Median of Numbers

In mathematics and statistics, data is a collection of numerical values or observations that help us understand patterns within. To summarise data effectively, we use specialised measured values, also known as the measures of central tendency—mean, median, and mode. Here, we will focus on the median of numbers — how to find it, its formula (for grouped and ungrouped data), and its importance in statistics and data analysis. 

1.0Introduction to Median

The median in statistics is basically the value of a dataset that lies exactly in the middle of the dataset, when arranged in an ascending or descending order. Simply put, the median tells us the central position of the data — the point that divides the dataset into two halves. 

It is unaffected by the outliers or the extreme values. That is, other values may vary widely, but they don’t affect the median unless they shift the order of the middle values.

2.0Why is the Median Important?

The median in statistics is particularly useful when working with real data. The median is not affected, unlike the mean, by very high or very low values (outliers). For instance, where the majority of students in a class got between 60 and 80 marks, but one student got 10 and another 100, the average will be inaccurate. The median is a better indication of what a "typical" score is like.

3.0How to Find the Median?

Just like other central tendencies, the Median is also calculated differently for grouped and ungrouped data. Here’s how to find the median for both types of data: 

Median of Ungrouped Data

Ungrouped data refers to a collection of individual numbers that are not sorted or organised in any specific class intervals. To find the median of ungrouped data, follow these easy steps:

Step 1: Arrange the data in order from smallest to largest (in increasing order) or largest to smallest (in decreasing order).

Step 2: Count the number of values in a data set, which is generally denoted by n.

Step 3: Check the nature of n, whether it is odd or even:

• For the odd number of values: If n is odd, the median is the middle value. Below is the formula to find the median if the number of values in a data set is odd, like 3, 5, 7, 9, 11,….
• $Median=(\frac{n+1}{2})$th value of the dataset
• For an even number of values: If n is even, the median is the average of the two middle values. Here’s how to find the median of even numbers of values: 
• $\text{Median}=\frac{{\left(\frac{n}{2}\right)}^{\text{th}}\text{ term of the dataset}+{\left(\frac{n}{2}+1\right)}^{\text{th}}\text{ term of the dataset}}{2}$

Median of Grouped Data

Grouped data is data that has been arranged into class intervals along with their corresponding frequencies. Since the exact values within each class are not given, we use a formula to estimate the median. Here’s the median formula for grouped data:

$\text{Median}=L+\left(\frac{\frac{n}{2}-\text{CF}}{f}\right)\times h$

Here:

• L is the Lower Limit of the median class
• n is the Total frequency
• CF is the Cumulative frequency before the median class
• f is the Frequency of the median class
• h is the Class width (width can be found using the formula: Upper Limit – Lower Limit)

Cumulative Frequency Table

Cumulative frequency means the total number of values added up one after another. It shows how many values are less than or equal to a certain number or class in a data set. Here’s how the frequency table column for the data set is formed: 

• Begin with the first frequency as the initial CF.
• Add each frequency to the prior CF to extend the column. For Example, if frequencies are 5, 8, 12 → CF becomes:
• • 5 → 5
  • 5 + 8 → 13
  • 13 + 12 → 25

Fun Fact: The sum of all the frequencies of a dataset is always equal to the last cumulative frequency of the last class interval. 

4.0Solved Examples for Median

Problem 1: The following table gives the frequencies of monthly incomes of workers. Find the median income.

Solution: Let’s first convert a frequency table: 

According to the newly calculated frequency table $\frac{n}{2}=25$, therefore, 

• l = 10,000
• f = 15
• CF = 12
• h = 5000

Now using the formula: 

$\text{Median}=L+\left(\frac{\frac{n}{2}-\text{CF}}{f}\right)\times h$

$Median=10,000+(\frac{25-12}{15})\times 5000$

$Median=10,000+(\frac{13000}{3})$

Median= 10,000+ 4333.33=Rs. 14,333.33

Problem 2: The table below shows the marks obtained by students in a test. Two frequencies are missing, represented by x and y. The total number of students is 60, and the median is given as 32.5. Find the values of x and y.

Solution: First, construct the cumulative frequency table:

According to the new frequency table  $\frac{n}{2}=30$, therefore: 

• l = 30
• h = 10
• f = y
• CF = 13+x

The total number of students (f) = 60

35 + x + y = 60 

x + y = 25  ………………………….(1)

Now, using the formula for median: 

$\text{Median}=L+\left(\frac{\frac{n}{2}-\text{CF}}{f}\right)\times h$

$32.5=30+(\frac{30-(13+x)}{y})\times 10$

$2.5=\frac{17-x}{y}\times 10$

y= (17-x)4

y=68-4x

4x+y=68 …(2)

Eliminating equation 2 from 1

x + y = 25

4x+y = 68

________

–3x = – 33

x = 11

Now, put this value of x in equation 1: 

11 + y = 25

y = 25 – 11 = 14

Hence, the value of x and y is 11 and 14, respectively. 

Problem 3: A group of 11 students appeared for a rapid-fire quiz. Their scores out of 20 are as follows, but one score is missing: 12, 15, 10, 13, 14, 16, 18, 11, 17, 19, x. If the median score is known to be 15, find the missing score x.

Solution: According to the question:

n = 11 

Median or the order: 15

Median of odd number of values =$(\frac{n+1}{2})$th value of the dataset 

Median = $(\frac{11+1}{2})$th=6th term

Now, arrange the given numbers in increasing order: 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, x

Let's put x in the end. But according to our observation, the 6th term should be 15, but according to our assumed order, it is 16. 

Hence, our assumption is incorrect, and x is the 6th term, which is 15, in the observation.