The three main ways of designating the average value of a list of integers in mathematics are mean, median, and mode. 

• The arithmetic mean is calculated by multiplying the sum of the numbers in the list by the number of numbers in the list. This is the most common meaning of the term “average.”
• The median is the value in the middle of a list which is ranked from smallest to largest. 
• The mode is the value in the list that appears the most frequently.

Mean : 

The average value of a data set is the value that falls between the highest and minimum values, however it is not the number in the data set.

Mean = Sum of All Data Points / Number of Data Points

Median : 

The median indicates where a data set’s midpoint is located. It’s applied in numerous real-life circumstances, such as bankruptcy legislation, where you can only file for bankruptcy if your income is below the state’s median income.

{(n + 1)÷ 2}th  is the median formula, where “n” is the number of items in the set and “th” simply indicates the (n)th number.

Mode : 

The most often occurring value in the data collection is referred to as mode. A normal data set, a group data set, and a non-grouped or ungrouped data set can all be used to determine the mode of a data set.

Mo=l+f!-fo2f1-fo-f2h

Here,

l= lower limit 

h= size of the class interval

f1=frequency of modal class

fo=frequency of class before modal class

f2=frequency of class after modal class

ALL ABOUT MEDIAN : 

The median of the data is the value of middle – most observations acquired after arranging the data in increasing order. In many cases, it is difficult to evaluate all of the data for representation, hence the median comes in handy. The median is a simple metric to calculate among the statistical summary metrics. Because the data in the middle of a sequence is regarded as the median, the median is also known as the Place Average.

Formulas to calculate Median : 

• If you have an odd number of data points to get the median for, first arrange them in order of least to largest, then count them. The median number is the one that is exactly in the middle of the range, with an equal number of numbers on either side. Or to ease it out, we can use the given formula where “n” is the total number of observations 

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

• The number of data points is even: There will not be one number in the middle if the set of numbers you’re dealing with has an even number of data points. In this scenario, the median must be calculated. To do so, arrange the numbers in ascending order, then add the two middle numbers together and divide by two.

Below is the given formula 

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

How to Calculate Median : 

The instructions below will help you apply the median calculation to ungrouped data.

1. Arrange the data in ascending or descending order in step one.
2. Count the total number of observations, which is ‘n’.
3. Determine whether ‘n’ is an even or odd number of observations.

Examples : 

Example 1: Determine the median of 10, 56, 89, 67 and 78

Sort the following in increasing order: 10, 56, 67, 78 and 89

The median is 67 because the middle number is 55.

Example 2 : 16, 24, 99, 45, 6, 77, 69, 5, 28, 10, 72, 84, 29, 18

When we put those numbers in order we have:

5, 6, 10, 16, 18, 24 ,28, 29, 45, 69, 72, 77, 84, 99

There are now fourteen numbers and so we don’t have just one middle number, we have a pair of middle numbers

In this example the middle numbers are 28 and 29

To find the value halfway between them, add them together and divide by 2:

28+29 = 57

then 44 ÷ 2 = 28.5

Median of Grouped Data : 

The median is found using the procedures below when the data is continuous and in the form of a frequency distribution.

Step 1: Count up how many observations there are in total (n).

Step 2: Determine the class size(h) and divide the data into groups.

Step 3: Compute each class’s cumulative frequency.

Step 4: Determine which class the median belongs to. (The median class is the one in which n/2 is found.)

Step 5: Determine the median class(llower )’s limit as well as the cumulative frequency of the class preceding the median class (c).

To find the median value, apply the formula below.

Conclusion: 

The median of the data is the value of the middle-most observation obtained after arranging the data in ascending or descending order. When describing a set of data, the data set’s middle position is determined and used in the median formula. This is referred to as the central tendency measure. The median is a useful indicator of central tendency.