How to Find Median for Odd Even and Grouped Data with Formula and Examples
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 Step by Step Guide with Formula
1. What is the median in maths?
The median is the middle value in a data set when the numbers are arranged in ascending or descending order. It divides the data into two equal halves.
- If the number of values is odd, the median is the middle number.
- If the number of values is even, the median is the average of the two middle numbers.
2. How do you find the median step by step?
To find the median, first arrange the numbers in order and then locate the middle value.
- Step 1: Arrange the data in ascending order.
- Step 2: Count the total number of values (n).
- Step 3: If n is odd, median = value at position (n+1)/2.
- Step 4: If n is even, median = average of values at positions n/2 and (n/2)+1.
3. What is the formula for median?
The formula for the median depends on whether the number of observations is odd or even.
- If n is odd: Median = value at (n+1)/2 position
- If n is even: Median = (Value at n/2 + Value at (n/2 + 1)) ÷ 2
4. How do you find the median of an odd set of numbers?
For an odd number of values, the median is the exact middle number after arranging the data in order.
- Example: 3, 7, 9
- Arranged data: 3, 7, 9
- Middle value = 7
5. How do you find the median of an even set of numbers?
For an even number of values, the median is the average of the two middle numbers after arranging the data.
- Example: 4, 8, 10, 12
- Middle numbers: 8 and 10
- Median = (8 + 10) ÷ 2 = 9
6. How do you find the median of grouped data?
The median of grouped data is found using the formula Median = L + [(N/2 − cf) / f] × h.
- Find cumulative frequencies.
- Locate N/2 to identify the median class.
- Substitute values into the formula.
7. What is the difference between mean and median?
The mean is the average of all values, while the median is the middle value of ordered data.
- Mean = (Sum of values) ÷ (Number of values)
- Median = Middle value after arranging data
- Median is less affected by extreme values (outliers).
8. Why is the median important in statistics?
The median is important because it represents the central value without being affected by extreme numbers. It is especially useful in skewed distributions.
- Used in income and salary analysis
- Better measure when outliers exist
- Represents the 50th percentile
9. Can you give a real-life example of finding the median?
A real-life example of the median is finding the middle salary in a company. Suppose salaries are 20,000; 25,000; 30,000; 35,000; 1,00,000.
- Arranged data: 20,000, 25,000, 30,000, 35,000, 1,00,000
- Middle value = 30,000
10. What happens to the median if there are outliers?
The median is generally not affected much by outliers because it depends only on the middle position. Extreme high or low values do not change the central position significantly.
- Example: 2, 3, 4, 5, 100
- Median = 4
