# Median

A **median** is a measure of central tendency which divides data into $2$ parts, separating the upper and lower half of the data by a value which is called the median value. Before calculating a median, we need to arrange all the values in an ascending order. After that, we can calculate the median for different types of series in different ways:

**Individual Series:**Arrange all the observations in an ascending order. See whether the number of observations is an odd or even number. If the number is odd, use the formula $\left(\frac{N+1}{2}\right)^\text{th}$ observation, where $N$ is the number of observations. When the number of observations are even, the median is calculated by taking the average of the $\left(\frac{N}{2}\right)^{\text{th}}$ observation and the next observation.**Discrete Series:**Take the cumulative frequency for all the observations by successively adding the previous frequencies and then use the same formula as stated above in individual series.**Continuous Series:**The formula is

$l+\frac{\frac{N}{2}-cf}{f}\times h,$ where

- $l$ represents the lower limit of the median class interval;
- $N$ represents the total number of frequency;
- $cf$ represents the cumulative frequency of the class interval preceding the median class;
- $f$ represents the corresponding frequency of the median class interval;
- $h$ represents the width of the median class interval.

For more information on the types of statistical series, see Statistical Series.

## What is the median of the following distribution: $1,2,3,5,6,8,9?$

Since the number of observations is odd, use the formula $\left(\frac{N+1}{2}\right)^{\text{th}}$. Here $N=7$. Putting this in the formula, we get that the median is the $4^{\text{th}}$ observation which is $5$. So, the median is $5$. $_\square$

## What is the median of the following distribution: $1,2,3,5,6,8,9,11?$

Since the number of observations are even in number, we have to take the average of the $\left(\frac{N}{2}\right)^{\text{th}}$ observation and the next observation. Here, $N=8$. So, the average of the $4^{\text{th}}$ and $5^{\text{th}}$ observations is the median of this distribution. So, the median is $\frac{5+6}{2}=5.5. \ _\square$