 # How to Find Median for Grouped and Ungrouped Data

Statistics refers to the collection, analysis, interpretation and presentation of masses of numerical data. Captain John Graunt of London is known as the father of vital statistics due to his studies on statistics of births and deaths. Median is an important topic in statistics. The median is a measure of central tendency, which denotes the value of the middle-most observation in the data. In this article, we will discuss how to find the median for grouped and ungrouped data.

## What is Median?

Median is the most middle value in a set of data.

## How to calculate Median step by step?

You can use the following steps to calculate the median.

For ungrouped data:

Step 1. Arrange the given values in the ascending order.

Step 2. Find the number of observations in the given set of data. It is denoted by n.

Step 3. If n is odd, the median equals the [(n+1)/2]th observation.

Step 4. If n is even, then the median is given by the mean of (n/2)th observation and [(n/2)+1]th observation.

For grouped data:

Step 1. Make a table with 3 columns. First column for the class interval, second column for frequency, f, and the third column for cumulative frequency, cf.

Step 2. Write the class intervals and the corresponding frequency in the respective columns.

Step 3. Write the cumulative frequency in the column cf. It is done by adding the frequency in each step.

Step 4. Find the sum of frequencies, ∑f. It will be the same as the last number in the cumulative frequency column.

Step 5. Find n/2. Then find the class whose cumulative frequency is greater than and nearest to n/2. This is the median class.

Step 6. Now we use the formula Median =$l+\left ( \frac{\frac{n}{2}-cf}{f} \right )\times h$

l = lower limit of median class

n = no. of observations

cf denotes cumulative frequency of the class preceding the median class

f = frequency of median class

h = class size (assuming classes are of equal size)

### Formula

For ungrouped data:

Median = [(n+1)/2]th observation, if n is odd.

Median = mean of (n/2)th observation and [(n/2)+1]th observation, if n is even.

For grouped data:

Median =$l+\left ( \frac{\frac{n}{2}-cf}{f} \right )\times h$

### Example

The median of a set of 9 distinct observations is 20.5. If each of the largest 4 observations of the set is increased by 2, then the median of the new set

(A) Is increased by 2

(B) Is decreased by 2

(C) Is two times the original median

(D) Remains the same as that of the original set

Solution:

Given n = 9

Median = 20.5

Median term = [(n+1)/2]th term

= [(9+1)/2]th term

= 10/2)th term

= 5th term

Given the largest 4 observations are increased by 2. Since the median is the 5th term, there will be no change in it.

Hence option (D) is the answer.