How to find median of grouped data

Median is a measure of central tendency. So, before we know what Median is and how to find it, we must know about the measures of central tendency. A measure of central tendency al

How to find median of grouped data

Median is a measure of central tendency. So, before we know what Median is and how to find it, we must know about the measures of central tendency. A measure of central tendency also known as a measure of central location can be described as a single value that is used to determine a set of data by identifying the central location within that given set of data. The measures of central tendency are also classed as summary statistics. So far, you might have heard about the mean or the average value and how to calculate it from a given set of data. But there are other measures of central tenancy as well. These are mode and median.

All these three measures of central tendency can be used under different conditions. We are going to discuss the median in this article and at the end of this article, you will learn what is median, how to find it, and under what conditions can we use it most appropriately.

What is Median?

The median of a given set of data is known as the middle score of that set of data. Since the Median is the middle point of the given dataset, this implies that half of the data points will be less than the value of the median, and half of the data points will be larger than the value of the median.

How to Find Median?

There are two methods to find the median of a given set of data. These methods are as follows:

Method 1:

To find the Median of a given set of data, you will need to arrange that set of data either in ascending order or in descending order. Once you do that, count all the data points (numbers) of that data set.

If there are odd numbers of data points in a data set, identify the middle value of that data set. That middle value or the data point in the middle will be the median.

If there are even numbers of data points in the given data set, identify the two data points in the middle, add those two and divide them by 2. The answer that you would get would be the average value of those two data points in the middle. This average value would be the median.

Method 2:

To find the median of a given set of data, we have a formula which is:

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

where n is the number of data points in the given set of data and the middle is the position of the data point.

This method differs when you have an even number of data points in the given set of data.

Lets now solve some examples to understand the concept of median completely.

Example: What is the Median of the following numbers?

5, 17, 8, 15, 7,

Solution: we will find the median of these numbers by both methods.

Method 1:

We will first arrange these numbers in ascending order, i.e.,`

5, 7, 8, 15, 17

Now we will identify the middle value of this data set.

Here, '8' is the middle value of this data set.

Hence, this number '8' is the median of this data set.

Method 2:

We will first count the number of data points in the data set.

The number of data points in this data set is 5, or we can say that n=5.

Now we will apply the formula, i.e.,

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

= \[\left ( \frac{(5+1)}{2} \right )^{th}\]

= \[\left ( \frac{6}{2} \right )^{th}\]

= 3rd

The 3rd number in the above data set is '8'.

Hence '8' is the median of this data set.

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 being 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:

Arrange the number in an order according to their size.

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

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 meaning of ungrouped data.

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

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\[\left ( \frac{n+1}{2} \right )^{th}\]

Example of 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 grouped data.

Problem Statement:

Given below is the data obtained about the time that four students took to complete 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 the mean of two middlemost numbers.

Mean = \[\left ( \frac{6.3+7.1}{2} \right )\]

=>\[\frac{13.4}{2}\]

=>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 \[\frac{n}{2}\]. Now, the median class is the group where the cumulative Frequency has an equal value to \[\frac{n}{2}\].

Given below is the Formula to find the Median of Grouped data:

Median = \[l+\left (\frac{h}{f} \right )\left ( \frac{n} {2-c} \right )\]

Here,

l= Lower class interval for the modal class

f = Median classs frequency

n = f = total frequency

c = Cumulative frequency of preceding class

h = Modal classs 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:

Solution:

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

n=36

n/2 = 36/2 => 18

Therefore, Median Class = 54.5  59.5

Now,

Median = \[l+\left (\frac{h}{f} \right )\left ( \frac{n} {2-c} \right )\]

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

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

=>54.5 + 1.25

=> 55.75

Therefore, median = 55.75

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