Median

The median is one of the three main measures used to describe the middle value of the data set. When we examine a group of numbers, we often need to know that the value lies at the center, and this idea is known as a measure of central tendency. The three most common measures are mean, median, and mode.

The idea of the median originated in the 17th century, when Joseph Boscovich used it in 1760 to analyze data with errors. In the 18th century, Francis Galton was officially introduced the word “median.” Since then, the median has become a meaningful way to find the middle value in data, especially when the numbers are uneven. Today, it is widely used in statistics, economics, social studies, and finance.

For example, a child scores the following marks in five tests: 30, 45, 50, 70, 95. What is the median of these scores?

Answer:

The numbers are already in ascending order: 30, 40, 50, 70, 95.

The middle value in this set is 50.

Median = 50

Understanding the median helps to identify the middle value in the dataset. It offers a clear picture of central tendency, even with skewed numbers. To find the median of a list of numbers, follow these easy steps:

Step 1: Arrange the numbers in order. Write the numbers from ascending to descending order.

Step 2: Count how many numbers it has. This helps you to know whether the total is odd or even.

Step 3: If the number of values is odd. The median is the middle number.

Use the formula:

\(\text{Median} = \left( \frac{n + 1}{2} \right)^\text{th} \text{ value}\)

Step 4: If the number of values is even

The median is the average of the two middle numbers.

Use the formula:

\(\text{Median} = \frac{\text{Value at } \frac{n}{2} \text{ position} + \text{Value at } \left( \frac{n}{2} + 1 \right) \text{ position}}{2}\)

Median works for both ungrouped lists and class-interval data. Ungrouped data uses the middle number, but grouped data requires a frequency-based formula.

Grouped Data: In grouped data, the information is arranged in class intervals with their frequencies and cumulative frequencies. The median is then found using this Formula:
 

Median Formula for Grouped Data

Median = \(l + \frac{\left(\frac{n}{2} - \text{cf}\right)}{f} \times h\)

Where:

l = lower limit of the median class

n = total number of observations

f = frequency of the median class

h = class size (class width)

cf = cumulative frequency of the class just before the median class

For example, find the median of the following data:
 

Solution: 
Step 1: First, find the cumulative frequency

 

Step 2: Total observations 
            n = 50

Step 3: Find \(\frac{n}{2} = \frac{50}{2} = 25\)

Locate the class where CF ≥ 25 → Median Class = 20–30

Step 4: Apply the Formula

l = 20
f = 14
cf = 14 (CF before median class)
h = 10

Median = \(20 + \left( \frac{25 - 14}{14} \right) \times 10\)
 =\(20 + \left( \frac{11}{14} \right) \times 10\)


\(= 20 + 7.86 = 27.86\)

Median ≈ 27.86.

Ungrouped Data: In ungrouped data, the information is listed as individual values rather than the class intervals. To find the median, the data is first arranged in ascending order, and then the median formula is applied depending on whether the number of observations (n) is odd or even.

Median of Ungrouped Data: Ungrouped data means that the numbers are listed individually, without grouping them into intervals. The way we find the median depends on whether n is odd or even.

Median Formula When n Is Odd

If the number of values is odd:

Median = \(\left( \frac{n + 1}{2} \right)^{\text{th}} \text{ value}\)


Example (Odd Number of Values)

Find the median of
12, 18, 25, 30, 45

Arrange in order → already sorted.

Number of values = 5 (odd)

Median = \(\frac{5 + 1}{2}\) = 3rd value

The 3rd value is 25

 Median = 25


Median Formula When n Is Even

If the number of values is even:

Median =\(\left( \frac{n + 1}{2} \right)^{\text{th}} \text{ value}\)

Example (Even Number of Values)

Find the median of:
\(10, 15, 20, 30, 40, 55\)

Arrange in order is already sorted.

Number of values = 6 (even)

Middle values = 3rd (20) and 4th (30)

Median = \(\frac{20 + 30}{2}\)

Median = 25

Understanding the differences among mean, median, and mode helps identify how the data behaves and which value best represents a given dataset. These three measures explain the average, the median, and the mode, making data interpretation easier.

 

Sometimes students get confused with the concept of median and the best way to cope with that confusion is to follow some tips and tricks, here are some of the tips and tricks mentioned below:

1. Formulas: If the student practices the formulas, it will help the student in understanding the concept of median. They should be able to differentiate between the odd and even datasets.

2. Handling Duplicates: Sometimes students can get confused with the number of duplicate values and forget to take the value that has been repeated. The median is calculated the same way, and the duplicates are included.
 

3. Large data sets: For large datasets, students can take the help of a software or a calculator to find the median, as it is the most effective way to find a median.

 

4. Arrange Data First: Always sort the data in ascending or descending order before finding the median.

 

5. Odd vs Even Reminder: Odd set → middle value, Even set → average of two middle values.
 

6. Create the Simple Practice Activities: You can give children tasks like sorting numbers, finding the middle value, or interpreting simple data. Regular practice may help them understand the median.

7. Encourage the Use of Online Tools: Allow students to verify their answers using a calculator for mean, median, and mode. It helps to boost both accuracy and confidence.
 

8. Compare Mean, Median, and Mode: Ask children to calculate all three for the same dataset. This helps them to understand the mean, median, and mode, and highlights how the median differs from the average.
 

9. Start With Simple, Ordered Lists: Give children a few sets to arrange in increasing order. Finding the middle value will help them grasp the idea of the median quickly.

10. Practice With an Even Number Sets: Explain how to take the average of the two middle numbers when the list has an even count. This clearly reinforces the distinction between median and average.

We use the concept of median in various fields and applications. Let us now see how median is used as a real world application.


Economics and Income: We use the median to calculate the income distribution of a typical population. The median gives a more accurate representation of what most people earn. It also helps us understand about the poverty line.


Real Estate: We use median in real estate to find out the housing prices of different datasets of houses.


Healthcare: In healthcare, we use the concept of median to understand the patient data and help us study about the different types of illness and help the doctors navigate through the data.


Education: We use the concept of median to calculate the average of the scores in a class of students by determining who scores more and who scores less.


Surveys & Polls: Median responses are used to reflect the central opinion when data has outliers.