How To Find Median?

Dhristi JEE 2022-24

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,

Balls

51-55

56-60

61-65

66-70

Frequency

2

7

8

4

 

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.

 

                                                (Image will be uploaded soon)

 

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,

Marks 

0-10

10-20

20-30

30-40

40-50

50-60

Number of Students

2

7

15

10

11

5

Answer: To find the solution:

Marks 

0-10

10-20

20-30

30-40

40-50

50-60

Number of Students

2

7

15

10

11

5

Cumulative Frequency (C)

2

9

24

34

45

50

 

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 Practical Application of the Median?

Most students know how to find out median but need to understand its necessity in the real world. Around us, we can find information which when collected into certain data sets, can be analyzed mathematically. Statistics formulates several basic techniques like mean, median, and mode to compare and contrast the data sets. For example, comparing the growth rate of two companies by analysis of the profits made by it in multiple years. Median is such a tool that measures the average of a group of data. Median refers to a point in the large group of data, where fifty percent of the data is above the median and fifty percent is below that value.

2. Is Median and Average the Same?

Median and average can sometimes be related, as the average is most commonly understood. However, the median is the middle value in a large set of data. A data set can have an equal number of values or repetitive numbers. This entire set needs to be first arranged from its lowest value to its highest value. From this, if we find that the majority of the data elements are clustered towards one extreme. Then merely finding the average or mean of the data does not give the middle point. In this case, the median needs to be found which is calculated depending on several factors of the set.

3. What are the Important Notes on Median?

The most important topics discussed in this article of Vedantu on median are as follows - 

  1. Median is the positionally centre most value of a given data set.

  2. The given data is first needed to be arranged in an increasing/decreasing order or ascending/descending to be able to calculate the median value.

  3. It is not necessary to consider every value to find the median.

  4. Median doesn’t take into account the two extremes of the data.

4. Calculate Median of Two Numbers and its relation with Mean?

In an ascending or descending ordered series, the median is the number that is midway between the two range extremes but does not always consider those two extremes and hence It is not usually identical to the mean. 


Let's understand how to find the median for only two numbers or just two extreme values of data. For a set of two values or numbers, the median will be the same as the mean, or arithmetic average. 


For example, for the numbers 2 and 8, both will have a mean and a median of 5. The students should note that the median is the value present at the midpoint of a given set of data, not the midpoint itself. The mean is the arithmetic average: (8 + 2)/2 = 5. What if there were two more numbers, let's say 3 and 5? The median will be 4, while the solution for arithmetic mean will be (2 + 3 + 5 + 8)/4 = 4.5.

5. What are the steps to calculate the median of grouped data?

When the given data is continuous and in the form of a frequency distribution (or simply grouped data is given), then in order to calculate the median we need to go through these steps written below:


Step 1: Count the total number of the data entries or the observations, denoted by “n”.

Step 2: Identify or define the class range or size (h) and drive the data in different classes.

Step 3: Now, calculate the cumulative frequency (c.f.) of each class.

Step 4: after the cumulative frequency, find the lower limit of the median class (l).

Step 5: Just apply the formula of the median with all the terms calculated before.

Median = I + [\frac{n/2 - c}{2}]\times h

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