What is Median and How to calculate it?

Median is another measure of central tendency. In our previous article on Mean, we explained what central tendency is. You can read is here.  In brief, central tendency is a numerical value that explains the central point of a data set.

In this article, we are going to study the second measure of tendency called Median.

What is Median?

Quite literally, median is the “middle-est” value in a data set when the data set is arranged in a chronological order. In other words, after you arrange a data set in either ascending or descending order, you find the value that is right in the middle of the data set. Now, there can be two kinds of data sets that you can be working with – even data set (where the number of values is even – divisible by 2) or odd data set (where the number of values is odd – not divisible by 2). The method to calculate median is slightly different for odd data set and even data set.

How to calculate Median?

For odd data set

It is relatively easier to calculate median for an odd data set than the even data set. Once you have arranged the data set in a chronological order, you simply take the value which is right in the middle of the data set. Let us understand this with the help of an example. Let’s say you want to calculate the median of the number of runs scored by your favorite cricketer in the last 11 innings he has played; you go to a cricketing stats website to find out the following details; since it is the ICC Champions Trophy season, it will be great fun to analyze. To make it even more fun, we have taken the data of an actual Indian Batsman for his last 11 innings:

Opposition Score
South Africa 23
Sri Lanka 7
Pakistan 53
England 45
England 150
England 15
South Africa 0
West Indies 55
West Indies 28
West Indies 16
Australia 12

 

 

Just glance through that table a little bit before you start doing anything with that data. How do you think the form of the batsman is? Is the batsman dependable? Now, you can argue that to find that out, you would need a lot more data – like the number of balls played, the position at which the batsman plays etc. You are right – to do an in depth analysis you will need that kind of data. But, can only the median tell you something about the form of the batsman? Yes, it can! Let’s begin.

Step 1: Arrange all the data into descending order. Use MS Excel for this and you will get the following output:

Opposition Score
England 150
West Indies 55
Pakistan 53
England 45
West Indies 28
South Africa 23
West Indies 16
England 15
Australia 12
Sri Lanka 7
South Africa 0

 

Step 2: Find the middle most value in the data set. In our data set, there are 11 data points. So, which one will be the middle most data point? The sixth value from the top (or the bottom) is the middle most value in this example. Mathematical formula to find out the position of the middle most value for an odd data set is given below:

Position of Middle Value = (N+1)/2

Where,

N is the number of data point in the data set.

In our example, N = 11. Putting the value of N in the formula, we get:

Median = (11+1)/2 = 12/2 = 6 = the data point in the sixth position = 23

The inference here is that nearly half of this batsman’s innings in the last 11 innings are above 23 runs or there is nearly a 50% probability that this batsman will score over 23 runs in the next innings he plays given his current form.

For even data set

We will take the same example. This time, we will take the last 10 innings and hence, we get the following data set:

Opposition Score
South Africa 23
Sri Lanka 7
Pakistan 53
England 45
England 150
England 15
South Africa 0
West Indies 55
West Indies 28
West Indies 16

 

Step 1: Sort the data in descending order. Use MS Excel to do this and you will get the following result:

Opposition Score
England 150
West Indies 55
Pakistan 53
England 45
West Indies 28
South Africa 23
West Indies 16
England 15
Sri Lanka 7
South Africa 0

 

Step 2: Since you can’t split an even data in 2 halves from the middle most value, for even data set, you find the 2 middle values and take the average of those values to calculate the median. The formula to find out the position of the 2 middle values for the even data set is:

Position of Middle Values = N/2 and (N/2)+1

In our example, N is 10. Putting the value of N, we get:

Position of Middle Values = 10/2 and (10/2)+1 = 5th and 6th position = 28 and 23.

To find out the median, calculate the average of 28 and 23. Read our article on Mean to understand how to calculate the average. The average of 28 and 23 is 25.5.

Hence, 25.5 is the median of the last 10 innings played by this batsman. You can take out the inference in the same way we did for the odd data set.

Hope you enjoyed learning what median is and how to calculate it.

Come back tomorrow for the third measure of central tendency – Mode.

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