Mean Median Mode Formula

The mean median mode formula helps us to find the measures of central tendency. The mean, median, and mode for grouped and ungrouped data are calculated by the formulas below:

Get into this article, to learn about the mean median mode formulas and to know what each variable in the above table stands for.

What is the Mean Median Mode Formula?

Let us first understand what does mean, median, and mode mean in statistics.

• Mean is also known as the arithmetic mean of the given data.
• Median is the middlemost value of the given ungrouped data if the data is arranged in ascending order.
• Mode is the value that appears most in the data.

The Mean, median, and mode formulas are explained below separately for the group of data.

To understand what each of the variables in the above formulas represent, go through the explanation below.

Mean Formula

The mean formula for ungrouped data is defined as the sum of the observations divided by the total number of observations. This will be helpful in solving a majority of the topics related to the arithmetic mean. The mean formula of given observations can be expressed as,

Mean Formula = (Sum of Observations) ÷ (Total Numbers of Observations)

Similarly, we have a mean formula for grouped data. Which is expressed as

x̄ = Σ f x /n

where,

• x = midpoint of each class
• f = frequency of the respective class
• n = total frequency

Hence, the average of all the data points is termed as the mean.

Median Formula

For finding the median of ungrouped data we need to arrange the data either in ascending order or descending order. Now after arranging the data, get the total number of observations in the data.

• If the number is odd, the median is ((n+1)/2)^th item in the given set.
• If the number is even, find the two middle terms which are (n/2)^th and ((n/2) + 1)^th values. Find the mean of these 2 middle terms. Thus the median formula for even numbers is given as: Median = ((n/2)^th term + ((n/2) + 1)^th term)/2

We have the median formula for grouped data defined as follows. To apply this formula, first compute the median class. The median class is the class containing (n + 1)/2^th item.

Median = l + [(n/2−c)/f] × h

Where,

• l = lower limit of the median class
• n = Total frequency
• c = Cumulative frequency of class before the median class
• f = Frequency of the median class
• h = Class width (Upper limit - Lower limit)

Mode Formula

For ungrouped data, the value or a number that appears most frequently in a data set is a mode. For data without any repeating values, all data values are considered to be modes.

Mode for grouped data is found using the following mode formula. Before applying this formula, first find the modal class. Modal class is the class with the greatest frequency.

Mode formula = L + h \(\dfrac{(f_{m} - f_{1})}{(f_{m} - f_{1}) + (f_{m} - f_{2})}\)

where,

• 'L' is the lower limit of the modal class.
• 'h' is the size of the class interval.
• '\(f_m\)' is the frequency of the modal class.
• '\(f_1\)' is the frequency of the class that comes just before the modal class.
• '\(f_2\)' is the frequency of the class that comes just after the modal class.

☛ Also Check:

• Mean Mode Median Worksheets
• Mean Median Mode Calculator

Examples on Mean Median Mode Formula

Let us solve some interesting problems using the mean median mode formula.

Example 1: Using the mean mode median formula find the mode of the data {14, 16, 16, 16, 17, 16, 18}

Solution:

Since there is only one value repeating itself, it is a unimodal list.

According to the mean median mode formula,

Mode = 16

Answer: Mode of {14, 16, 16, 16, 17, 16, 18} is 16.

Example 2: The ages of the members of a community centre have been listed below: {42, 38, 29, 37, 40, 33, 41}. Using the mean median mode formula, calculate the median of the given data.

Solution:

To find the median of the given set.

Given: Set of ages for different members: {42, 38, 29, 37, 40, 33, 41}
Arranging the set in ascending order: {29, 33, 37, 38, 40, 41, 42}
Number of observations, n = 7 (odd)
Using the median formula,
Median = (7 + 1)/2 ^th term
= 4^th term
= 38

Answer: Median of the given data = 38

Example 3: Using the mean median mode formula find the mean of the first five natural numbers, using the mean formula.

Solution:

The first five natural numbers = 1, 2, 3, 4, 5

Using mean median mode formula

Mean = {Sum of Observations} ÷ {Total numbers of Observations}

Mean = (1 + 2 + 3 + 4 + 5) ÷ 5 = 15/5 = 3

Answer: The mean of the first five natural numbers {1, 2, 3, 4, 5} is 3.