How to find Median in Statistics

First, what is Median in Statistics?

In this article, we will learn how to find Median in Statistics. Before that let us have a look at what the median is.

The median is the data point that falls in the middle of a data set arranged in ascending or descending order. These numbers either belong to a sample data set or a population. There is no universal symbol for the median, but x͂ or Mdn generally represent it.

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Let us continue.

Median as a measure of central tendency

Median is one of the main measures of the central tendency of a data set, and the other two popular ones are mean and mode. Measures of central location and summary statistics are some synonyms for measures of central tendency.

 

We use measures of central tendency to get more insights into the nature of the data set. The median may not represent the average value or the data points with the highest frequencies. Still, it captures the value of the data points representing the center of the spread of the sample or population.

 

In many cases, there is no single suitable measure of central tendency to reveal everything about the nature of the data. In many cases, the mean or mode represents the sample or population more accurately. However, the median has the advantage that it is less affected by outliers and skewed data as it is the middle value of the data set.

Is the median the same as the 50th percentile?

Yes! The Median and the 50th percentile value are the same things. The Median represents the middle value of the given data points, and so does the 50th percentile value of a data set with 50% values smaller and 50% values larger than itself.

 

Refer to the blog posts ‘How to find mean’ and ‘How to find mode‘ to learn how to calculate mean and mode.

 

How to find median in statistics?

How to find the median of a data set

As mentioned above, the median is nothing but the value of the data points that occur in the middle of a data set sorted in order of magnitude.

Step 1

We must arrange the given data in ascending or descending order before calculating the median. Let us take n to

Step 2

• If n is odd, the middle value is the median. We calculate its position by the formula (n + 1)/2.
• If n is even, the average of the two middle values is the median with positions n/2 and (n/2) + 1.

How to find the median from a histogram

A histogram represents a frequency table visually as a bar chart. It usually has ordered data representing the class intervals on one axis. We can calculate the median by locating the central class intervals in the histogram. If we have grouped data, we must follow the method illustrated in example 3.

How to find the median using Excel

The native ‘MEDIAN’ function returns the median value in Excel. We can select MEDIAN from the formula dropdown menu containing formulas or type MEDIAN in the formula bar. Then, we select the cells containing the data. For example, if we want to calculate the median for column K, rows three through 23, the formula will be ‘MEDIAN(K3:K23).’

Sample problems on Median

Example 1

What is the median of the ungrouped sample data set containing 3, 5, 9, 2, 16, 14, 3, 15, 5, 17, 26, 2, and 21?

The data points arranged in increasing order = 2, 2, 3, 3, 5, 5, 9, 14, 15, 16, 17, 21, 26

The data set has 13 values. Hence, n =13. Here, n is an odd number.

 

The position of the median value = (n+1)/2 = (13+1)/2 = 7

Median = 9

 

Example 2

What is the median of the ungrouped sample data set containing 9, 4, 7, 13, 5, 15, 23, 18, 4, 35, 46, 51, 13, and 22?

The data points in increasing order = 4, 4, 5, 7, 9, 13, 13, 15, 18, 22, 23, 35, 46, 51

n = 14

 

The positions of the data points occurring in the middle of the data set = n/2 and (n/2) + 1 = 7 and 8

Median = (13 + 15)/2 = 14

 

The data can be ungrouped or grouped. In the case of grouped data, we may not know the exact values of the data points in the sample or the population for grouped data. However, we know the frequency of the data points occurring in the groups or intervals. For such data, we calculate the median differently.

 

Example 3

How to find the Median in Statistics for grouped data?

The time (in minutes) taken by participants of a group to finish a task is

19

22

18

27

14

34

32

36

9

22

24

13

20

17

11

21

14

16

26

20

 

Frequency distribution table of the time taken by the participants to finish a task

Time taken (in minutes)	Frequency (fi)	Cumulative frequency (c)
1 ≤ t < 10	1	1
10 ≤ t < 15	4	5
15 ≤ t < 20	4	9
20 ≤ t < 25	6	15
25 ≤ t < 30	2	17
30 ≤ t < 35	2	19
35 ≤ t < 40	1	20

 

Median = l + [(n/2−c)/f] × h

 

In the given data, the 4th class interval (20 ≤ t < 25) is the median class interval as it is the middle value of the data set.

l = lower limit of the median class interval = 20

n = number of observations = 20

c = cumulative frequency of the class interval before the median class interval = 9

f = frequency of the median class interval = 6

h = class interval size = 5

 

Hence, median = 20 + [(20/2−9)/6] × 5 = 20.83

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