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Median of Even Numbers – Procedure & Examples


The median is a positional average that tells us the value of the middlemost observation among all of the given data values. Since the median lies exactly in the middle exactly 50% of observations lie above the median and the remaining 50% of observations lie below it.

In this article, we explain how to find the median when we are given an even number of data values in raw form.

How to calculate the median of an even number of observations?

  1. Arrange the data in either ascending or descending order.
  2. Count the total number of terms. Suppose that there are a total of ‘N’ data values. Obviously, we assume that N is even since there are an even number of observations.
  3. Find the two middlemost values. They are the (N/2)th term and the (N/2 + 1)th term.
  4. Take the average of the above two values, that is, add the two numbers and divide by 2. The resulting number is the median of the given data values.

Formula for Finding Median of Even Numbers:

Suppose we are given ‘N’ number of observations (where N is even) arranged in ascending or descending order. Then the median of the observations can be calculated using the formula, \text{Median } = \frac{1}{2}\left[ \left(\frac{N}{2}\right)^{th}\text{ term } + \left(\frac{N}{2}+ 1 \right)^{th}\text{ term }\right]

From the above formula, it is clear that the median can be easily obtained by simply taking the average of the two middlemost terms. We now give some examples illustrating how we can use the above formula to calculate the median.

Example 1:

Suppose we are given the following data values: 2, 4, 18, 21, 11, 9.

Step 1: We first arrange the data in ascending order as follows: 2, 4, 9, 11, 18, 21

Step 2: We count the total number of terms. Here, N=6 which is even.

Step 3: We apply the above formula to calculate the median,

\text{Median } = \frac{1}{2}\left[ \left(\frac{N}{2}\right)^{th}\text{ term } + \left(\frac{N}{2}+ 1 \right)^{th}\text{ term }\right] \text{Median } = \frac{1}{2}\left[ \left(\frac{6}{2}\right)^{th}\text{ term } + \left(\frac{6}{2}+ 1 \right)^{th}\text{ term }\right] \text{Median } = \frac{1}{2}[3^{rd}\text{ term } + 4^{th}\text{ term }] = \frac{1}{2}[9+11] = 10.

So the median of the given data values is 10. We notice that three data values lie above 10 and three data values lie below 10. So we conclude that 10 is indeed the middlemost value.

Example 2:

Suppose we are given the following data values: 12, 13, 22, 25, 31, 49, 56, 58, 60, 64.

Since the data is already arranged in ascending order we can apply the formula directly. We are given N=10 number of terms.

\text{Median } = \frac{1}{2}\left[ \left(\frac{N}{2}\right)^{th}\text{ term } + \left(\frac{N}{2}+ 1 \right)^{th}\text{ term }\right] \text{Median } = \frac{1}{2}[5^{th}\text{ term } + 6^{th}\text{ term }] = \frac{1}{2}[31+49] = 80/2 = 40.

So the median of the given data values is 40.

Further Reading:

  1. Advantages, Disadvantages & Uses of Median in Statistics.
  2. How to find the Median of Odd Numbers?
  3. Median Absolute Deviation – Calculation, Formula, Examples.

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