# 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 \[ \frac{5+6}{2}=5.5. \ _\square\]

I have 3 distinct positive integers.

The sum of these three integers is either 5, 6 or 7.

The product of these three integers is either 8, 9 or 10.

What is the median of these three integers?

Inspiration.