

Median: The median is the middle value between the smallest and largest of a set of numbers.

Mode: The mode is the number that appears most often in a data set. It only has significance if numbers are repeated.

Range: The range is the difference between the largest and smallest numbers.

Mean, Median, Mode, and Range Problem


Find the mean, median, mode, and range of the following set of values:

11, 19, 13, 16, 12, 12, 18, 14, 20

Solution:

It is often helpful to arrange the numbers from lowest to highest:

11, 12, 12, 13, 14, 16, 18, 19, 20

Part 1 - How To Find the Mean


The mean of a set of numbers is the average. The mean is calculated by finding the sum of all the values and dividing by the number of values.

11+12+12+13+14+16+18+19+20 = 135

There are 9 numbers in the series, so the mean is:

Mean = 135/9 = 15

A quick way to check if your answer is reasonable is to see if your answer is somewhere between the lowest and greatest number in the series.

In our case, 11 is the lowest and 20 is the greatest. 15 is between the two.

Part 2 - How To Find the Median


The median of a series of numbers is the number that appears in the middle of the list when arranged from smallest to largest. For a list with an odd number of members, the way to find the middle number is to take the number of members and add one.

Then divide that value by two. In our case, there are 9 numbers in the series. 9+1 = 10 and half of 10 is 5. The fifth number in the series is the median or 14.

If the number of members of the series was even, the average of the two middle numbers would be the median.

Part 3 - How To Find the Mode


The mode is the number in the series that appears the most often. If there is no single number that appears more than any other number in the series, there is no value for the mode.

The number 12 appears twice in the series. The mode of this series is 12.


Part 4 - How To Find the Range


The range is the difference between the largest number and the smallest number. The range is an indication of how spread out the values of the series are. The smaller the range, the closer the numbers in the series are to each other.

The range of our series is 20-11 = 9.

Answer:

For the series:

11, 12, 12, 13, 14, 16, 18, 19, 20

The mean is 15. The median is 14. The mode is 12 and the range is 9.

To summarize:

Mean = average value
Median = middle value
Mode = most common value
Range = the "spread" of the series. This is the difference between largest and smallest numbers in the series.
 

More Homework Help

For more example problems, use the Worked Chemistry Problems.

It contains over a hundred different worked example problems useful for chemistry students.

Learn More

From here, you can review details and examples of how to calculate the mean or average or move on to learn how to calculate standard deviation. Related concepts include accuracy and precision as well as significant figures.