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# Median of Grouped Data

In Statistics, we know that the three measures of central tendencies are mean, median and mode. The mean represents the average of the dataset, the median represents the middle value of the data set, and the mode represents the repeated value in the dataset. In this article, we will learn in detail about the method of finding the median of the grouped data with an example.

## Finding Median of Ungrouped Data

As discussed above, the median is one of the measures of central tendency, which gives the middle value of the given data set. While finding the median of the ungrouped data, first arrange the given data in ascending order, and then find the median value.

If the total number of observations (n) is odd, then the median is (n+1)/2 th observation.

If the total number of observations (n) is even, then the median will be average of n/2th and the (n/2)+1 th observation.

For example, 6, 4, 7, 3 and 2 is the given data set.

To find the median of the given dataset, arrange it in ascending order.

Therefore, the dataset is 2, 3, 4, 6 and 7.

In this case, the number of observations is odd. (i.e) n= 5

Hence, median = (n+1)/2 th observation.

Median = (5+1)/2 = 6/2 = 3rd observation.

Therefore, the median of the given dataset is 4.

## How to Find the Median of Grouped Data?

In a grouped data, it is not possible to find the median for the given observation by looking at the cumulative frequencies. The middle value of the given data will be in some class interval. So, it is necessary to find the value inside the class interval that divides the whole distribution into two halves. In this scenario, we have to find the median class.

To find the median class, we have to find the cumulative frequencies of all the classes and n/2. After that, locate the class whose cumulative frequency is greater than (nearest to) n/2. The class is called the median class.

After finding the median class, use the below formula to find the median value.

$$\begin{array}{l}Median = l+ \left ( \frac{\frac{n}{2}-cf}{f} \right )\times h\end{array}$$

Where

l is the lower limit of the median class

n is the number of observations

f is the frequency of median class

h is the class size

cf is the cumulative frequency of class preceding the median class.

Now, let us understand how to find the median of a grouped data using the formula with the help of an example.

### Median of Grouped Data Example

Example:

The following data represents the survey regarding the heights (in cm) of 51 girls of Class x. Find the median height.

 Height (in cm) Number of Girls Less than 140 4 Less than 145 11 Less than 150 29 Less than 155 40 Less than 160 46 Less than 165 51

Solution:

To find the median height, first, we need to find the class intervals and their corresponding frequencies.

The given distribution is in the form of being less than type,145, 150 …and 165 gives the upper limit. Thus, the class should be below 140, 140-145, 145-150, 150-155, 155-160 and 160-165.

From the given distribution, it is observed that,

4 girls are below 140. Therefore, the frequency of class intervals below 140 is 4.

11 girls are there with heights less than 145, and 4 girls with height less than 140

Hence, the frequency distribution for the class interval 140-145 = 11-4 = 7

Likewise, the frequency of 145 -150= 29 – 11 = 18

Frequency of 150-155 = 40-29 = 11

Frequency of 155 – 160 = 46-40 = 6

Frequency of 160-165 = 51-46 = 5

Therefore, the frequency distribution table along with the cumulative frequencies are given below:

 Class Intervals Frequency Cumulative Frequency Below 140 4 4 140 – 145 7 11 145 – 150 18 29 150 – 155 11 40 155 – 160 6 46 160 – 165 5 51

Here, n= 51.

Therefore, n/2 = 51/2 = 25.5

Thus, the observations lie between the class interval 145-150, which is called the median class.

Therefore,

Lower class limit = 145

Class size, h = 5

Frequency of the median class, f = 18

Cumulative frequency of the class preceding the median class, cf = 11.

We know that the formula to find the median of the grouped data is:

$$\begin{array}{l}Median = l+ \left ( \frac{\frac{n}{2}-cf}{f} \right )\times h\end{array}$$

Now, substituting the values in the formula, we get

$$\begin{array}{l}Median = 145+ \left ( \frac{25.5-11}{18} \right )\times 5\end{array}$$

Median = 145 + (72.5/18)

Median = 145 + 4.03

Median = 149.03.

Therefore, the median height for the given data is 149. 03 cm.

### Practice Problems

1. The median of the following data set is 525. Find the values of x and y, if the total frequency is 100.
 Class Interval Frequency 0 – 100 2 100 – 200 5 200 – 300 x 300 – 400 12 400 – 500 17 500 – 600 20 600 – 700 y 700 – 800 9 800 – 900 7 900 – 1000 4
2. The following frequency distribution table shows the monthly consumption of electricity of 68 consumers of a locality. Find the median of the given data.
 Monthly consumption of electricity (in units) Number of consumers 65 – 85 4 85 – 105 5 105 – 125 13 125 – 145 20 145 – 165 14 165 – 185 8 185 – 205 4

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## Frequently Asked Questions on Median of Grouped Data

### What is meant by the median in statistics?

In statistics, the median is the middle value of the given dataset.

### How to find the median value if the number of observations is odd?

If the number of observations (n) is odd, the median is the (n+1)/2th observation.

### How to find the median value if the number of observations is even?

If the number of observations (n) is even, the median is the average of n/2th and (n/2)+1th observation.

### What is the formula to find the median of grouped data?

The formula to find the median of grouped data is:
Median = l+ [((n/2) – cf)/f] × h
Where l = lower limit of median class, n = number of observations, h = class size, f = frequency of median class, cf = cumulative frequency of class preceding the median class.

### What is the median class?

The median class is the class interval whose cumulative frequency is greater than (and nearest to) n/2.