Median

Median represents the middle value for any group. It is the point at which half the data is more and half the data is less. Median helps to represent a large number of data points with a single data point. The median is the easiest statistical measure to calculate. For calculation of median, the data has to be arranged in ascending order, and then the middlemost data point represents the median of the data.

Further, the calculation of the median depends on the number of data points. For an odd number of data, the median is the middlemost data, and for an even number of data, the median is the average of the two middle values. Let us learn more about median, calculation of median for even-odd number of data points, and median formula in the following sections.

Median is one of the three measures of central tendency. When describing a set of data, the central position of the data set is identified. This is known as the measure of central tendency. The three most common measures of central tendency are mean, median, and mode.

Median Definition

The value of the middle-most observation obtained after arranging the data in ascending order is called the median of the data. Many an instance, it is difficult to consider the complete data for representation, and here median is useful. Among the statistical summary metrics, the median is an easy metric to calculate. Median is also called the Place Average, as the data placed in the middle of a sequence is taken as the median.

Median Example

Let's consider an example to figure out what is median for a given set of data.

• Step 1: Consider the data: 4, 4, 6, 3, and 2. Let's arrange this data in ascending order: 2, 3, 4, 4, 6.
• Step 2: Count the number of values. There are 5 values.
• Step 3: Look for the middle value. The middle value is the median. Thus, median = 4.

Using the median formula, the middle value of the arranged set of numbers can be calculated. For finding this measure of central tendency, it is necessary to write the components of the group in increasing order. The median formula varies based on the number of observations and whether they are odd or even. The following set of formulas would help in finding the median of the given data.

Median Formula for Ungrouped Data

The following steps are helpful while applying the median formula for ungrouped data.

• Step 1: Arrange the data in ascending or descending order.
• Step 2: Secondly, count the total number of observations 'n'.
• Step 3: Check if the number of observations 'n' is even or odd.

Median Formula When n is Odd

The median formula of a given set of numbers, say having 'n' odd number of observations, can be expressed as:
Median = [(n + 1)/2]^th term

Median Formula When n is Even

The median formula of a given set of numbers say having 'n' even number of observations, can be expressed as:
Median = [(n/2)^th term + ((n/2) + 1)^th term]/2

Example: The age of the members of a weekend poker team has been listed below. Find the median of the above set.

{42, 40, 50, 60, 35, 58, 32}

Solution:

Step 1: Arrange the data items in ascending order.

Original set: {42, 40, 50, 60, 35, 58, 32}

Ordered Set: {32, 35, 40, 42, 50, 58, 60}

Step 2: Count the number of observations. If the number of observations is odd, then we will use the following formula: Median = [(n + 1)/2]^th term

Step 3: Calculate the median using the formula.

Median = [(n + 1)/2]^th term

= (7 + 1)/2^th term = 4^th term = 42

Median = 42

Median Formula for Grouped Data

When the data is continuous and in the form of a frequency distribution, the median is calculated through the following sequence of steps.

• Step 1: Find the total number of observations(n).
• Step 2: Define the class size(h), and divide the data into different classes.
• Step 3: Calculate the cumulative frequency of each class.
• Step 4: Identify the class in which the median falls. (Median Class is the class where n/2 lies.)
• Step 5: Find the lower limit of the median class(l), and the cumulative frequency of the class preceding the median class (c).

Now, use the following formula to find the median value.

Application of Median Formula

Let us use the above steps in the following practical illustration to understand the application of the median formula.

Illustration: There are 5 top management employees in an organization. The salaries given to the employees are $5,000, $6,000, $4,000, $8,000, and $7,500. Using the median formula calculates the median salary.

Solution: We will follow the given steps to find the median salary.

• Step 1: Sorting the given data in increasing order, $4,000, $5,000, $6,000, $7,500, and $8,000.
• Step 2: Total number of observations = 5
• Step 3: The given number of observations is odd.
• Step 4: Using median formula for odd observation, Median = [(n + 1)/2]^th term
• Median = [(5+1)/2]^th term. = 6/3 = 3^rd term. The third term is $6,000.

The median salary is $6,000.

We use a median formula to find the median value of given data. For a set of ungrouped data, we can follow the below-given steps to find the median value.

• Step 1: Sort the given data in increasing order.
• Step 2: Count the number of observations.
• Step 3: If the number of observations is odd use median formula: Median = [(n + 1)/2]^th term
• Step 4: If the number of observations is even use median formula: Median = [(n/2)^th term + (n/2 + 1)^th term]/2

Example: The height (in centimeters) of the members of a school football team have been listed below.

{142, 140, 130, 150, 160,135, 158,132}

Find the median of the above set.

Solution:

Step 1:

Arrange the data items in ascending order.

Original set: {142, 140, 130, 150, 160, 135, 158,132}

Ordered Set: {130, 132, 135, 140, 142, 150, 158, 160}

Step 2:

Count the number of observations.

Number of observations, n = 8

If number of observations is even, then we will use the following formula:

Median = [(n/2)^th term + ((n/2) + 1)^th term]/2

Step 3:

Calculate the median using the formula.

Median = [(n/2)^th term + ((n/2) + 1)^th term]/2

Median = [(8/2)^th term + ((8/2) + 1)^th term]/2

= (4^th term + 5^th term)/2

= (140 + 142)/2

= 141

For a set of grouped data, we can follow the following steps to find the median:

When the data is continuous and in the form of a frequency distribution, the median is calculated through the following sequence of steps.

• Step 1: Find the total number of observations(n).
• Step 2: Define the class size(h), and divide the data into different classes.
• Step 3: Calculate the cumulative frequency of each class.
• Step 4: Identify the class in which the median falls. (Median Class is the class where n/2 lies.)
• Step 5: Find the lower limit of the median class(l), and the cumulative frequency(c).
• Step 6: Apply the formula for median for grouped data: Median \(= l + [\dfrac {\dfrac{n}{2}-c}{f}]\times h\)

We have seen examples to find median for ungrouped data in the previous section. Here's an example to calculate median for grouped data.

Example: Calculate the median for the following data:

Solution:

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

N = \(\sum f_i\) = 70

N/2 = 70/2 = 35

Median Class is 40 - 60

l = 40, f = 35, c = 25, h = 20 \(\)

Using Median formula:

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

= 40 + [(35 - 25)/35] × 20

= 40 + (10/35) × 20

= 40 + (40/7)

Median of Two Numbers

In an ordered series, the median is the number that is mid-way between the range extremes. It is not usually identical to the mean. Let's understand how to find the median. For a set of two values, the median will be the same as the mean, or arithmetic average. For example, the numbers 2 and 10, both have a mean and a median of 6. Note that the median is the value at the mid-point of the dataset, not the mid-point of the values. The mean is the arithmetic average: (10 + 2)/2 = 6. What if we add two more numbers, say 3 and 4? The median will be 3.5, but the mean will be (2 + 3 + 4 + 10)/4 = 4.75

Important Notes on Median:

The above content to find the median has been summarized in the form of the following points.

• Median is the central value of data (Positional Average).
• Data has to be arranged in ascending/descending order to find the middle value or median.
• Not every value is considered while calculating the median.
• Median doesn't get affected by extreme points.

Thinking Out of the Box:

Now it's time to apply the learned concepts of the median. Here's a question for you!

Question: Determine the median of the first five whole numbers. In a company, for each of the 10 employees working in a service upgrade process, the number of service upgrades sold are as follows: 34, 26, 30, 21, 25, 12, 18, 20, 19, 15. Find out the median number of service upgrades sold by the 10 employees?

Related Topics on Median:

• Mode
• Average
• Mean
• Geometric Mean

What Is Meant By Median in Statistics?

The value of the middle-most observation obtained after arranging the data in ascending or descending order is called the median of the data. When describing a set of data, the central position of the data set is identified and used further in the median formula. This is known as the measure of central tendency. Median is an important measure of central tendency.

What is the Median Formula for Ungrouped Data?

The median formula for ungrouped data is totally dependent on the number of observations (n). If the number of observation is odd then the median formula is [Median = {(n + 1)/2} ^th term]. If the number of observation is even then the median formula is [Median = ((n/2)^th term + (n/2 + 1)^th term)/2]

What is the Median Formula When Number of Observations is Even?

When the number of observation is even (n = even) then the median formula is [Median = ((n/2)^th term + (n/2 + 1)^thterm)/2]

What is the Median Formula When Number of Observations is Odd?

When the number of observation is odd (n = odd) then the median formula is [Median = {(n + 1)/2} ^th term]

What is the Difference Between Mean, Median, Mode, and Range?

The mean is the arithmetic average of a given dataset. The median is the middle score in a set of given numbers. The mode is the most frequently occurring value in a set of given numbers. The range is the difference between the highest and the lowest values.

How to Find Mean, Mode, and Range?

The mode refers to the most repetitive number in the given dataset. The mean is the average of all numbers: Add all the values and divide the sum by the number of values. The range is the difference between the highest and the lowest values.

How To Calculate Median?

The median of a dataset is calculated by following two simple steps. First, arrange the given data in ascending order. Next, we need to pick the middlemost data.

• For an even number of data points, there are two middle values, and we need to take the average of those two middle values.
• For the odd number of data points, there is only one middle data point and we can take it as the median of the data.

What is Mean vs Median?

The mean of the data is the average of the data and is equal to the sum of all the data values divided by the number of data points. The median of the data is the mid-value of the data, after arranging the data in ascending order.

Is the Median Same as the Average?

The median of the data is different from the average. The median is the mid-value of the given data points, and the average is the value obtained by dividing the sum of the data values by the number of data points. But for equally spaced numbers such as 2, 4, 6, 8, 10, the median and the average are the same, that is 6.

What are the Applications of Median?

Median is an important statistical measure that helps in representing a single value for a large number of data points. As an example, the data of height or the age of the students in a class is represented by a single median value of the data.

How to Arrange Data in Ascending Order?

For arranging the data in ascending order we need to write the data starting with the smallest values and further include the data points with increasing order of their value.

Why Median is Called Positional Average?

The median falls in the middle when the data is arranged in an increasing or decreasing order. Hence the median is referred to as positional average. The median is the exact middle value, in case of odd number of data points whereas for even number of data points, the median is the average of the two middle values.

What Does n Represent In Median Formula?

In the median formula 'n' represent the number of observations. When 'n' is odd number of observations then the median formula is [Median = {(n + 1)/2} ^th term]. When 'n' is is even number of observations then the median formula is [Median = ((n/2)^th term + (n/2 + 1)^th term)/2].