The median formula is used to find the middle value of a dataset arranged in ascending or descending order. For an odd number of observations, the median formula is Median = [(n+1)/2]th term, while for an even number of observations, Median = [(n/2)th term + ((n/2)+1)th term] ÷ 2.
Median is one of the important measures of central tendency in statistics that divides data into two equal parts. Students use the median formula in Class 9, Class 10, Class 11, Class 12 Maths, and data-related problems.
In this guide, you will learn the median formula for odd numbers, even numbers, grouped data, median class formula, and mean median mode relationship with simple explanations and solved examples.
What is Median in Maths?
Median is the middle value of a dataset when all observations are arranged in ascending or descending order.
It divides the complete data into two equal parts:
- 50% values are below the median
- 50% values are above the median
For example:
Data:
5, 8, 10, 15, 20
Here, the middle value is 10.
Therefore,
Median = 10
Median is commonly used in statistics to understand the central position of data.
Do Check: Reference Guide of Maths Formulas
Median Formula Table
Different types of data require different median formulas. The table below explains all important median formulas with their uses.
| Type of Data | Median Formula | When to Use |
|---|---|---|
| Odd Number of Observations | Median = [(n + 1) / 2]th term | When the total number of values is odd |
| Even Number of Observations | Median = [(n/2)th term + {(n/2)+1}th term] / 2 | When the total number of values is even |
| Grouped Data Median Formula | Median = L + [(N/2 − CF) / f] × h | Continuous frequency distribution |
| Median Position Formula | Position of Median = (n+1)/2 | To find the location of median value |
| Median Class Formula | N/2 | To identify median class |
| Mean Median Mode Formula | Mode = 3 Median − 2 Mean | For moderately skewed distribution |
Median Formula for Ungrouped Data
Ungrouped data means simple raw values that are not divided into groups or class intervals.
There are two cases:
- Odd number of observations
- Even number of observations
Before finding the median, always arrange the data in increasing order.
Also Check: Complete Class 10 Maths Formulas Chapterwise
Median Formula for Odd Number of Observations
When the total number of observations is odd, the median is the exact middle value.
Formula:
Median = [(n + 1) / 2]th observation
Where:
n = Total number of observations
Example:
Find the median:
12, 18, 8, 25, 15
Step 1: Arrange the data
8, 12, 15, 18, 25
Step 2: Count observations
n = 5
Step 3: Apply median formula
Median position:
= (5 + 1) / 2
= 6 / 2
= 3rd observation
The 3rd value is 15.
Answer: Median = 15
Median Formula for Even Number of Observations
When the number of observations is even, there are two middle values. The median is calculated by taking the average of these two values.
Formula:
Median = [(n/2)th value + {(n/2)+1}th value] / 2
Example:
Find the median:
10, 20, 30, 40, 50, 60
Step 1: Count observations
n = 6
Step 2: Find middle positions
First middle value:
n/2 = 6/2 = 3rd value
Second middle value:
(n/2)+1 = 4th value
Values:
3rd value = 30
4th value = 40
Step 3: Calculate median
Median:
= (30 + 40) / 2
= 70 / 2
= 35
Answer: Median = 35
Also Read: CBSE Class 12 Maths Formulas
Median Formula for Grouped Data
Grouped data is arranged in class intervals with frequencies.
The median formula for grouped data is:
Median = L + [(N/2 − CF) / f] × h
Where:
| Symbol | Meaning |
|---|---|
| L | Lower limit of median class |
| N | Total frequency |
| CF | Cumulative frequency before median class |
| f | Frequency of median class |
| h | Class interval size |
How to Find Median of Grouped Data?
Follow these simple steps:
- Find the total frequency (N)
- Calculate N/2
- Prepare the cumulative frequency column
- Find the median class
- Apply the grouped data median formula
Median Formula for Grouped Data Example
Find the median from the following frequency table:
| Class Interval | Frequency | Cumulative Frequency |
|---|---|---|
| 0-10 | 5 | 5 |
| 10-20 | 8 | 13 |
| 20-30 | 12 | 25 |
| 30-40 | 10 | 35 |
| 40-50 | 5 | 40 |
Total frequency:
N = 40
Find:
N/2 = 40/2 = 20
The cumulative frequency just greater than 20 is 25.
Therefore,
Median class=20-30
Values:
- L = 20
- CF = 13
- f = 12
- h = 10
Apply formula:
Median = 20 + [(20 − 13)/12] × 10
Median = 20 + (7/12 × 10)
Median = 20 + 5.83
Median = 25.83
Mean Median Mode Relationship Formula
Mean, median, and mode are three important measures of central tendency.
For a moderately skewed distribution:
Mode = 3 Median − 2 Mean
The formula can also be rearranged:
Median Formula Using Mean and Mode
Median = (2 Mean + Mode) / 3
Mean Formula Using Median and Mode
Mean = (Mode + 2 Median) / 3
Difference Between Mean, Median and Mode
| Point | Mean | Median | Mode |
|---|---|---|---|
| Meaning | Average value | Middle value | Most repeated value |
| Formula | Sum of values ÷ Number of values | Depends on number of observations | Value with highest frequency |
| Effect of extreme values | More affected | Less affected | Not affected much |
| Best Used For | Normal data | Skewed data | Repeated values |
Median Formula Revision
| Data Type | Formula |
|---|---|
| Odd observations | Median = [(n+1)/2]th value |
| Even observations | Median = Average of two middle values |
| Grouped data | Median = L + [(N/2−CF)/f] × h |
| Median position | (n+1)/2 |
| Mean median mode relation | Mode = 3 Median − 2 Mean |
Applications of Median Formula
The median formula is useful in many areas such as:
1. Mathematics and Statistics
Students use median to solve statistics questions in school exams.
2. Data Analysis
Median helps understand large datasets.
3. Business Research
Companies use median values to study:
- Salary data
- Customer behaviour
- Market trends
4. Competitive Exams
Median questions are commonly asked in:
- School exams
- Aptitude tests
- Entrance exams
Important Points to Remember About Median
- Arrange the data before calculating the median.
- Median represents the central value of data.
- Median is a measure of central tendency.
- Median is less affected by very high or very low values.
- For grouped data, always identify the correct median class first.
- Mean, median, and mode are equal in a perfectly symmetrical distribution.
Conclusion
The median formula is an important concept in statistics that helps students find the middle value of a dataset. For simple data, the formula changes depending on whether the number of observations is odd or even. For grouped data, students use the formula Median = L + [(N/2 − CF)/f] × h. Understanding median, mean, and mode formulas helps in solving maths problems, analysing data, and preparing for exams effectively.