Finding the median in a set of numbers is a common question in exams like the SAT, ACT, PSAT/NMSQT, GED, GRE, GMAT, AP Exams, PERT, Accuplacer, and the MCAT. The median is simply the middle value in a list of data when the numbers are ordered from least to greatest.

Calculating the median is straightforward if you know the right formulas and how to apply them. In this article, we’ll walk you through how to find the median for both ungrouped data—whether there’s an odd or even number of terms—and how to approach grouped data. We’ll also go over how to calculate the median when there are two numbers and discuss some real-world applications. To make sure you understand everything clearly, we’ll include solved examples to show you exactly how to handle these problems when they pop up on exams. Ready to become a median pro? Let’s get started!

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.

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.

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.

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).

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.

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

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

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

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

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

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 21stchild to 35ndchild is 54kg.  

Therefore, 

The weight of the 28thchild = 54kg 

Hence, 

The Median weight = 54kg 

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:

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 23rd child is 150cm and the height of each child after 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. 

Conclusion

In conclusion, understanding how to find the median is essential for exams like the SAT, ACT, PSAT, GRE, and more. By knowing the formulas for both ungrouped and grouped data, you can easily calculate the median, whether the number of terms is odd or even. With real-world applications, mastering this concept can help you ace exam questions and apply it in various practical scenarios. Keep practicing, and you'll be a median expert in no time!