How to Find the Median? 

For all sets of numbers, there is one number that holds the others in balance — extremely low or high values don't influence it and will often provide a more stable snapshot than average. This is called the Median, and it has significant importance in statistics. Working with test scores, daily temperatures, or survey answers, discovering the median can give you valuable insight into your data. Here, we will explore this important entity of statistics from formulas and examples to how to find the median value.

1.0What is Median?

In statistics, the median refers to the middle value of a data set when the values are arranged in “ascending order”. It is one of the central tendency measures (such values that are close to every value of the data set), along with the mean and the mode. The median is particularly useful since extreme values or outliers do not influence it, and hence, it is a better representative of data in such situations.  

2.0How to Find the Median of Data? 

How to find the median in statistics specifically depends upon the type of dataset to be analysed, which can be divided into two types of datasets: grouped and ungrouped datasets.  Now, let’s understand the methods involved in finding the median of grouped and ungrouped data:

How to Find the Median of Ungrouped Data? 

Ungrouped Data is a list of random/raw numbers or other data that are not arranged according to a certain pattern. Generally, this type of data is not arranged in a frequency distribution table. Here are the steps for you to find the median of ungrouped data: 

• Step 1: Put the data in order from smallest to largest (in increasing order).
• Step 2: Count the number of data values denoted by n.
• Step 3: Check whether n is odd or even:
• If n is odd, the median is the middle value.
• If n is even, the median is the average of the two middle values.

Formula to Find Median (Ungrouped Data): 

• For the odd number of values: Below is the formula to find the median if the number of values in a data set is odd, like 3, 5, 7,….

$\text{ Median }=\left(\frac{n+1}{2}\right)\text{ th value of the dataset }$

• For the even number of values: Here’s how to find the median of even numbers of values:

$\text{ Median }=\frac{\left(\frac{n}{2}\right)\text{ th term of the dataset }+\left(\frac{n}{2}+1\right)\text{ th term of the dataset }}{2}$

How to Find the Median for Grouped Data? 

Grouped data is data which has been categorised into class intervals along with the respective frequencies. As the precise values for each class interval are unknown, a formula is applied to estimate the median. 

Formula to Find Median for Grouped Data: 

In grouped data, the precise values for each class interval are unknown, which is why a formula is applied to estimate the median. This formula is: 

$\text{ Median }=L+\left(\frac{\frac{n}{2}-CF}{f}\right)\times h$

Here:

• L = Lower Limit of the median class
• n = Total frequency
• CF = Cumulative frequency before the median class
• f = Frequency of the median class
• h = Class width (width can be found using the formula: Upper Limit – Lower Limit)

Steps To Use the Formula: 

1. If not given already, create a frequency distribution table of the dataset with class intervals and their respective frequencies. 
2. Form a cumulative frequency table column for the data set like this: 
3. 1. Begin with the first frequency as the initial CF.
   2. Add each frequency to the prior CF to extend the column. For Example, if frequencies are 5, 8, 12 → CF becomes 5, 13, 25.

Note: The sum of all the frequencies of a dataset is always equal to the last cumulative frequency of the last class interval. 

3. Find the total frequency denoted either by n or $\sum f$. Then, calculate half of the total frequency, that is $\frac{n}{2}$. 
4. Identify the median class of the data. This is the lower limit of the class interval where the cumulative frequency is equal to or just exceeds the value of $\frac{n}{2}$. 
5. Substitute each value discussed above in the formula to find the median of grouped data.   

3.0Solved Example to Find Median

Problem 1: The ages (in years) of five children in a dance class are 7, 3, 9, 1, and 5. Find the median age.

Solution: To find the median, first, arrange the scores in ascending order: 

$\to 1,3,5,7,9$

Here, the number of values is odd, n = 5, so the formula to be used is: 

$\begin{array}{c}\text{ Median }=\left(\frac{n+1}{2}\right)\text{ th value of the dataset }\\ \text{ Median }=\left(\frac{5+1}{2}\right)\text{ th value }\\ \text{ Median }=3\text{ rd value }\end{array}$

This means that the median age of the five children in the class is 5.

Problem 2: A teacher records the scores of 4 students in a math quiz: 10, 2, 8, and 4. Find the median score. 

Solution: Firstly, arrange the score in ascending order: 

→ 2, 4, 8, 10

Here, n = 4 (even), hence the formula to find the median used will be: 

$\begin{array}{c}\text{ Median }=\frac{\left(\frac{n}{2}\right)\text{ th term of the dataset }+\left(\frac{n}{2}+1\right)\text{ th term of the dataset }}{2}\\ \text{ Median }=\frac{\left(\frac{4}{2}\right)\text{ th term }+\left(\frac{4}{2}+1\right)\text{ th term }}{2}\\ \text{ Median }=\frac{2nd\text{ term }+3\text{ rd term }}{2}\\ \text{ Median }=\frac{4+8}{2}=\frac{12}{2}\\ \text{ Median }=6\end{array}$

Problem 3: Find the median of the following table, which shows the marks obtained by students in a class:

Solution: 

Since n = 35, $\frac{n}{2}=17.5$

From the frequency distribution table, we can see that 25 is the closest value to 17.5. Hence, the median class is 20 – 30. 

L = 20, CF = 13, f = 12, h = 10

Using the formula, 

$\begin{array}{c}\text{ Median }=L+\left(\frac{\frac{n}{2}-CF}{f}\right)\times h\\ \text{ Median }=20+\left(\frac{17.5-13}{12}\right)\times 10\\ \text{ Median }=20+\left(\frac{4.5}{12}\right)\times 10=20+3.75\\ \text{ Median }=23.75\end{array}$