Median is defined as the middle term of the given set of data if the data is arranged either in ascending or descending order.
Example: Suppose we have the height of 5 friends as 167 cm, 169 cm, 171 cm, 174 cm, 179 cm. Now clearly observing the data we see that 171 cm is the middle term in the given data thus, we can say that the median height of the friends is, 171 cm.
In this article, we will learn about the median, its formula for grouped and ungrouped data, examples of the median, and others in detail.
The median is the middle value of the dataset when arranged in ascending or descending order. If the dataset has an odd number of values, the median is the middle value. If the dataset has an even number of values, the median is the average of the two middle values.
A median divides the data into two halves. Median is among one of the three measures of central tendency and finding the median gives us very useful insight into the given set of data.
The three measures of the central tendency are,
Various examples of the median are:
Example 1: Median salary of five friends, where the individual salary of each friend is,
- 74,000,
- 82,000,
- 75,000,
- 96,000, and
- 88,000.
First arranged in ascending order 74,000, 75,000, 82,000, 88,000, and 96,000 then by observing the data we get the median salary as 82,000.
Example 2: Median Age of a Group- Consider a group of people's ages:
25, 30, 27, 22, 35, and 40.
First, arrange the ages in ascending order: 22, 25, 27, 30, 35, 40. The median age is the middle value, which is 30 in this case.
As we know median is the middle term of any data, and finding the middle term when the data is linearly arranged is very easy, the method of calculating the median varies when the given number of data is even or odd.
For example,
- If we have 3 (odd-numbered) data 1, 2, and 3 then 2 is the middle term as it has one number to its left and one number to its right. So finding the middle term is quite simple.
- But when we are given with even number of data (say 4 data sets), 1, 2, 3, and 4, then finding the median is quite tricky as by observing we can see that there is no single middle term then for finding the median we use a different concept.
Here, we will learn about the median of grouped and ungrouped data in detail.
The median formula is calculated by two methods,
- Median Formula (when n is Odd)
- Median Formula (when n is Even)
Now let's learn about these formulas in detail.
If the number of values (n value) in the data set is odd then the formula to calculate the median is,
If the number of values (n value) in the data set is even then the formula to calculate the median is:
Grouped data is the data where the class interval frequency and cumulative frequency of the data are given. The median of the grouped data median is calculated using the formula,
Median = l + [(n/2 – cf) / f]×h
where,
- l is the Lower Limit of the Median Class
- n is the Number of Observations
- f is Frequency of Median Class
- h is Class Size
- cf is the Cumulative Frequency of Class Preceding Median Class
We can understand the use of the formula by studying the example discussed below,
Example: Find the Median of the following data,
If the marks scored by the students in a class test out of 50 are,
| Marks | 0-10 | 10-20 | 20-30 | 30-40 | 40-50 |
|---|
| Number of Students | 5 | 8 | 6 | 6 | 5 |
|---|
Solution:
For finding the Median we have to build a table with cumulative frequency as,
| Marks | 0-10 | 10-20 | 20-30 | 30-40 | 40-50 |
|---|
| Number of Students | 5 | 8 | 6 | 6 | 5 |
|---|
| Cumulative Frequency | 0+5 = 5 | 5+8 = 13 | 13+6 = 19 | 19+6 = 25 | 25+5 = 30 |
|---|
n = ∑fi = 5+8+6+6+5 = 30(even)
n/2 = 30/2 = 15
Median Class = 20-30
Now using the formula,
Median = l + [(n/2 – cf) / f]×h
Comparing with the given data we get,
- l = 20
- n = 30
- f = 6
- h = 10
- cf = 13
Median = 20 + [(15 - 13)/6] × 10
= 20 + (2/6) x 10
= 60/3 + 10/3
= 20 + 3.3333 = 23.33 (approx)
Thus, the median mark of the class test is 23.33
To find the median of the data we can use the steps discussed below,
Step 1: Arrange the given data in ascending or descending order.
Step 2: Count the number of data values(n)
Step 3: Use the formula to find the median if n is even, or the median formula when n is odd, accordingly based on the value of n from step 2.
Step 4: Simplify to get the required median.
Study the following example to get an idea about the steps used.
Example: Find the median of given data set 30, 40, 10, 20, and 50
Solution:
Median of the data 30, 40, 10, 20, and 50 is,
Step 1: Order the given data in ascending order as:
10, 20, 30, 40, 50
Step 2: Check if n (number of terms of data set) is even or odd and find the median of the data with respective ‘n’ value.
Step 3: Here, n = 5 (odd)
Median = [(n + 1)/2]th term
Median = [(5 + 1)/2]th term = 33r term = 30
Thus, the median is 30.
The median formula has various applications, this can be understood with the following example, in a cricket match the scores of the five batsmen A, B C, D, and E are 29, 78, 11, 98, and 65 then the median run of the five batsmen is,
First arrange the run in ascending order as, 11, 29, 65, 78, and 98. Now by observing we can clearly see that the middle term is 65. thus the median run score is 65.
For two numbers finding the middle term is a bit tricky as for two numbers there is no middle term, so we find the median as we find the mean by adding them and then dividing it by two. Thus, we can say that the median of the two numbers is the same as the mean of the two numbers. Thus, the median of the two numbers a and b is,
Median = (a + b)/2
Now let's understand this using an example, find the median of the following 23 and 27
Solution:
Median = (23 + 27)/2
Median = 50/2
Median = 25
Thus, median of 23 and 27 is 25.
Read More,
Example 1: Find the median of the given data set 60, 70, 10, 30, and 50
Solution:
Median of the data 60, 70, 10, 30, and 50 is,
Step 1: Order the given data in ascending order as:
10, 30, 50, 60, 70
Step 2: Check if n (number of terms of data set) is even or odd and find the median of the data with respective ‘n’ value.
Step 3: Here, n = 5 (odd)
Median = [(n + 1)/2]th term
Median = [(5 + 1)/2]th term = 3rd term
= 50
Example 2: Find the median of the given data set 13, 47, 19, 25, 75, 66, and 50
Solution:
Median of the data 13, 47, 19, 25, 75, 66, and 50 is,
Step 1: Order the given data in ascending order as:
13, 19, 25, 47, 50, 66, 75
Step 2: Check if n (number of terms of data set) is even or odd and find the median of the data with respective ‘n’ value.
Step 3: Here, n = 7 (odd)
Median = [(n + 1)/2]th term
Median = [(7 + 1)/2]th term = 4th term
= 47
Example 3: Find the Median of the following data,
If the marks scored by the students in a class test out of 100 are,
| Marks | 0-20 | 20-40 | 40-60 | 60-80 | 80-100 |
|---|
| Number of Students | 5 | 7 | 9 | 4 | 5 |
|---|
Solution:
For finding the Median we have to build a table with cumulative frequency as,
| Marks | 0-20 | 20-40 | 40-60 | 60-80 | 80-100 |
|---|
| Number of Students | 5 | 7 | 9 | 4 | 5 |
|---|
| Cumulative Frequency | 0+5 = 5 | 5+7 = 12 | 12+9 = 21 | 21+4 = 25 | 25+5 = 30 |
|---|
n = ∑fi = 5+7+9+4+5 = 30(even)
n/2 = 30/2 = 15
Median Class = 40-60
Now using the formula,
Median = l + [(n/2 – cf) / f]×h
Comparing with the given data we get,
- l = 40
- n = 30
- f = 9
- h = 10
- cf = 12
Median = 20 + [(15 - 12)/6]×10
= 40 - (3/9) x 20
= 40 +6.6667
= 46.6667
Thus, the median mark of the class test is 46.67.
Example 4: Find the median number of hours studied per week
The following table shows the distribution of the number of hours spent studying per week by a group of students:
Hours Studied (Per week) | 0 - 5 | 5 - 10 | 10 - 15 | 15 - 20 | 20 - 25 |
|---|
Frequency | 8 | 15 | 25 | 12 | 10 |
|---|
Solution:
For finding the Median we have to build a table with cumulative frequency as,
Hours Studied (Per week) | 0 - 5 | 5 - 10 | 10 - 15 | k)15 - 20 | 20 - 25 |
|---|
Frequency | 8 | 15 | 25 | 12 | 10 |
|---|
Cumulative Frequency | 0 + 8 = 8 | 8 + 15 = 23 | 23 + 25 = 48 | 48 + 12 = 60 | 60 + 10 = 70 |
|---|
n = ∑fi = 8 + 15 + 25 + 12 + 10 = 70(even)
n/2 = 70/2 = 35
Median Class = 10 - 15
Now using the formula,
Median = l + [(n/2 – cf) / f]×h
Comparing with the given data we get,
- l = 10
- n = 70
- f = 25
- h = 5
- cf = 23
Median = 10 + [(35 - 23)/25]×5
= 10 - (12/15) x 5
= 10 - (0.48) x 5
= 10 + 2.4
= 12.4
Thus, the median number of hours per week is 12.4 hours.
Conclusion
The median is an important statistical measure that helps us find the middle value of a dataset. It is especially useful when the data contains extreme values, as it provides a better representation of the central tendency compared to the mean. Calculating the median is simple and offers an easy way to understand the distribution of values in a set.
Similar Reads
Statistics in Maths Statistics is the science of collecting, organizing, analyzing, and interpreting information to uncover patterns, trends, and insights. Statistics allows us to see the bigger picture and tackle real-world problems like measuring the popularity of a new product, predicting the weather, or tracking he
3 min read
Introduction to Statistics
Mean
Mean in StatisticsIn statistics, three measures are defined as central tendencies that are: Mean, Median, and Mode, where the mean provides the average value of the dataset, the median provides the central value of the dataset, and the most frequent value in the dataset is the mode.Calculation of central tendency, su
15+ min read
How to find Mean of grouped data by direct method?Statistics involves gathering, organizing, analyzing, interpreting, and presenting data to form opinions and make decisions. Applications range from educators computing average student scores and government officials conducting censuses to demographic analysis. Understanding and utilizing statistica
8 min read
How to Calculate Mean using Step Deviation Method?Step Deviation Method is a simplified way to calculate the mean of a grouped frequency distribution, especially when the class intervals are uniform. In simple words, statistics implies the process of gathering, sorting, examining, interpreting and then understandably presenting the data to enable o
6 min read
Median
Mode
Frequency Distribution - Table, Graphs, Formula A frequency distribution is a way to organize data and see how often each value appears. It shows how many times each value or range of values occurs in a dataset. This helps us understand patterns, like which values are common and which are rare. Frequency distributions are often shown in tables or
11 min read
Cumulative frequency Curve In statistics, graph plays an important role. With the help of these graphs, we can easily understand the data. So in this article, we will learn how to represent the cumulative frequency distribution graphically.Cumulative FrequencyThe frequency is the number of times the event occurs in the given
9 min read
Mean Deviation The mean deviation (also known as Mean Absolute Deviation, or MAD) of the data set is the value that tells us how far each data is from the center point of the data set. The center point of the data set can be the Mean, Median, or Mode. Thus, the mean of the deviation is the average of the absolute
12 min read
Standard Deviation - Formula, Examples & How to Calculate Standard deviation is a statistical measure that describes how much variation or dispersion there is in a set of data points. It helps us understand how spread out the values in a dataset are compared to the mean (average). A higher standard deviation means the data points are more spread out, while
15+ min read
Variance Variance is a number that tells us how spread out the values in a data set are from the mean (average). It shows whether the numbers are close to the average or far away from it.If the variance is small, it means most numbers are close to the mean. If the variance is large, it means the numbers are
12 min read