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Calculation of Median for Different Types of Statistical Series

Last Updated : 06 Sep, 2023
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What is Median?

When elements in the data set are organised sequentially, that is, in either an ascending or descending order of magnitude, the median can be referred to as the middle value of the data set. Its value is located in a distribution in such a way that 50% of the items are below it and 50% are above it. It focuses on the center or middle of a distribution.

Median in Individual Series

The steps required to determine the median of an individual series are as follows:

Step 1: Firstly, arrange the given data in ascending or descending order.

Step 2: Apply the following formula of the Median:

Median(M) = Size~of~[\frac{N+1}{2}]^{th}~item

Where, 

N is the Number of Items

Median in case of Odd and Even Number of Items

  • In the case of an odd number of items, Median is the Middle term of the observation.
  • In the case of an even number of items, the Median is the average of two middle terms and is determined by using the following formula:

Median(M)= \frac{Size~of~[\frac{N}{2}]^{th}~item+Size~of~[\frac{N}{2}+1]^{th}~item}{2}

Example 1: (When there is an even number of items in the series)

Given below is the age of some students. Find out the median of their age: 40, 32, 38, 28, 20, 44, 22, 18.

Solution:

Median in Individual Series

 

The number of items is 8. So, 

Median = \frac{Size~of~[\frac{8}{2}]^{th}~item+Size~of~[\frac{8}{2}+1]^{th}~item}{2}

Median = \frac{Size~of~4^{th}~item+Size~of~5^{th}~item}{2}

\frac{28+32}{2}     = 30

Median = 30

Example 2: (When there is an odd number of items in the series)

From the following data of the weekly wages of 7 employees (in ₹), compute the median wage.

450600800

250

430

200

400

Solution:

Median in Individual Series

 

The number of items is 7. So, 

Median = Size~of~[\frac{N+1}{2}]^{th}~item

Median = Size~of~[\frac{7+1}{2}]^{th}~item = Size of 4th item = 430

Median = 430

Median in Discrete Series

The steps required to determine the median of a discrete series are as follows:

Step 1: Arrange the given distribution in either ascending or descending order.

Step 2: Denote the variables as X and frequency as f.

Step 3: Determine the cumulative frequency; i.e., cf.

Step 4: Calculate the median item using the following formula:

Median(M)= Size~of~[\frac{N+1}{2}]^{th}~item

Where, 

N = Total of Frequency

Step 5: Find out the value of  [\frac{N+1}{2}]^{th}~item    . We can find it by first locating the cumulative frequency, which is equal to [\frac{N+1}{2}]^{th}~item     or next higher to it, and then finding the value corresponding to this cumulative frequency. This value will be the Median value of the series.

Example:

Calculate the median of the following data.

Median in Discrete Series

 

Solution:

Median in Discrete Series

 

Median = Size~of~[\frac{N+1}{2}]^{th}~item

Size~of~[\frac{35+1}{2}]^{th}~item     = Size of 18th item

Since the 18th item falls under the cumulative frequency 28 the size of the distribution against this cumulative frequency value is 6.

Median = 6

Median in Continuous Series

The value of the median cannot be easily located for continuous series. In this situation, the median is located between the lower and upper limits of a class interval. A formula is used to interpolate (guess) the median to obtain the exact value. However, it should be kept in mind that when the median class of a series is first class, then the c.f. in the formula will be taken as zero. The steps required to determine the median of a continuous series are as follows:

Step 1: Arrange the given data in either descending or ascending order.

Step 2: Determine the cumulative frequency; i.e., cf.

Step 3: Calculate the median item using the following formula:

Median(M) = Size~of~[\frac{N}{2}]^{th}~item          

Where, 

N = Total of Frequency

Step 4: Now inspect the cumulative frequencies and find out the cf which is either equal to or just greater than the value determined in the previous step.

Step 5: Now, find the class corresponding to the cumulative frequency equal to or just greater than the value determined in the third step. This class is known as the median class. 

Step 6: Now, apply the following formula for the median:

Median = l_1+\frac{\frac{N}{2}-c.f.}{f}\times{i}          

Where,

l1 = lower limit of the median class

c.f. = cumulative frequency of the class preceding the median class

f = simple frequency of the median class

i = class size of the median group or class

Note: While calculating the median of a given distribution, we have to assume that every class of the distribution is uniformly distributed in the class interval.

Example: 

Calculate the median of the following data.

Median in Continuous Series

 

Solution:

Median in Continuous Series

 

Median(M) = Size~of~[\frac{N}{2}]^{th}~item     Size~of~[\frac{50}{2}]^{th}~item     Siz~of~25^{th}~item

Hence, the median lies in the class 30-40.

l1 = 30, f = 14, i = 10, c.f. = 18

Now apply the following formula:

Median = l_{1}+\frac{\frac{N}{2}-c.f.}{f}\times{i}     30+\frac{\frac{50}{2}-18}{14}\times{10}

Median = 35

Median in Special Cases

The calculation process of the median is different under some special circumstances. There are five types of special cases including:

  • Cumulative Frequency Distribution
  • Mid-Values are given
  • Inclusive Class-Intervals
  • Open-End Series
  • Unequal Class Intervals

1. Cumulative Frequency Distribution (Less than or More than)

When the data is given in the form of less than or more than for all items in the series, then such data has to be converted into a simple frequency distribution, to find out the frequency of the median class. Once it is done, the rest of the procedure is the same as in any other continuous series.

Example: 

Calculate the median from the following data.

Median in Cumulative Frequency Distribution

 

Solution:

Median in Cumulative Frequency Distribution

 

Median(M) = Size~of~[\frac{N}{2}]^{th}~item     Size~of~[\frac{200}{2}]^{th}~item     Siz~of~100^{th}~item

Hence, the median lies in the class 20-25.

l1 = 20, f = 50, i = 5, c.f. = 60

Now apply the following formula:

Median = l_{1}+\frac{\frac{N}{2}-c.f.}{f}\times{i}     20+\frac{\frac{200}{2}-60}{50}\times{5}

Median = 24

2. Mid-Values are given

When mid values of class intervals are given, then the class intervals are found; i.e., to calculate the median, we need to first convert it into a continuous series.

Step 1: First of all, calculate the difference between the two mid-values.

Step 2: Then, half of the difference is subtracted and added to each mid-value to find the lower and upper limits respectively of the class intervals.

Example:

Compute the median from the following data:

Median when Mid-values are given

 

Solution:

Step 1: The difference between the two mid-values is 10.

Step 2: Half of the difference is 5. Now, 5 is reduced and added to each mid value to determine the lower limit and upper limit.

Median when Mid-values are given

 

Median(M) = Size~of~[\frac{N}{2}]^{th}~item     Size~of~[\frac{194}{2}]^{th}~item     Siz~of~97^{th}~item

Hence, the median lies in the class 150-160.

l1 = 150, f = 58, i = 10, c.f. = 75

Now apply the following formula:

Median = l_{1}+\frac{\frac{N}{2}-c.f.}{f}\times{i}     

150+\frac{\frac{194}{2}-75}{58}\times{10}

Median = 153.7 or 154

3. Inclusive Class-Intervals

Step 1: Find the difference between the upper limit of a class interval and the lower limit of the next class interval.

Step 2: Add half of this difference to the upper limit of a class interval and subtract the remaining half from the lower limit of each class interval. This procedure fills up the gap between the two classes and thereby we get the exclusive classes.

Example:

Calculate the median from the following data:

Median in Inclusive Class Intervals

 

Solution:

Median in Inclusive Class Intervals

 

Median(M) = Size~of~[\frac{N}{2}]^{th}~item     Size~of~[\frac{530}{2}]^{th}~item     Siz~of~265^{th}~item

Hence, the median lies in the class 79.5-89.5.

l1 = 79.5, f = 120, i = 10, c.f. = 210

Now apply the following formula:

Median = l_{1}+\frac{\frac{N}{2}-c.f.}{f}\times{i}     79.5+\frac{\frac{530}{2}-210}{120}\times{10}

Median = 84.

4. Open-End Series

In the case of open-end classes, the lower limit of the first class and the upper limit of the last class are not given. Median is known to be the best average in open-end class-interval series. In this case, there is no need to complete the class intervals and the formula remains the same.

Example:

Calculate the median from the following data:

Median in Open-end Series

 

Solution:

Median in Open-end Series

 

Median(M) = Size~of~[\frac{N}{2}]^{th}~item = Size~of~[\frac{100}{2}]^{th}~item = Siz~of~50^{th}~item  

Hence, the median lies in the class 20-30.

l1 = 20, f = 36, i = 10, c.f. = 32

Now apply the following formula:

Median = l_{1}+\frac{\frac{N}{2}-c.f.}{f}\times{i}     20+\frac{\frac{100}{2}-32}{36}\times{10}

Median = 25

5. Unequal Class Intervals

When the class intervals are unequal, there is no need to make the class intervals equal. The frequencies need not be adjusted and the same formula will be applied.

Example:

Calculate the median from the following data:

Median in Unequal Class Intervals

 

Solution:

Median in Unequal Class Intervals

 

Median(M) = Size~of~[\frac{N}{2}]^{th}~item     Size~of~[\frac{120}{2}]^{th}~item     Siz~of~60^{th}~item

Hence, the median lies in the class 30-60.

l1 = 30, f = 60, i = 30, c.f. = 40

Now apply the following formula:

Median = l_{1}+\frac{\frac{N}{2}-c.f.}{f}\times{i}     30+\frac{\frac{120}{2}-40}{60}\times{30}

Median = 40

Missing Frequency

Steps to determine the missing frequency if the value of median is already given, are as follows:

Step 1: Represent missing frequencies by f1 or f2 as the case may be.

Step 2: Apply the formula for the calculation of median. In this process, we get an equation that gives us the missing frequencies.

Example:

The following table gives the distribution of the monthly salary of 1,800 employees. However, the frequencies of the classes 40-50 and 60-70 are missing. If the median of the distribution is ₹59.25, find the missing frequencies.

Missing Frequency

 

Solution:

Let f1 and f2 be the frequencies of the classes 40-50 and 60-70, respectively.

Missing Frequency

 

As the median of the above series is given as 59.25, the median lies in the class 50-60 

l1 = 50, c.f. = 240+f1, f = 400, i = 10, and N = 1,800 (given)

Median = l_{1}+\frac{\frac{N}{2}-c.f.}{f}\times{i}     = 50+\frac{\frac{1800}{2}-(240+f_1)}{400}\times{10}

59.25 = 50+\frac{\frac{1800}{2}-240-f_1}{400}\times{10}

9.25 × 40 = 660 – f1

f1 = 660 – 370

f1 = 290

Also, 1,010 + f1 + f2 = 1,800 (given)

1,010 + 290 + f2 = 1,800

f2 = 1,800 – 1,300

f2 = = 500



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