Most of you have learned about the median formula in your maths classes from the chapter, measures of central tendency. We will talk about it in detail and learn some of the important properties of the median formula**.** But before that, let us understand the significance of central tendency. Mathematically, the central tendency may be defined as the tendency of a given set of observations to gather or cluster around a single middle or central value, and the single value that represents a given set of observations is described as a measure of central tendency. It plays an important role in our everyday life as it provides a basis for comparison between different distributions. In this article, we will discuss specifically the median, median formula, and various properties of the median.

## What is a Median?

Median can be defined as the middle-most value when the observations are arranged either in descending order or ascending order of magnitude. It can be called a positional average which means that the value of the median is dependent upon the position of the given set of observations for which the median is desired. If you want to learn it live from the best Maths tutors, visit Cuemath. Online Math classes from Cuemath help you understand the concept in detail.

## Median Formula for Simple Frequency Distribution and Grouped Frequency Distribution

### Median Formula for Simple Frequency Distribution

It is very easy to calculate the median of a simple distribution.

For an odd number of observations, the median is the middlemost value that is arrived at after arranging an observation either in an ascending order or a descending order. Let us see an example: Marks of 5 students are 15, 10, 20, 18, and 30. We need to find the median mark.

Solution: Let us first arrange these observations in ascending order i.e., 10, 15, 18, 20, and 30. Since the third term that is 18 is the middlemost value, the median mark = 10.

For an even number of observations, the median is calculated by taking the arithmetic mean of the two middlemost values of the observations.

Here, Median = [ (n + 1) / 2] th observation.

Let us see an example. Marks of 6 students are 10, 25, 18, 16, 20, and 30. We need to find the median mark.

Solution: Let us first arrange the given observations in ascending order i.e., 10, 16, 18, 20, 25, 30. Now, 18 and 20 are the middle-most terms.

Thus, median = (18 + 20) / 2 = 19.

### Median Formula for Grouped Frequency Distribution

We find the median with the help of the cumulative frequency distribution of the variable under consideration in the case of a grouped frequency distribution. The formula is given below:

M = l_{1 }+ [(N/2 – N_{1}) / (N_{u }– N_{1})] × C

Where,

I_{1 }= lower class boundary of the median class

N = total frequency

N_{1 }= less than cumulative frequency corresponding to l_{1}

N_{u }= less than cumulative frequency corresponding to l_{2}

l_{2 }is the upper-class boundary of the median class

C = length of the median class

## Properties of Median

- If x and y are two variables, to be related by y = a + bx for any two constants a and b, then the median of y is given by

Median of y = a + b × (Median of x)

- For a given set of observations, the sum of absolute deviations is minimum when the deviations are taken from the median.