Class 7 - Median: Formula, Properties, and Solved Examples

Median is an important concept in Class 7 mathematics under data handling that helps us find the middle value of a dataset when the numbers are arranged in order. It's one of the three measures of central tendency alongside the mean and the mode. Median is a useful measure of central tendency, especially when dealing with data that contains extreme values. In this guide, you will learn the definition of median and the method to calculate it for both odd and even sets of data, along with step-by-step solved examples for quick and easy understanding of the concept.

Table of Contents

• What Is Median?
• Median Formula
• How to Find the Median
• Properties of Median
• Solved Examples on Median
• Median vs Mean vs Mode
• Advantages and Limitations of Median
• Practice Questions on Median

What Is Median?

The median is the middle number when the values of a numerical data set are written in numerical order. When a data set has an even number of values, the median is the mean of the two middle values. 'Median' refers to the middle value of the observations when arranged in ascending or descending order. It divides the data into two equal halves; exactly half the values lie below it, and exactly half lie above it. The median is also called the positional average or place average, because its value is determined entirely by the position of the middle term, not by the actual numerical values of all the data points.

Median Formula

The formula for finding the median depends on whether the count of observations is odd or even.

For a set with an odd number of values,
Median =  (n+12)th term

For a set with an even number of values,
Median = average of the (n/2)th and (n/2 + 1)th terms
Median =  (n2)thterm+(n2+1)thterm2

How to Find the Median

Case 1: Odd Number of Observations

Example: Find the median of 17, 8, 25, 4, 13, 32, 21
Step 1: Arrange in ascending order: 4, 8, 13, 17, 21, 25, 32
Step 2: Count observations: n = 7 (odd)
Step 3: Apply the formula: Median = (n+12)thterm
Median = [(7 + 1) / 2]th term = 4th term
Step 4: Identify the 4th term: The fourth term in 4, 8, 13, 17, 21, 25, 32 is 17
Therefore, Median = 17

Case 2: Even Number of Observations

Example: Find the median of 22, 35, 14, 48, 9, 27, 41, 56
Step 1: Arrange in ascending order: 9, 14, 22, 27, 35, 41, 48, 56
Step 2: Count observations: n = 8 (even)
Step 3: Apply the formula: Median = average of the (n/2)th and (n/2 + 1)th terms
Median = average of (8/2)th and (8/2 + 1)th terms
= average of 4th and 5th terms
Step 4: Identify the 4th and 5th terms: The  4th and 5th terms in 9, 14, 22, 27, 35, 41, 48, 56 are 27 and 35
Median = (27 + 35) / 2 = 62 / 2 = 31
Therefore, Median = 31

Properties of Median

• The median is not affected by extreme values (outliers).

• The median divides data into exactly two equal halves. By definition, 50% of observations lie below the median and 50% lie above it.

• The median can be determined graphically using an ogive (cumulative frequency curve).

• The median is uniquely defined. Every dataset has exactly one median.

Solved Examples on Median

Example 1: The tail lengths of Asian grass lizards were measured to the nearest centimetre. The results in centimetres were as follows: 25, 19, 22, 24, 21, 20, 19, 17, 18, 25, 23,21. Find the median of the given data.
Solution: Arrange the given data in ascending order.
17, 18, 19, 19, 20, 21, 21, 22, 23, 24, 25, 25
The number of values is 12, which is an even number.
Median = average of the (n/2)th and (n/2 + 1)th terms
Median = average of (12/2)th and (12/2 + 1)th terms
= average of 6th and 7th terms
= (21 + 21)/2 = 21
Therefore, the median is 21.

Example 2: The data in ascending order is 3, 7, 11, x, 22, 28. The median is 16. Find x.
Solution: The data in ascending order is 3, 7, 11, x, 22, 28. 
Given median = 16
The number of values is 6, which is an even number.
Median = average of the (n/2)th and (n/2 + 1)th terms
Median = average of (6/2)th and (6/2 + 1)th terms
= average of 3rd and 4th terms
16 = (11+x)/2
32 = 11 + x
x = 21
Therefore, the median is 21.

Example 3: Find the median of 11, 5, 19, 3, 15, 8, 22.
Solution: The data in ascending order is 3, 5, 8, 11, 15, 19, 22
n = 7,
Median = [(7 + 1) / 2]th term = 4th term
Median = 4th term = 11

Example 4: Every 30 minutes, workers at an amusement park counted the tickets sold. The following tickets were sold: 125, 142, 99, 79, 133,114, 92, 89 and 94. What was the median number of tickets sold?
Solution: The data in ascending order are 79, 89, 92, 94, 99, 114, 125, 133, 142
n = 9
Median = [(n + 1) / 2]th term = [(9 + 1) / 2]th term = 5th term
The 5th value in the ordered list is 99
The median number of tickets sold is 99.

Example 5: The data of the prices of different cereals at a store are ₹23, ₹25, ₹36, ₹65, ₹52, ₹56, ₹71, ₹38, ₹67, ₹78, ₹45, ₹56. Find the median.
Solution: The data in ascending order is ₹23, ₹25, ₹36, ₹38, ₹45, ₹52, ₹56, ₹56, ₹65, ₹67, ₹71, ₹78
n = 12
Median = average of the (n/2)th and (n/2 + 1)th terms
Median = average of (12/2)th and (12/2 + 1)th terms
= average of 6th and 7th terms
= (52 + 56)/2 = 54
The median price is ₹54.

 

Median vs Mean vs Mode

Median, mean, and mode are measures of central tendency used to represent a dataset with a single value. While the mean gives the average, the median shows the middle value, and the mode identifies the most frequently occurring number.

Advantages and Limitations of Median

Advantages of Median :

• Not influenced by extreme values or outliers

• Easy to understand conceptually. 'Median' simply means the middle value.

• Can be calculated even for open-ended class intervals (e.g., "above 100"), unlike the mean.

• Works well for both quantitative and qualitative ordinal data (e.g., ranking customers as "satisfied", "neutral", or "dissatisfied").

• It can be located graphically through an ogive.

Limitations of Median:

• It doesn't use all values in the dataset. Only the position matters. 

• For large datasets, arranging data in order takes time.

• It's not suitable for further algebraic manipulations.

Practice Questions on Median

1. Every 30 minutes, workers at an amusement park counted the tickets sold. The following tickets were sold: 125, 142, 99, 79, 133, 114, 92, 89 and 94. What was the median number of tickets sold?

2. The weights (kg) of the girls in a class are as follows: 25, 34, 30, 40, 29, 36, 44, 36, 42, 31, 29, 35, 43, 24. Find the median.

3. Find the median of the given data. 7, 3, 5, 7, 9, 11, 3, 8, 5, 44, 40, 55, 48, 44, 58, 67

4. Find the median for the following data: 12, 72, 3, 18, 22, 33, 4, 34, 16, 41, 53

5. Find the median of 6, 18, 12, 24, 9, 15.

6. Eight values arranged in ascending order are 4, 9, 13, x, x + 4, 28, 35, and 41. The median is 22. Find x.