The mean median mode calculator is the easiest way to learn how to find the mean, median, mode, and even more! In this article, you'll find the definitions and formulas of:

  • mean
  • median
  • mode
  • range & midrange

This calculator not only tells you the answer, but also shows you all the steps to help you learn how to calculate the mean, median, mode, range, and midrange by yourself.

What is mean, median, and mode?

The mean, median, and mode are three of the most common measures of the center of a distribution (whether it's a binomial distribution, Poisson distribution or any other distribution), also called the central tendency.

Mean, median, and mode definitions:

  • The mean is the average of all numbers.

  • The median is the "middle" number, where there is an equal number of values less than and greater than the median.

  • The mode is the most frequently occurring number.

What is range and midrange?

The range (a measure of spread) and midrange (another measure of central tendency) are also commonly used to describe a set of numbers.

Definition of range and midrange:

  • The range is the difference between the minimum and maximum values.

  • The midrange is the value exactly halfway between the minimum and maximum values.

Detailed definitions and formulas for mean, median, mode, range, and midrange are described below.

How to find the mean

When people talk about the mean, they are usually referring to the arithmetic mean, which is the average of all numbers in the dataset.

The formula for the mean (arithmetic) is:

μ = ∑X / N

where:

  • μ represents the population mean (you can use the letter M to represent the mean of a sample instead, but the calculation is the same),
  • ∑X represents the sum of all numbers,
  • N is the total number of numbers.

For example, you would calculate the mean for the numbers 2, 5, 6, 8, 9 as follows:

μ = (2 + 5 + 6 + 8 + 9) / 5

μ = 30 / 5

μ = 6

Try it yourself

Enter your numbers into the mean median mode calculator and select mean in the steps to show option to see the calculation steps for the mean.

Although the arithmetic mean is the most common mean calculation, you can learn about other types of means by checking out the geometric mean calculator, the harmonic mean calculator, and the weighted mean calculator.

Median definition

The median is defined as the "middle" number in a sorted list of numbers. There is an equal amount of numbers above and below the median.

When determining how to calculate median, we need to know if the dataset is even or odd:

  • If there is an odd number of numbers, such as the dataset 4, 6, 9, the median is simply the middle number, in this case, 6.

  • If there is an even number of numbers, such as the dataset 3, 6, 7, 9, the median is the average of the two middlemost numbers, in this case, 6 and 7, so the median is 6.5.

How to find the median

As an example, let's find the median of the set of numbers 2, 7, 4, 8, 6, 9:

1. Sort the numbers from least to most

Our list, from least to more, is:

2, 4, 6, 7, 8, 9

Note that sorting from most to least is also acceptable.

2. Determine the "center" of the numbers

You can probably see at a glance that the middle two numbers are 6 and 7, but, in a large dataset, it will help to know how many numbers there are.

To find out the "center" of the dataset, use the equation:

center = (N + 1) / 2

Since there are 5 numbers, the center is calculated as follows:

center = (5 + 1) / 2 = 3.5

Since the "middle" number is 3.5, the median will be the average of the 3rd and 4th numbers in our list.

3. Find the median

Counting from left to right (or right to left), we find that the 3rd and 4th numbers are 6 and 7:

2, 4, 6, 7, 8, 9

The median is the average of 6 and 7. Therefore, the median is 6.5.

Mode definition

The mode is the number that occurs most frequently. There can be a few scenarios:

  • If one number occurs more frequency than all the others, there is one mode, and the distribution is called unimodal.

  • If there is more than one number that occurs at the highest frequency, then there are multiple modes, and the distribution is called multimodal.

  • If there are exactly two modes, the distribution is still multimodal, but you can call it bimodal to be more specific.

  • If all numbers occur the same number of times (e.g., each number occurs once), then there is no mode.

How to find the mode

Let's find the mode for the list of numbers 2, 7, 6, 4, 7, 7, 6, 8, 7, 4, 6 using three methods.

Method 1. Tick marks on a number line

One easy way to find the mode with pen and paper is to create a number line and add a tick mark or a dot on the number line for each number:

number line to find the mode

The number 7 has the highest column and is therefore the mode.

Method 2. Arrange the numbers from least to greatest

Alternatively, you can sort the numbers from least to greatest. Sorting makes it easier to see the mode visually:

2, 4, 4, 6, 6, 6, 7, 7, 7, 7, 8

Note how many times each number occurs. The number 7 occurs the most frequently - a total of 4 times.

Method 3. Create a frequency table

Finally, for large datasets, it can be useful to organize the data in a frequency table. You can fill in the frequency table after sorting the numbers from least to greatest:

ValueNumber of Occurences
21
42
63
74
81

How to calculate range and midrange

The range is the simplest measure of spread (how varied a set of numbers is). To find the range, calculate the difference between the maximum and minimum values using the range formula:

range = max - min

The midrange is yet another measure of the central tendency, like the mean, median, and mode. The midrange is exactly halfway between the maximum and minimum values. In other words, it is the average of the maximum and minimum values:

midrange = (max + min) / 2

For the set of numbers 2, 7, 6, 4, 7, 7, 6, 8, 7, 4, 6, the minimum is 2 and the maximum is 8. Therefore, the range is 8 - 2 = 6 and the midrange is (8 + 2) / 2 = 5.

Jasmine J Mah