Median of Ungrouped Data

The Median of Ungrouped Data is when you have numbers in a list and have to find the middle number. Usually, if there are an even number of numbers, the median is the average of the two in the middle.

Example of Ungrouped Data

The median of ungrouped data is calculated by finding which number falls at the exact centre point in an ordered set. This means that half (or exactly 50% or 0.5) of your data will be on one side and half on the other. Let's see in detail about median and median of ungrouped data with frequency

What is the Median Formula for Ungrouped Data?

Firstly, we have to arrange the given data in order to use the median formula.

• Step 1. Arrange the given values in 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 $\dfrac{(n+1)}{2}^{\text {th }}$ observation.

• Step 4. If $n$ is even, then the median is given by the mean of $\dfrac{n}{2}^{\text {th }}$ observation and $[\dfrac{n}{2}+1]^{\text {th }}$ observation.

Median Problems and Median Solutions

To calculate the Median Of Ungrouped Data, data needs to be put in order. Also, one must first group them and find the total number of items in a set. The next step is to add 1 and divide this total by two; the answer is the data point.

i.e., Median = Value of the (\[\frac{n+1}{2}\])\[^{th}\]

Solved Examples

Q 1 The heights (in cm) of 11 players of a team are as follows: 173, 158, 158, 159, 160, 160, 165, 166, 171, 170, 163.

Ans: On arranging the variates in ascending order, we get 158, 158, 159, 160, 160, 163, 165, 166, 170, 171, 173

There are 11 variates, and the number is odd.

Therefore, median = $\left[\dfrac{(11+1)}{2}\right]^{th}$ variate = 6th variate = 163.

Q 2 The median of 17, 13, 10, 15, x is an integer x then find x.

Ans: There are five variates. So, $\left[\dfrac{(5+1)}{2}\right]^{th}$ variate, is 3rd variate when written in ascending order will be the median x.

On arranging the variates in ascending order, we get 10, 13, x, 15, 17.

Therefore, 13 < x < 15.

But x is an integer. So, x = 14.

Q 3 Find the median of the first ten even numbers

Ans: First ten odd integers = 2,4,6,8,10,12,14,16,18,20

The number of variates = 10

Since n is even, we take two numbers in the middle, add them, and then divide by 2.

So, 10+12=22. We get 11 by dividing 22 by 2. So, the median is 11.

Q 4 Find the median of 1,2,3,4,5.

Ans: n= 5

Since the number of variates is odd, we have to add 1 and divide by 2. Now, n=5, on adding 1, it is 6. On dividing 6 by 2, we get 3, which is the data point. 3 falls on the 3rd point. So, the median is 3.

Q 5 Find the median of the following: 16, 24, 8, 12, 19

Ans: On arranging the data in ascending order, we get 8, 12, 16, 19, 24

There are 5 variates; the number is odd.

Therefore, median = $\left[\dfrac{(5+1)}{2}\right]^{th}$ variate = 3rd

Median = 16

Practice Questions

Q1. Find out the median of the following ungrouped data

1,3,9,5,3,7,9,2,6,9,5,3

Ans: 5

Q2. Find the Median: 36, 44, 86, 31, 37, 44, 86, 35, 60, 51

Ans: 44

Q3. The median of observations 11, 12, 14, 18, x + 2, x + 4, 30, 32, 35, 41 arranged in ascending order is 24. Find the values of x.

Ans: x = 21

Q4. Find whether the data is grouped or Ungrouped. The ages of children at a music show are as follows: 10, 9, 10, 11, 12, 8, 8, 9, 9

Ans: Ungrouped data

Q5. State whether the following is True or False. A list of the heights of every student in a class would be an example of data.

Ans: True

Summary

The median of ungrouped data helps to find the middle number in an ordered list or the average of two numbers that are put together. For example, if you have 5 numbers and your data looks like this 5, 6, 7, 8, 10, then you would calculate your mean as 5 because it uses all of your data. Then take these values and divide them by two and one to get your median of 4.6 because it used half of this group and one-half from another group.