In statistics, the median is the middle value of a set of numbers when they are arranged in order, from the smallest to the largest. It divides the data into two equal parts. Half of the numbers are smaller than the median, and half are larger. If the total number of values is odd, the median is the exact middle number. For example, in the set {3, 5, 7}, the median is 5. If the total number of values is even, the median is found by taking the average of the two middle numbers. For example, in {2, 4, 6, 8}, the two middle numbers are 4 and 6, so the median is (4 + 6)/2 = 5. The median is very useful because it shows the centre of the data and is not affected by very high or very low numbers. It is often used in exams and practical data analysis to quickly understand the central value of a data set.

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What is Median?

The median of a given set of data is the value of the middle term in that data set when those values in the set are arranged in ascending or descending order. The set of data can be given in two forms. 

Firstly, you can be given a raw list of values that represent a particular quantity. This is known as the raw data form. Secondly, you can be given a list of values that represent a particular quantity along with their frequency distribution. This is known as the tabulated form. 

Median Definition

Amongst the three most familiar central tendencies; mean, median and mode we are reading about median here. In simple terms, it is defined as the middle-most observation fetched after arranging the given data. If we consider a simple example: 4,5,7 as the data set, the median for this is 5.

Median Example

Let us step through a quick example to understand what is median?

If we are given with the below data set:

6, 4, 7, 7, and 9.

The number of values is 5.

First arrange this information in ascending order: 4, 6, 7, 7, 9.

Point out the middle term/value as it is the required median. Therefore, median = 7.

What is Median in Statistics

In statistics, the median is the middle value of a group of numbers. To find the median, the numbers must first be arranged in order from smallest to largest. If there is an odd number of values, the median is the one that comes exactly in the middle. If there is an even number of values, the median is found by adding the two middle numbers and dividing the total by two. The median helps us understand the centre of the data. It is very useful because very large or very small values do not change it much. This makes the median easy to use and reliable for simple data.

Median Formula

The formula under this heading can be categorised as ungrouped data and grouped data. Again inside ungrouped data, we would learn about the formula when n is odd and even. So lets us proceed with the formulas now:

To find the median of a given list of observations, you have to first arrange the list in either ascending or descending order (if necessary) and then apply the formula:

Median Formula for Ungrouped Data

The median is similar to a type of average, that we utilised to locate the central value. Ungrouped data means the data which is not grouped or arranged properly. To work with such a group of data the below points are to be followed:

• We will start with arranging the data in increasing or decreasing order.
• Next thing is to calculate the total number of observations(denoted by n).
• Lastly, check if the number of observations estimated is even or odd. Depending on them we will apply the formula.

Check out this article on Variance and Standard Deviation.

Median Formula When n is Odd

The formula when the given set of numbers is odd or when the total number of observations tends to be odd is:

Median=\(({n+1\over{2}})^{th}\text{term}\)

Median Formula When n is Even

An opposite case to the above condition, if the total number of observations tends to be even, then the formula is:

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

Median Formula for Grouped Data

To compute the median of a grouped or continuous frequency distribution, we follow the below steps:

• For the total number of observations(n) obtain the class size(h), followed by dividing the data into separate classes.
• Next, compute the cumulative frequency per class and locate the class where the median would fall(i.e. n/2).
• Get the value of the l-lower limit of the median class and the value of the c- cumulative frequency of the class that precedes the median class.
• End up the calculation by substituting the values in the below formula.
• \(\text{Median}=l+\left(\frac{\frac{n}{2}-c}{f}\right)\times h\)
• Where ‘l’ is the lower limit of the median class, ‘f’ denotes the frequency of the median class, ‘h’ is the width of the median class, ‘c’ denotes the cumulative frequency of the class preceding the median class.

How to Find Median?

So far we have learnt the definition, of all types of formulas along with the application. It’s time to learn how we can find the median. For example in the beginning we saw a small example but what if a large number data set is given to us. Finding the median manually for such a case is not possible. So here comes the role of formulas.

For ungrouped data:
We first sort the given data in an ordered manner. Next, as per the observation we use the odd or even number formulas as shown:

When n is even:

\(\text{Median}=\frac{\left(\left(\frac{n}{2}\right)^{\text{th}}\text{term}+\left(\frac{n}{2}+1\right)^{\text{th}}\text{term}\right)}{2}\)
When n is odd.

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

For grouped data we use the below formula: .
\(\text{Median}=l+\left(\frac{\frac{n}{2}-c}{f}\right)\times h\)
We will discuss related examples in the solved example heading concerning these formulas for more understanding.

Example 1: Median of Ungrouped Data

The heights (in centimetres) of some students in a class are given below:

{145, 138, 150, 142, 155, 140, 148, 135}

Find the median of the data.

Solution:

Step 1: Arrange the numbers in ascending order.

Ordered data:
{135, 138, 140, 142, 145, 148, 150, 155}

Step 2: Count the total number of values.
Total number of observations, n = 8

Since n is even, we take the average of the 4th and 5th terms.

Step 3: Find the median.
4th term = 142
5th term = 145

Median = (142 + 145) ÷ 2
Median = 143.5

So, the median height is 143.5 cm.

Finding Median for Grouped Data

When data is given in groups, we follow these steps:

1. Find the total number of observations (n).
2. Find the cumulative frequency.
3. Find n/2.
4. Identify the median class (the class where n/2 lies).
5. Use the median formula for grouped data.

Example 2: Median of Grouped Data

Find the median for the following data:

Marks:
0–10, 10–20, 20–30, 30–40, 40–50

Number of students:
4, 10, 18, 6, 2

Solution:

First, find the cumulative frequency.

0–10: 4 → cumulative frequency = 4
10–20: 10 → cumulative frequency = 14
20–30: 18 → cumulative frequency = 32
30–40: 6 → cumulative frequency = 38
40–50: 2 → cumulative frequency = 40

Total number of students, n = 40

n/2 = 40 ÷ 2 = 20

The median class is 20–30 because the value 20 lies in this class.

Lower limit of median class (l) = 20
Cumulative frequency before median class (c) = 14
Frequency of median class (f) = 18
Class size (h) = 10

Median = l + [(n/2 − c) ÷ f] × h

Median = 20 + [(20 − 14) ÷ 18] × 10
Median = 20 + (6 ÷ 18) × 10
Median = 20 + 3.33
Median = 23.33

So, the median marks are 23.33.

Mean, Median and Mode 

Before we explain the properties of median, let’s quickly see an example of mean, median and mode.

Suppose we have this set of numbers:
3, 5, 7, 8, 8

Median – This is the middle value when numbers are arranged in order. Here, the middle number is 7, so the median = 7.

Mean – This is the sum of all numbers divided by the total number of numbers.
Mean = (3 + 5 + 7 + 8 + 8) / 5 = 31 / 5 = 6.2

Mode – This is the number that appears most often. In this case, 8 appears twice, so mode = 8.

Median of Two Numbers

In addition to the median, the additional two central tendencies are mean and mode. We define mean as the ratio of the summation of all observations to the total observations. On the other hand, the mode is the value in the given data-set that is repeated for the maximum time. Hence, these are called measures of central tendency.

The median for an ordered series is the digit that lies midway between the range extremes. It is generally not the same as the mean. But for a set of two numbers median=mean. This is due to the reason that the median is not the midpoint of the values but it is the value at the midpoint for a given dataset.
For example, two numbers say 4 and 8:
Mean=(4 + 8)/2 = 6
\(\text{Median}=\frac{\left(\left(\frac{n}{2}\right)^{\text{th}}\text{term}+\left(\frac{n}{2}+1\right)^{\text{th}}\text{term}\right)}{2}\)

\(=\frac{\left(\left(\frac{2}{2}\right)^{\text{th}}\text{term}+\left(\frac{2}{2}+1\right)^{\text{th}}\text{term}\right)}{2}\)

\(=\frac{\left(\left(1\right)^{\text{th}}\text{term}+\left(2\right)^{\text{th}}\text{term}\right)}{2}\)

\(=\frac{\left(4+8\right)}{2}=6\)

Properties of Median

The median is the middle value of a set of numbers when they are arranged in order. It has some important properties that help us understand data better. Here are the main properties:

1. Middle Value
   The median divides the data into two equal parts. Half of the numbers are smaller than the median, and half are larger. For example, in the set {2, 5, 7, 10, 12}, the median is 7. Two numbers are below it (2 and 5), and two numbers are above it (10 and 12).
2. Not Affected by Extreme Values
   Unlike the mean, the median does not change much if there is a very large or very small number in the set. For example, in {2, 5, 7, 10, 1000}, the median is still 7, even though 1000 is very big. This makes the median useful for data with extreme values.
3. Unique Value
   For any set of numbers, the median is a single value. If the number of observations is odd, it is the exact middle number. If it is even, it is the average of the two middle numbers.
4. Position Based
   The median depends on the position of numbers, not their exact values. For example, in {1, 3, 5, 7, 9}, the median is 5, but even if we change the numbers to {2, 4, 6, 8, 10}, the middle position still gives the median as 6.
5. Always Lies Between Minimum and Maximum
   The median is always greater than or equal to the smallest number and less than or equal to the largest number. It cannot go beyond the range of the data.
6. Divides the Data Symmetrically
   If the data is symmetric, the median is equal to the mean. In asymmetric data, it still splits the data into two equal halves in terms of count, not value.
7. Useful in Skewed Data
   The median gives a better idea of the ‘central’ value when data is skewed. For example, in income data, a few very high salaries can make the mean too large, but the median shows the typical income.

Application of Median Formula

As per the definition, the median is the central value of a provided data set. Whether it be grouped or ungrouped data set in statistics there are formulas for them as discussed in the previous section along with the definitions and related terms. Moving forward let us look at some of the related applications for the same.

• It is operated in many real-life circumstances, like bankruptcy law, where one can claim default if the individual income is below the median income.
• This is further useful for representing things like wages/earning, where large figures can throw off the mean.
• This is also applied when the data set comprises very high and low values of grouped and ungrouped data sets.
• These are applied even in the case of a historical data set(data set that arrives over time).

Important Notes on Median

• The median is the middle value of a group of numbers. It shows the centre of the data, so it is also called a positional average.
• To find the median, the data must first be arranged in order. The numbers can be written from smallest to largest or from largest to smallest. Only after arranging the data can the middle value be found.
• While finding the median, not all values are used in the calculation. Only the position of the middle number matters, not the total of all numbers.
• The median is not affected by very large or very small values. Even if one value is much higher or lower than the rest, the median usually stays the same. This makes the median useful when the data has extreme values.

Solved Examples on Median

Throughout the article, we learnt about the definition, formulas for grouped and ungrouped data and also how we can find the median that is one of the measures of central tendency using the formulas. The only thing we are left with is the solved examples related to the topic which is important for exam purposes. Let us proceed with the same.

Solved Example 1: Find the median of 8, 13, 16, 7, 21, 9, 5, and 11.

Solution: First let us arrange the list in ascending order (or descending order)

5, 7, 8, 9, 11, 13, 16, 21

Total number of terms n=8(even)

Median=\({({n\over2})^{th} term+{({n\over2}+1})^{th} term\over{2}}\)

=\({1\over{2}}[5^{th} term+4^{th} term]={9+11\over{2}}\)

=10

Hence, the median of the given list of numbers is 10. 

Solved Example 2: The following list of numbers is arranged in the ascending order of their values:
19, 23, 29, 30, 32 x-10, x-8, x-5, 54, 66, 70, 78.
If their median is 42, find the value of x. 

Solution: Number of terms n=12 (even)

Therefore, 

Median=\({({n\over2})^{th} term+{({n\over2}+1})^{th} term\over{2}}\)

Median=\({({12\over2})^{th} term+{({12\over2}+1})^{th} term\over{2}}\)

42=\({1\over{2}}[6^{th} term+7^{th} term]\)

84=x-10+x-8

2x=102

x=51

Also, read about covariance here.

In the case of tabulated data, we need to first calculate the cumulative frequency, to find the median of the given data. After calculating cumulative frequency, apply the median formula to find the correct answer from the table accordingly. 

Solved Example 3: The weights of 55 children in a class were recorded, to the nearest kg, as follows:

Calculate the median weight of the class. 

Solution: For tabulated data, we have to first calculate the cumulative frequency (c.f.). The c.f. of the first observation will remain the same. For the second observation, we will add the frequency of the first observation and the frequency of the second observation. The result of this addition will be added to the frequency of the third observation, to make the c.f. for the third observation. Repeat the process for the rest of the observations. 

The total number of children (n)=55, which is odd, 

Therefore, 

Median=\(({n+1\over{2}})^{th} term\)

=\(({55+1\over{2}})^{th} term=28^{th} term\)

Median weight =weight of 28thchild

In the above table, look at the column of cumulative frequency, and observe that the weight of each child from 21st child to 35nd child is 54kg.

Therefore, 

The weight of the 28thchild = 54kg 

Hence, 

The Median weight = 54kg 

Learn more about the Geometric Mean here.

Solved Example 4: The heights of 46 children in a class were recorded, to the nearest cm, as follows: 

Calculate the median height of the class. 

Solution: First calculate the cumulative frequency of the given data as follows:

The total number of children (n)=46, which is even, 

Therefore, 

Median=\({({n\over2})^{th} term+{({n\over2}+1})^{th} term\over{2}}\)

Median=\({({46\over2})^{th} term+{({46\over2}+1})^{th} term\over{2}}\)

=\({1\over{2}}[{23^{rd} term+24^{th} term}]\)

From the cumulative frequency column, we observe that the height of the 23rd child is 150cm and the height of each child after the 23rdchild to 34thchild is 152cm. So the height of 24thchild is 152cm. 

Hence, 

Median=(150+152)/2=302/2

=151cm

Hence, the median height of the class is 151 cm. 

Hope you have understood the concept of Median by now. You can check out other related articles as well. If you have any issues with this topic or any other topic, then you can contact us. You may download the Testbook App, for free, to access study material for your preparation for competitive exams.