How to Find Median

How to Find Median
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As the name suggests, the 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 this webpage.

Table of Contents

How to Find Median?

Can you observe some important features of data by considering only certain representatives of the data? This is possible by using the measures of central tendency or averages. A measure of central tendency describes a set of data by identifying the central position in the data set as a single value. We can think of it as a tendency of data to cluster around a middle value. In statistics, the three most common measures of central tendencies 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. Do you know how to find the median of data? Let's consider an example to figure out how to find the median. 

  • 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

Median of Data

How to Find the 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 with 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

Formula for Median

The formula to find the median depends on the kind of data and the amount of data. The following set of formulae would help in finding the median of the given data. 

Case 1: Ungrouped Data

The following steps are helpful to find the median of 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.

If n is odd, then use the formula:  

Median for Odd Data

And If n is even, then use the formula: 

Median for Even Data

Case 2: 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 median class (c).

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

Median for Grouped Data

Important Notes

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.
  • Every value is not considered.
  • It doesn't get affected by extreme points.

Think Tank 

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?


Topics Related to How to Find Median

Given below is the list of topics that are closely connected to How to Find Median. These topics will also give you a glimpse of how such concepts are covered in Cuemath. 


FAQs on How to Find Median

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 Application 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.


Solved Examples on How to Find Median

Example 1: Annie noted the number of cakes she baked every day over the past week. The numbers were: 1, 2, 2, 3, 4, 3, 3. What is the median value of the cakes she baked? 

Solution:

To find the median value of the number of caked baked by Annie we first arrange the numbers of cakes baked per day in a sequence and then pick the middlemost value.

  • The original set of number of cakes baked per day: 1, 2, 2, 3, 4, 3, 3. 
  • The Ordered Set of the cakes baked per day: 1, 2, 2, 3, 3, 3, 4.
  • Count the number of observations(n) = 7.
  • Since the number of observations is odd, median = middle value i.e. 4th value. Thus, median = 3.

Therefore, the median value of cakes she baked is 3.

Example 2: Raffle tickets were being sold during a Carnival. Jack, a Carnival worker, was taking count of the sales each hour. The number of tickets sold every hour was as follows: 130, 123, 146, 109, 112, 111 What was the median number of tickets sold? 

Solution:

To find the median number of raffle tickets sold during the carnival, we arrange the data in a sequence, and then pick the middlemost

  • The original set of the number of tickets sold every hour during the carnival: 130, 123, 146, 109, 112, 111.
  • Ordered Set of the number of tickets sold during the carnival: 109,111, 112, 123, 130, 146.
  • Number of observations= 6 i.e, even.
  • Use the median formula for even numbers, Median ={(n/2) + (n/2 + 1)}/2. Thus, median = (3rd observation + 4th observation)/2 = (112 + 123)/2 = 235/2. 

Therefore, the median number of tickets sold was 117.5.

Practice Questions on How to Find Median

Here are a few activities for you to practice. Select/Type your answer and click the "Check Answer" button to see the result.

 
 
 
 
 
 
  
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