How to Find The Median?

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Median, to find the Median for grouped data, and to find the Median for ungrouped data:

Starting with the median finding procedure, let us first understand the grouped and ungrouped data.


Grouped Data:

It is the data categorized into groups after getting collected. We use a frequency table for classifying the raw data into several groups.


Ungrouped Data:

Ungrouped data, or raw data, is the information, not placed under any group or category after getting collected. This data is in the form of characteristics or numbers.

Now let us move on to what a median is, and how to find Median, class 10 introduction.

Median is the central element of a group of numbers arranged in a proper sequence according to their size. For the even number of terms, we calculate the Median using the mean of the two middle numbers. In general, we use the below-mentioned sequence of steps for finding the Median:

  1. Arrange the number in an order according to their size.

  2. For the odd number of terms, the Median is the middle element of the sequence.

  3. For the even number of terms, to find the Median, we must add the two middle elements and then divide the sum by 2. This is taking a mean of the numbers.

With this, let us first understand how to get the mean of ungrouped data?

To do so, the mean of data = sum of all elements under consideration/ total number of elements.

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How to Find the Median of Ungrouped Data?

Starting with how to find median for ungrouped data, there is a general formula above for the same, i.e.,

Median = Value of the ((n+1)/2)th term. 

Example on how to find the Median of ungrouped data:

Now, let us solve an example to get a brief on how to find the Median of a grouped data.


Problem Statement:

Given below is the data obtained about the time that four students took for completing a running race. Find the Median of the racing time.

9.7hrs, 6.3hrs, 2.5hrs, 7.1hrs


Solution:

First, let us arrange the data in ascending order:

2.5, 6.3, 7.1, 9.7

As the number of observations on the data set is even, we can calculate the Median by taking mean of two middlemost numbers.

Mean = (6.3+7.1)/2

=>13.4/2

=>6.7

Therefore, the median race time here is 6.7 hrs.


How to Solve the Median in Grouped Data?

For this, we must know how to find the median class of grouped data.

To do so, we are required to find the cumulative frequencies first and then calculate the value of n/2. Now, the median class is the group where the cumulative Frequency has equal value to n/2.

Given below is the formula to find the Median of grouped data:

Median = l+ (h/f)(n/2-c)

Here,

  • l= Lower class interval for the modal class

  • f = Median class’s frequency

  • n = ∑f = total frequency

  • c = Cumulative frequency of preceding class

  • h = Modal class’s Interval size

Example on How to Calculate Median from a Frequency Table for Grouped Data:

Examples help in understanding the concept in a better manner. Here is an example of finding the Median for grouped data:


Problem Statement:

Calculate the Median for the following data:


Class Interval

Frequency

40-44

1

45-49

5

50-54

9

55-59

12

60-64

7

64-69

2

 

Solution:


First, let us make the final table for making the calculation easier:


Class Interval

Frequency

Final Class Interval

Cumulative Frequency

40-44

1

39.5-44.5

1

45-49

5

44.5-49.5

6

50-54

9

49.5-54.5

15

55-59

12

54.5-59.5

27

60-64

7

59.6-64.5

34

65-69

2

64.5-69.5

36


n=36

n/2 = 36/2 => 18

Therefore, Median Class = 54.5 – 59.5

Now, 

Median = l+ (h/f)(n/2-c)

=>54.5 + (5/12)(18-15)

=>54.5 + (5/12)(3)

=>54.5 + 1.25

=> 55.75

Therefore, median = 55.75

FAQ (Frequently Asked Questions)

Q1: Explain How to Calculate the Median for Ungrouped Data. Solve the Given Problem Using the Steps on How to Get the Median of Ungrouped Data: Find the Median of the Given Data Set: $1.59, $1.31, $1.96, $3.09, $1.64, $1.55, $2.61

Ans: Here are the steps to be followed for finding the median of ungrouped data:

  1. Arrange the values in ascending order

  2. Use the locator’s formula, i.e., c=(n+1)/2 (n=number of observations)

  3. Search for the value available at (n+1)/2.

If the c is a fractional value, the median is calculated after taking the average of the two values. For Example, for the location coming out as 3.5, the median is the average value of 3rd and 4th position.

For the above-given data, arranging it in ascending order:

$1.31, $1.55, $1.59, $1.64, $1.96, $2.61, $3.09

The median = ((7+1)/2)th term

=>4th term

=>Median = $1.64

Q2: Explain How to Find the Median of Discrete Data.

Ans: Discrete data set is the one where the observations do not belong to any data classes; they are discrete and distinct observations. When it comes to following the discrete data set, the median can be calculated using the formula given below:


Median for discrete data = ((n+1)/2)th term of the data set.


Further, the median is the particular value that corresponds to the cumulative frequency where the observation value ((n+1)/2) lies.


For Example, let the discrete data set be:


Element

Frequency

14

2

36

5

45

1

70

3

105

12

145

0

 

Median = Value of ((n+1)/2)th term


=>((6+1)/2)th term


=> 3.5th term


=> value of(3rd +4th )/2


=> (45+70)/2


=> 57.5