Finding Medians: Definition, Formulas and Illustrations

Finding Medians: When a data collection is in a mathematical order, the median is the value that falls precisely in the middle. The median is a rigid value. It is simple to understand and calculate, and in some cases, it can be found simply by looking at it. The extreme values have zero effect on the median and can be calculated for distributions with open-ended classes.

The median is most often associated with quantitative data, where the values are numerical, although it can also be found in ordinal data, where the values are ranked categories. Medians indicate when one set of data exceeds the other. In extreme cases, it can be ignored. Consider finding a cricketer’s performance where his worst and best performances can be ignored to determine his consistent performance. Let us learn more about how to find medians from grouped and ungrouped data in this article.

Measures of Central Tendency

Data is represented in different ways, including frequency distribution tables, bar graphs, histograms, and frequency polygons. The question now is whether we always need to study all of the data to make sense of it or if we can extract some essential features of it by focusing on only a subset of the data.

Using measures of central tendency or averages, this is possible.The measures of central tendency in statistics are:

• Mean
• Median
• Mode

They each tell us what value in a data set is typical or indicative of the data set in different ways.

• Calculating the mean is the same as determining the average value of a set of data. The sum of the data is divided by the count of the data.
• The median is the value in the middle of a data set. This is the halfway point in organised data.
• The mode is the most frequently occurring observation in a data set. Count how many times each observation appears in the data set. The number that is most repeated is the mode. Data could have more than one mode.

What is a Median?

When data are arranged in ascending or descending order, the median is the value of the middle observation.

• In many cases, it is difficult to consider the entire data set for representation, and in these cases, the median is useful.
• The median is a simple metric to calculate the central tendency among statistical summary metrics.
• It is also called ‘Place Average’ as the middle data of a sequence is its median.

How to Calculate Median?

Let us look at how to calculate median for grouped and ungrouped data:

Median for Ungrouped Data

The formulas to find the median for grouped and ungrouped data is different. To get the median of ungrouped data, follow the steps below:

Step 1: Arrange the data in ascending/descending order.

Step 2: Count the total number of observations.

Step 3: Check whether the number of observations \(‘n’\)is even or odd.

Step 4: Use the formulas given below according to the \(n\) value from Step \(3\).

When \(n\) is Odd

If the number of observations, \(n\) of the data set is odd, then the formula to calculate median is

Median \(=\left(\frac{(n+1)}{2}\right)^{\text {th }}\) observation

Example:

Consider the data: \(2,3,5,4,1\).

We need to find the median of this data. So, first, we need to arrange the data in ascending or descending order and count the number of observations. We can see that there are \(5\) observations.

Now, see the middle observation. This middle value is the median of the observations, which is \(4\) in this case.

When \(n\) is Even

If the number of observations, \(n\) of the data set is even, then the formula to calculate median is

Median \(=\frac{\left(\frac{n}{2}\right)^{\text {th }} \text { observation }+\left(\frac{n}{2}+1\right)^{\text {th }} \text { observation }}{2}\)

Example: Consider the data set \(2,5,6,3,1,7\). We need to find the median of this data.
First, we need to arrange the data in ascending or descending order and count the number of observations.

\(1,2,3,5,6,7\)

We can see that there are \(6\) observations.

So, the median \(=\frac{\left(\frac{6}{2}\right)^{\text {th }} \text { observation }+\left(\frac{6}{2}+1\right)^{\text {th }} \text { observation }}{2}\)

\(=\frac{3^{\text {rd }} \text { observation }+(4)^{\text {th }} \text { observation }}{2}\)

\(=\frac{3+5}{2}\)

\(\therefore\) Median \(=4\)

Median for Grouped Data

When the given data is grouped and given in the form of a frequency distribution, the median is calculated using the steps listed below.

Step 1: Count the number of observations \((n)\).
Step 2: Choose the class size \((h)\) and divide the data into classes.
Step 3: Determine the total frequency of each class.
Step 4: Determine which class the median belongs to. The Median Class is the class in which \(\frac{n}{2}\) is found.
Step 5: Determine the lower limit of the median class \((l)\) and the cumulative frequency of the preceding class \((c)\).
Now, using the formula below, we can calculate the median value.

Median \(=l+\left[\frac{\frac{n}{2}-c}{f}\right] \times h\)

Solved Examples – How to Find Medians

Let us look at some of the solved examples about how to find medians:

Q.1. Find the median of the data set given below: \(102,56,34,99,89,101,10\)
Sol: First, we need to sort the data from the smallest number to the highest number.
\(10,34,56,89,99,101,102\)
There are \(7\) observations, which means \(n\) is odd. Hence, median will be the middle value. Hence, the median is \(89\) .

Q.2. The heights measured in \(\mathrm{cm}\) of \(9\) students of a class are given below. \(155,160,145,147,150,149,151,144,148\)
Find the median
Sol: First of all we have to arrange the data in ascending or descending order.
\(144,145,147,148,149,151,152,155,160\)
The number of students is \(9\), which is the \(n\), and its odd number.
Then, the median of the given data set will be \(\left(\frac{(n+1)}{2}\right)^{\text {th }}\) observation
\(=\left(\frac{(9+1)}{2}\right)^{\text {th }}\) observation \(=5^{\text {th }}\) observation \(=149\)
Hence, the median, or the medial height, is \(149 \mathrm{~cm}\)

Q.3. The points scored by a Khokho team in a series of matches played are given below:
\(16,2,7,27,15,5,14,18,10,24,48,10,8,7,8,28\)
Find the median of the points secured by them.
Sol: First, arrange the points scored by the team in ascending order.
\(2,5,7,7,8,8,10,10,14,15,16,18,24,27,28,48\)
There are \(16\) observations that is \(n\) is even.
Hence, the median can be calculated as \(=\frac{\left(\frac{n}{2}\right)^{\text {th }} \text { observation }+\left(\frac{n}{2}+1\right)^{\text {th }} \text { observation }}{2}\)
\(=\frac{\left(\frac{16}{2}\right)^{\text {th }} \text { observation }+\left(\frac{16}{2}+1\right)^{\text {th }} \text { observation }}{2}\)
\(=\frac{(8)^{\text {th }} \text { observation }+(9)^{\text {th }} \text { observation }}{2}\)
\(=\frac{10+14}{2}\)
\(=\frac{24}{2}\)
\(\therefore\) Median \(=12\)
Hence, the medial point scored by the Khokho team is \(12.\)

Q.4. Calculate the median for the following grouped data:

Sol:
We need to calculate the cumulative frequencies to find the median.

\(N=\sum f_{i}=70\)
\(\frac{N}{2}=\frac{70}{2}=35\)
Median class is \(40-60\)
\(l=40, f=35, c=25, h=20\)
We can use the formula
Median \(=l+\left[\frac{\frac{n}{2}-c}{f}\right] \times h\)
\(=40+\left[\frac{(35-25)}{35}\right] \times 20\)
\(=40+\left(\frac{10}{35}\right) \times 20\)
\(=40+\left(\frac{40}{7}\right)\)
\(\therefore\) Median \(=45.71\)

Q.5. The following table contains the marks scored by the students in the internal assessments. 

Find the median marks. 
Sol:

Median class \(=\left(\frac{N}{2}\right)^{\text {th }}\) value
\(=\left(\frac{50}{2}\right)^{\text {th }}\) value
\(=25^{\text {th }}\) value
Median class \(20-30\)
\(l=20, \frac{N}{2}=25, m=9, f=15\) and \(c=10\)
We can use the formula
Median \(=l+\left[\frac{\frac{n}{2}-c}{f}\right] \times h\)
\(=20+\left(\frac{[25-9]}{15}\right) \times 10\)
\(=20+\left(\frac{16}{15}\right) \times 10\)
\(=20+10.6\)
\(=30.6\)
\(\therefore\) Median \(\approx 31\)
Hence, the median marks is \(31\).

Summary

Median is one of the central tendencies deduced for a given data set. It is the central or middle value of the provided data. To find the middle value or the median, data must be sorted or arranged in ascending or descending order. The median of the arranged numbers can be calculated using the median formula.

The median formula is different depending on how many observations there are and whether they are odd or even. The formulas to find the median for grouped and ungrouped data are also different. It is easy to calculate, and for small data sets, we can find the median by merely looking at them.

Frequently Asked Questions (FAQs)

Let us look at some of the frequently asked questions about finding medians:

Q.1. What is the easiest way to find the median? 
Ans: The easiest way to find the median in an ungrouped data set is to locate the middle value after arranging the data points in ascending or descending order.

Q.2. How do you find medians when the number of observations is odd?
Ans: If the number of observations is odd, then the median will be the middle most value of the data set. it is the \(\left(\frac{(n+1)}{2}\right)^{t h}\) observation in the data.

Q.3. How do you find your median if the number of observations is even?       
Ans: If the number of observations, of the data set is even, then the formula to calculate the median is
Median \(=\frac{\left(\frac{n}{2}\right)^{\text {th }} \text { observation }+\left(\frac{n}{2}+1\right)^{\text {th }} \text { observation }}{2}\)

Q.4. What is the formula of the median for grouped data?
Ans: We can calculate the median value of grouped data using the formula,
\(\operatorname{Median}=l+\left[\frac{\frac{n}{2}-c}{f}\right] \times h\)

Q.5. Is median a central tendency?
Ans: Yes, the median is one of the central tendencies.

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