Median Calculator

Want to find the mean of sample or population data instead? Use the mean calculator.

How to use the median calculator

This median calculator allows you to quickly find the median value of any dataset. It not only gives you the answer, but also shows you exactly how the median was calculated for your dataset.

To use the calculator:

1. Enter the dataset. You can separate values using commas, spaces, tabs, or line breaks. Alternatively, you can copy-paste values directly from Excel
2. Click Calculate

The calculator will instantly return the median and show you how to find the median for your dataset, step-by-step.

What is Median

The median is the middle value after arranging the values either in ascending or descending order. It is one of the measures of central tendency and represents the 50th percentile of the data.

Median Formula

The median formula varies based on whether you’re dealing with an odd or an even number of observations. If you’re working with an odd number of observations (i.e., n is odd), the median is the value located exactly in the middle.

Therefore, the median formula for an odd number of observations is: Median = Value at position (n+1​)/2

Where:

• n is the total number of observations

In other words, for a dataset with an odd number of observations, the median is simply the observation at position (n+1)/2 after sorting the data.

However, if you’re working with an even number of observations (i.e., n is even), there will be no single middle value. Instead, the median is the average of the two middle values.

Therefore, the median formula for an even number of observations is

Median = [Value at position (n/2) + Value at position (n/2 +1)]/2

Where:

• n = total number of observations

Simply stated, the median of an even number of observations lies halfway between the two central values. Thus, you should calculate the mean of these two middle values.

How to Find the Median by Hand

Want to learn how to find the median manually? The method depends on whether the dataset has an odd or even number of observations.

1. Finding the Median for an Odd Number of Observations

When the number of observations is odd, the median is the value in the middle position of the ordered dataset.

To find the median by hand, when n is odd:

1. Arrange the data in ascending order.
2. Count the number of observations to get n.
3. Find the middle position using the formula: Median position = (n + 1) / 2
4. Identify the value in the median position. This is the median of your dataset.

Example 1

Find the median of the following data: 12, 5, 9, 3, 15

Solution

Step 1: Arrange the data in ascending order.

Arranging the values from smallest to largest gives: 3, 5, 9, 12, 15

Step 2: Count the number of observations.

There are 5 observations in the dataset. Thus: n = 5

Since 5 is odd, we use the median method for an odd number of observations.

Step 3: Find the middle position.

For an odd number of observations, the median position is: Median position = (n + 1) / 2

Substituting n = 5 gives: Median position = (5 + 1) / 2

= 6 / 2

= 3

This means the median is the value in the 3rd position of the ordered dataset.

Step 4: Identify the value in the middle position.

In the ordered dataset: 3, 5, 9, 12, 15, the value in the 3rd position is 9.

Therefore, the median is 9.

2. Finding the Median for an Even Number of Observations

When the number of observations is even, there is no single middle value. In this case, the median is the average of the two middle values.

To find the median by hand, when n is even:

1. Arrange the data in ascending order.
2. Count the number of observations to get n.
3. Find the two middle positions using the formulas:
   • First middle position = n / 2
   • Second middle position = (n / 2) + 1
4. Identify the two middle values.
5. Add the two middle values and divide by 2 to get the median

Example 2

Find the median of the following data: 8, 4, 10, 6

Solution

Step 1: Arrange the data in ascending order.

Arranging the values from smallest to largest gives: 4, 6, 8, 10

Step 2: Count the number of observations.

There are 4 observations in the dataset. Thus, n = 4

Since 4 is even, we use the median formula for an even number of observations.

Step 3: Find the two middle positions.

For an even number of observations (n = 4), the two middle positions are:

• First middle position = 4 / 2 = 2
• Second middle position = (4 / 2) + 1 = 3

This means the two middle values are in the 2nd and 3rd positions of the ordered dataset.

Step 4: Identify the two middle values.

In the ordered dataset: 4, 6, 8, 10, the value in the 2nd position is 6, and the value in the 3rd position is 8.

Therefore, the two middle values are 6 and 8.

Step 5: Find the average of the two middle values to get the median

Since the number of observations is even, the median is the average of the two middle values.

Thus, Median = (6 + 8) / 2

= 14 / 2

= 7

Therefore, the median is 7.

Frequently Asked Questions