Process to find the Median
The middle value of a set of numbers is called the median. It is an integral concept in analysing Statistical data. The calculation of the median value varies for an odd or even number of values. First of all, the list of numbers needs to be arranged in the order in an ascending sequence or a descending sequence. If there is an odd list of numbers, the midpoint of the list is the median. However, in the case of an even count of numbers, the two numbers in the middle of the list are considered. The process of calculating the median is a little complex. The examples will help the student to understand better.
What is Median?
The Median represents one of the three important measures of central tendency of grouped or ungrouped data. These three most commonly used central tendency is known as Mean (Arithmetic mean), Median and Mode. In order to describe a set of data, the central position of the data set is identified which is just known as the measurement of central tendency.
In an ascending or descending ordered series, Median is the positionally centre most value of a provided set of data (either grouped or ungrouped) or The median can be also be defined as the value present at the midpoint of a given set of data, not the midpoint of the values itself. It is the point from where half of the data entry is more and half of the data entry is less. For the calculation of the median, all the data is first written in ascending or descending order and then the centre most data point is identified and that will be our median.
How to find the Median of Ungrouped Data?
How to Find the Median with ODD Number of observations?
The formula of the median for a given set of numbers, having the odd number of observations can be expressed as such
Median = [(n+1)/2]th term
Let us understand with an Example
The age of seven participants of a golf tournament has been listed below. Find the median of this given set. {44, 39, 51, 63, 36, 57, 31}
Ordered set of data : {31, 36, 39, 44, 51, 57, 63}
As the count of all the observations is seven, i.e. n = 7
By the formula of Median = [(n+1)/2]th term
Median = [(7+1)/2] = 4th term = 44, hence the median is 44.
How to find the Median with an Even Number of Observations?
The formula of the median for a given set of numbers, having the even numbers of observations can be expressed as such
Median = [n/2]th term
Let us understand with an example
Suppose the set has an even count of numbers for example, 8,1, 3, 5, 22,17,12,13. It is a set of 8 numbers. The list when sorted in ascending order 1, 3, 5, 8, 12, 13, 17, 22.
8 and 12 are the two middle numbers here.
Therefore, adding 8 and 12 and dividing the result by 2 = (8 + 12) / 2
= 10
Here, 10 is the median of the given list of numbers. 84 199.
How to find the Median in Maths for Grouped Frequencies?
When we have grouped data, calculating the median becomes a little more complicated. Students must be careful during this calculation. Here we consider the following grouped data table for a set of balls,
Here, we find out the class interval that has the maximum frequency, 61 - 65.
Now, we need to find the midpoint of this interval. Using the formula,
Estimated Median = L+
[(n/2)−C]
[(n/2)−C / F ]*W
Where,
L is the lower class boundary of the group that contains the median.
n is the total number of values in the interval.
B is the cumulative frequency of all the groups before the median group.
F is the frequency of the group containing the median.
W is the width of each group.
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Solved Examples
1. How to find the median of the following set = {11, 22, 33, 55, 66, 99}
Answer: The given set {11, 22, 33, 55, 66, 99} is in ascending order.
The number of terms contained in the given list = 6 terms
Thus, the set contains an even number of elements.
The middle two terms of the list are 33 and 55.
Hence, the median of the set of numbers is = (33 + 55)/2
= 42.50
2. How to find the median of the marks scored by the students in an exam, as given below,
Answer: To find the solution:
n = 50
Median Class = n/2th value
= (50/2)th value
= 25th value
= 20 - 30
L = 20, n/2 = 25, C = 9, F = 15, W = 10
Median = L+
(n/2)−C
[(n/2)−C / F ]*W
= 30.6.
FAQs on How To Find Median?
1. What is the real-world application of the median in statistics?
The median is crucial for understanding data by identifying the middle value in a sorted dataset. This is particularly useful in real-world scenarios where data might be skewed by extreme values (outliers). For example, when analyzing typical household incomes, the median provides a more accurate representation than the average, as it isn't heavily influenced by a few very high or very low incomes. It's widely used in economics, social sciences, and demographics to describe distributions effectively.
2. How is the median different from the mean or average?
While both median and mean (or average) are measures of central tendency, they represent different aspects of a dataset. The mean is the sum of all values divided by the number of values, making it sensitive to extreme outliers. The median, however, is the middle value in an ordered list of numbers. If there are outliers, the median provides a more robust representation of the "typical" value because it is not affected by them.
3. What are the fundamental characteristics of the median value in a data set?
The median is the middlemost value that divides a data set into two equal halves when the data is arranged in ascending or descending order. Key characteristics include:
- It is a positional average, meaning its value depends on its position in the sorted data, not on the magnitude of all values.
- It is not affected by extreme values (outliers), making it suitable for skewed distributions.
- It can be determined for both quantitative and qualitative data (if it can be ordered).
4. How do you calculate the median for a simple set of ungrouped data?
To calculate the median for ungrouped data, follow these steps:
- Arrange the data in ascending or descending order.
- Count the number of observations (n) in the dataset.
- If n is odd, the median is the value at the `((n+1)/2)`th position.
- If n is even, the median is the average of the values at the `(n/2)`th and `((n/2)+1)`th positions.
5. What is the specific method for finding the median when there are an even number of observations?
When a dataset has an even number of observations, there isn't a single middle value. In such cases, after arranging the data in order, the median is calculated as the average of the two middle values. For example, if there are 6 observations, you would take the 3rd and 4th values, add them together, and then divide by 2 to find the median. This ensures the median remains the central point of the data.
6. Can you show an example of calculating the median for a given list of numbers?
Certainly. Let's find the median for the set of numbers: 4, 5, 9, 2, 6, 8, 7.
- First, arrange them in ascending order: 2, 4, 5, 6, 7, 8, 9.
- Count the observations: There are 7 numbers (n=7), which is an odd number.
- The median position is `(7+1)/2 = 4`th.
- The value at the 4th position is 6.
Therefore, the median of the given set is 6.
7. What formula is used to calculate the median for grouped data?
For grouped data, especially in a continuous frequency distribution, the 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 total number of observations (sum of frequencies).
- cf is the cumulative frequency of the class preceding the median class.
- f is the frequency of the median class.
- h is the class size (width of the median class).
8. What are the step-by-step instructions to find the median of grouped frequency distribution?
To find the median of grouped data, follow these steps:
- Prepare a cumulative frequency table for the given data.
- Calculate `n/2`, where `n` is the total number of observations.
- Identify the median class, which is the class interval whose cumulative frequency is just greater than or equal to `n/2`.
- Note the lower limit (L), frequency (f), and class size (h) of the median class.
- Note the cumulative frequency (cf) of the class just preceding the median class.
- Apply the median formula: Median = L + [((n/2) - cf) / f] × h to get the final value.
9. How does the understanding of median evolve from Class 7 to Class 10 Maths?
The concept of median is introduced gradually in CBSE Maths.
- In Class 7, students learn about the median for ungrouped data, focusing on arranging numbers and finding the middle value for simple sets.
- By Class 9, the understanding deepens with more complex ungrouped data sets, including how to handle even numbers of observations.
- In Class 10, the curriculum advances to calculating the median for grouped data and frequency distributions, involving formulas and cumulative frequency concepts. This progression ensures a thorough grasp of the topic as per CBSE 2025–26 syllabus.
10. Why is the median sometimes a more appropriate measure of central tendency than the mean?
The median is often preferred over the mean in situations where the data contains extreme values or outliers. Since the mean is calculated by summing all values, a single very large or very small number can significantly pull the mean towards that extreme, misrepresenting the typical value. The median, being the middle value in a sorted list, is resistant to these outliers and provides a more accurate central point for skewed distributions, such as income distribution or property prices.
