Mean Median Mode Formula

The mean median mode formula tells us the measures of central tendency. In this article, we will learn about the mean median mode formula along with solved examples.

What is the Mean Median Mode Formula?

Mean is also known as the arithmetic mean of the given data. Median is the middlemost value of the given grouped data if the data is grouped and 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. 

Mean Formula

The mean formula 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 = \(\dfrac{Σ f_{x}} {N}\)
Where,

• x = the mean value of the set of given data.
• f = frequency of the individual data
• N = sum of frequencies 

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

Median Formula

For finding the median 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.  If the number is even, find the two middle terms using the formula n/2 and (n/2) + 1. 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

Similarly, we have median formula for grouped data. The median formula for grouped data is given as,

Median = \(L_{m} + [\dfrac{\dfrac{n}{2} - F} {f_{m}}]i\)

Where,

• n = the total frequency.
• F = The cumulative frequency of before class median
• \(f_{m}\) = the frequency of the class median
• i = the class width
• \(L_{m}\) = the lower boundary of the class median

Mode Formula

Value or a number that appears most frequently in a data set is a Mode. In cases where we need to find the most occurred value, we find the mode vale for the set of given data. For data without any repeating values, there is no mode at all. The mode value depends on the given dataset. Mode for grouped data is found using the following mode formula.

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 preceding the modal class.
• '\(f_2\)' is the frequency of the class succeeding the modal class.

Examples Using Mean Median Mode Formula

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

Example 1: Using 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 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 center 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 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 Observation} ÷ {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.