How to Find the Median of Ungrouped Data Step by Step with Formula and Solved Examples
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.
FAQs on Median of Ungrouped Data Explained Clearly
1. What is the median of ungrouped data?
The median of ungrouped data is the middle value of a data set arranged in ascending or descending order. It divides the data into two equal halves, with 50% of values below it and 50% above it. Unlike the mean, the median is not affected much by extreme values (outliers). It is a measure of central tendency used for raw, individual observations.
2. What is the formula for the median of ungrouped data?
The formula for the median of ungrouped data depends on whether the number of observations (n) is odd or even.
- If n is odd: Median = value of the (n + 1)/2th observation.
- If n is even: Median = average of the n/2th and (n/2 + 1)th observations.
3. How do you find the median of ungrouped data step by step?
To find the median of ungrouped data, first arrange the data in order and then locate the middle value.
- Step 1: Arrange the data in ascending order.
- Step 2: Count the total number of observations (n).
- Step 3: If n is odd, use (n + 1)/2.
- Step 4: If n is even, take the average of the n/2 and (n/2 + 1) values.
4. How do you find the median when the number of observations is odd?
When the number of observations is odd, the median is the value at position (n + 1)/2. For example, in the data set 3, 5, 7, 9, 11 (n = 5):
- (5 + 1)/2 = 3
- The 3rd value is 7
5. How do you find the median when the number of observations is even?
When the number of observations is even, the median is the average of the n/2th and (n/2 + 1)th observations. For example, in 4, 6, 8, 10 (n = 4):
- n/2 = 2 and n/2 + 1 = 3
- Values are 6 and 8
- Median = (6 + 8)/2 = 7
6. Why is the median important in ungrouped data?
The median is important because it gives the central value of ungrouped data without being affected by extreme values. In data sets with outliers, the median provides a better measure of central tendency than the mean. It is commonly used in income statistics, test scores, and skewed distributions where the average may be misleading.
7. What is the difference between mean and median in ungrouped data?
The main difference is that the mean is the average of all values, while the median is the middle value of ordered data.
- Mean = (Sum of observations) / n
- Median = Middle value after arranging data
- The mean is affected by outliers, but the median is not significantly affected.
8. Can you give an example of finding the median of ungrouped data?
Yes, here is a simple example of finding the median of ungrouped data. Consider the data: 12, 7, 15, 10, 9.
- Arrange in ascending order: 7, 9, 10, 12, 15
- n = 5 (odd)
- Median position = (5 + 1)/2 = 3
- The 3rd value is 10
9. What are the properties of the median of ungrouped data?
The median of ungrouped data has several important properties.
- It divides the data into two equal halves.
- It is not much affected by extreme values.
- It can be determined for ordinal data.
- It does not depend on every value in the data set.
10. What are common mistakes when finding the median of ungrouped data?
Common mistakes when calculating the median of ungrouped data usually involve skipping key steps.
- Not arranging the data in ascending or descending order.
- Using the wrong formula for odd or even values of n.
- For even n, forgetting to take the average of the two middle numbers.
- Miscalculating the position using (n + 1)/2.
