What is Median?
The median is the middle value in a dataset when arranged in ascending or descending order. It divides the dataset into two equal halves and is a measure of central tendency that is less affected by extreme values (outliers) compared to the mean.
Complete Median Formulas Table
| Formula Type | Formula | When to Use | Explanation | Example |
|---|---|---|---|---|
| Basic Median (Odd n) | Median = Value at position (n+1)/2 | When total observations (n) is odd | The middle value after arranging data in order | For 5, 7, 9, 12, 15: n=5, Position = (5+1)/2 = 3rd value = 9 |
| Basic Median (Even n) | Median = [Value at n/2 + Value at (n/2)+1]/2 | When total observations (n) is even | Average of two middle values | For 4, 6, 8, 10: n=4, Median = (6+8)/2 = 7 |
| Median for Grouped Data (Continuous) | Median = L + [(n/2 – CF)/f] × h | For continuous frequency distribution | L = Lower boundary of median class, CF = Cumulative frequency before median class, f = frequency of median class, h = class interval | See detailed example below |
| Median for Discrete Series | Median = Value corresponding to (n+1)/2th observation | For discrete data with frequencies | Find cumulative frequency and locate middle position | If n=20, find value at 10.5th position |
| Position Formula | Position of Median = (n+1)/2 | To find position before calculating median | Determines which observation is the median | For n=9, position = (9+1)/2 = 5th observation |
| Median Class Formula | n/2 | To identify median class in grouped data | Compare with cumulative frequencies to find median class | If n=40, find class containing 20th observation |
| Mean-Median-Mode Relationship | Mode = 3 Median – 2 Mean | For moderately skewed distribution | Empirical relationship between measures of central tendency | If Mean=10, Median=12, then Mode = 3(12) – 2(10) = 16 |
| Alternative Mean-Mode Formula | Mean = (Mode + 2 Median)/3 | Rearranged empirical formula | Calculate mean when mode and median are known | If Mode=8, Median=10, then Mean = (8 + 2×10)/3 = 9.33 |
| Median from Mode and Mean | Median = (2 Mean + Mode)/3 | Another rearrangement of empirical formula | Calculate median when mean and mode are known | If Mean=15, Mode=18, then Median = (2×15 + 18)/3 = 16 |
Detailed Explanations by Category
1. Ungrouped Data (Raw Data)
For Odd Number of Observations:
- Formula: Median = (n+1)/2th term
- Step 1: Arrange data in ascending order
- Step 2: Find position using (n+1)/2
- Step 3: The value at that position is the median
For Even Number of Observations:
- Formula: Median = [n/2th term + (n/2+1)th term]/2
- Step 1: Arrange data in ascending order
- Step 2: Find two middle positions
- Step 3: Take average of values at these positions
2. Grouped Data (Frequency Distribution)
For Continuous Series:
Median = L + [(n/2 - CF)/f] × h
Where:
- L = Lower boundary of median class
- n = Total number of observations
- CF = Cumulative frequency of class before median class
- f = Frequency of median class
- h = Class interval (width)Steps to Calculate:
- Find n/2
- Create cumulative frequency column
- Identify median class (where cumulative frequency ≥ n/2)
- Apply the formula
3. Relationship Formulas
The empirical relationship between mean, median, and mode:
- In normal distribution: Mean = Median = Mode
- In skewed distribution: Mode = 3 Median – 2 Mean
Practical Examples
Example 1: Ungrouped Data (Odd n)
Data: 12, 15, 18, 22, 25, 28, 30
- n = 7 (odd)
- Position = (7+1)/2 = 4th term
- Median = 22
Example 2: Ungrouped Data (Even n)
Data: 10, 14, 16, 20, 24, 28
- n = 6 (even)
- Middle positions: 3rd and 4th terms
- Values: 16 and 20
- Median = (16+20)/2 = 18
Example 3: Grouped Data
| Class | Frequency | Cumulative Frequency |
|---|---|---|
| 0-10 | 5 | 5 |
| 10-20 | 8 | 13 |
| 20-30 | 12 | 25 |
| 30-40 | 10 | 35 |
| 40-50 | 5 | 40 |
- n = 40, n/2 = 20
- Median class: 20-30 (CF = 25 ≥ 20)
- L = 20, CF = 13, f = 12, h = 10
- Median = 20 + [(20-13)/12] × 10 = 20 + 5.83 = 25.83
Points to Remember
- Always arrange data in order before finding median for ungrouped data
- Median is not affected by extreme values, making it useful for skewed distributions
- For grouped data, ensure you identify the correct median class
- The empirical relationship between mean, median, and mode works best for moderately skewed, unimodal distributions
- Median divides the dataset into two equal halves by frequency, not by value range
Common Applications
- Statistics Class 10-12: Basic median calculations and relationships
- Competitive Exams: Quick calculation techniques
- Research & Analysis: When dealing with skewed data or outliers
- Business Analytics: Salary analysis, market research
- Quality Control: Process monitoring and control charts
Formula
| Data Type | Quick Formula |
|---|---|
| Raw Data (Odd n) | (n+1)/2th value |
| Raw Data (Even n) | Average of n/2th and (n/2+1)th values |
| Grouped Data | L + [(n/2-CF)/f] × h |
| From Mean & Mode | (2×Mean + Mode)/3 |
| Position Only | (n+1)/2 |
This comprehensive guide provides all essential median formulas with clear explanations, practical examples, and step-by-step procedures for accurate calculations across different data types and academic levels.
Frequently Asked Questions about Median Formulas
Q. What is the median formula and how do you calculate it?
The median formula depends on whether you have an odd or even number of observations. For odd numbers, use: Median = (n+1)/2th term after arranging data in ascending order. For even numbers, use: Median = [n/2th term + (n/2+1)th term]/2. Simply arrange your data from smallest to largest, find the middle position, and identify the median value. For example, in the dataset {3, 7, 9, 12, 15}, n=5, so the median is the 3rd value = 9.
Q. How do you find the median of grouped data or frequency distribution?
For grouped or continuous data, use the formula: Median = L + [(n/2 – CF)/f] × h, where L is the lower boundary of the median class, n is the total frequency, CF is the cumulative frequency before the median class, f is the frequency of the median class, and h is the class width. First, calculate n/2, then find the class where the cumulative frequency equals or exceeds n/2. This is your median class. Apply the formula using the values from that class to get the exact median.
Q. What is the difference between mean, median, and mode formulas?
Mean is the average calculated by summing all values and dividing by the count (Mean = Σx/n). Median is the middle value when data is arranged in order. Mode is the most frequently occurring value. The key difference: mean is affected by extreme values, median represents the central position, and mode shows the most common value. For moderately skewed distributions, they relate through: Mode = 3 Median – 2 Mean. Use mean for symmetric data, median for skewed data, and mode for categorical data.
Q. How to calculate median when there are even numbers of observations?
When you have an even number of observations, the median is the average of the two middle values. Follow these steps: (1) Arrange data in ascending order, (2) Find the two middle positions using n/2 and (n/2)+1, (3) Add these two middle values and divide by 2. For example, with data {4, 8, 12, 16}, the middle positions are 2nd and 3rd values (8 and 12), so Median = (8+12)/2 = 10. This ensures the median truly represents the center of the dataset.
Q. What is the median class formula and how do you find the median class?
The median class is the class interval in grouped data that contains the median value. To find it: (1) Calculate n/2 (where n is the total frequency), (2) Create a cumulative frequency column, (3) The median class is where cumulative frequency first equals or exceeds n/2. Once identified, use the formula Median = L + [(n/2 – CF)/f] × h where all values come from the median class. For instance, if n=50, find the class where cumulative frequency ≥ 25.
Q. When should you use median instead of mean in statistics?
Use median instead of mean when: (1) Data has outliers or extreme values (like income data where few very high earners skew the average), (2) Data is skewed rather than normally distributed, (3) You want a value that represents the typical middle rather than mathematical average, (4) Dealing with ordinal data (ranked data), or (5) The data has open-ended classes (like “above 100”). For example, median household income is more meaningful than mean because billionaires don’t skew it. Median provides a better representation of central tendency for non-normal distributions.