 # Median

Median Definition

So let’s say for example in a tough exam, you and your friends performed very averagely. However, this one brainiac scored 100%.  Now when you calculate the average marks of the class, it’s higher due to his score. But, this is not a fair representation of reality. Here is where the concept of Median will help you out.

Often the median is a better representative of a standard group member. If you take all the values in a list and arrange them in increasing order, the median will be the number located at the center. The median is a quality that belongs to any member of the group. Based on the value distribution, the mean may not be particularly close to the quality of any group member. The mean is also subjected to skewing, as few as, one value significantly different from the rest of the group can change the mean dramatically. Without the skew factor introduced by outliers, the median gives you a central group member. If you have a normal distribution, a typical member of the population will be the median value.

What is the Median?

In simplest terms, the Median is the middlemost value of a given data set. It is the value that separates the higher half of the data set, from the lower half. It may be said to be the “middle value” of any given population. To calculate the Median the data should be arranged in an ascending or descending order, the middlemost number from the so arranged data is the Median.

In order to calculate the Median of an odd number of terms, we have to arrange the data in an ascending or descending order. The middlemost term is the Median. And to calculate the Median of an even number of terms, we arrange the data in an ascending or descending order and take an average of the two middle terms.

• So, if the odd number is n, then M = ($\frac{n+1}{(2)}$)$^{th}$ term

• And if the even number is n, then M = [($\frac{n}{(2)}$ + 1)$^{th}$ term + $\frac{n}{(2)}$th term ] ÷ 2

Relationship between Mean, Median, And Mode

Karl Pearson explains the relationship between mean median and mode with the help of his Formula as:

(Mean – Median) = $\frac{1}{(3)}$ (Mean – Mode)

3(Mean – Median) = (Mean – Mode)

Mode = Mean - 3 (Mean – Median)

Mode = 3 Median - 2 Mean

Therefore, the equation given above can be used if any of the two values are given and you need to find the third value.

Median of Grouped Data

Since the data is divided into class intervals, we cannot just pick the middle value anymore. Hence, we have to follow a few steps which are listed below:

Step 1: Firstly, we have to prepare a table with 3 columns.

Step 2: Class Intervals will be written in column 1.

Step 3:  Corresponding frequencies denoted by fi will be written in column 2.

Step 4: Calculated Cumulative Frequency denoted by cf will be written in column 3.

Step 5: In this step, we have to find the total of fi denoted by N, and calculate N/2.

Step 6: here, we have to locate the Cumulative Frequency which is greater than or equal to N/2. Note down its corresponding Median Class as well.

Step 7: We have to use the formula given below to calculate the Median.

M = L + ($\frac{n}{(2)}$ - cf)$\frac{h}{(f)}$

Where,

L = Lower limit of Median Class

n = Total frequency

cf =The cumulative frequency of a class preceding the Median Class

f = The frequency of the Median class

h = The width of the Median Class

Median from an Ogive Curve

We have already seen in the Cumulative Frequency topic that the Median of data can also be calculated. This can be done by plotting the Less than frequency curve and More than the frequency curve. They point at which they intersect and the corresponding value on the X-axis would be the Median of the given data.

Solved Example

## Question 1: Find the Median Class for the Following Table:

 x f Class Boundaries c.f 8  - 10 2 7.5 - 10.5 2 11 - 13 4 10.5 - 13.5 6 14 -16 6 13.5 - 16.5 12 17 - 19 4 16.5 - 19.5 16 20 - 22 3 19.5 - 22.5 19 23 - 25 1 22.5 - 25.5 20 20

Solution: First we have to find the total number of frequencies in order to find the median class.

So, here frequency (f) = 20

The formula to obtain the Median Class is $\frac{Total Number of Frequencies + 1}{2}$

Thus, Median Class = $\frac{20 + 1}{2}$ = 10.5

Therefore, the Median class of 10.5 lies between the intervals 10.5 – 13.5

Question 2: Find the median of  3, 13, 7, 5, 21, 23, 39, 23, 40, 23, 14, 12, 56, 23, 29.

Solution: Firstly we have to organize the data in an increasing order  3, 5, 7, 12, 13, 14, 21, 23, 23, 23, 23, 29, 39, 40, 56

Secondly, we have to get the middle number so, if there are 15 numbers, the middle number would be the eight number i.e., 23. So our median is 23.

NOTE: In case we get a pair of middle numbers then we have to add those two numbers and then divide to get the median.