Mean Formula: How to Calculate, Examples with Solution

The formula of Mean: In statistics, "mean" is a measure of central tendency, calculated by summing up all the values in a dataset and dividing by the number of data points. The single numerical value calculated via Mean formula represents the average or typical value in the dataset which is a key descriptive statistic. In this article, we will be going through the meaning of mean, mean definition and formula for grouped and ungrouped data, properties of mean, solved examples and practice questions.

Mean Definition: What is Mean?

Mean Definition: According to NCERT textbook, “ Mean is the value which is derived by summing all the values and dividing it by the number of observations”.

Therefore, mean is the average of the values given in a dataset.

Arithmetic Mean

When we talk about statistical mean, there are majorly three types of mean, namely, Arithmetic mean, Geometric Mean and Harmonic mean. Arithmetic mean, the most common, is the sum of all values given divided by the count of all values. 

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Mean Formula

Mean = (sum of values in dataset/ count of values) 

In a concise way, if sum of values in a dataset is x and the count of values is n, we can write 

Mean formula = ∑x/ n

Mean Symbol

The symbol of mean is X̄ or X Bar. 

Hence, when you see the symbol X̄, it means:

X̄ = mean = (sum of values in dataset/ count of values) 

How to Find Mean

It is easy to find the mean of a given dataset by using the mean formula and substituting with the right values. However, the process to find the mean of ungrouped data is slightly different than finding the mean value of grouped data.

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How to Find Mean for Ungrouped Data?

In ungrouped data, values are provided individually and not grouped in any class or intervals.

To find the mean of an ungrouped data,

Step 1: Calculate the sum of all the values (x1 + x2 + x3 + … + xn)

Step 2: Divide by the number of values provided (n)

Solved Example

Calculate the mean of the first 5 natural numbers.

Solution:

Dataset: 1,2,3,4,5

Mean = ∑x/ n

= (1+2+3+4+5)/ 5

= 15/5

= 3

How to Find Mean for Grouped Data?

Grouped data is when a dataset is divided into class intervals. Mean for grouped data can be calculated using direct method, assumed mean method and step deviation method.

Find Mean Using Direct Method

Step 1: Create a table containing four columns: Column 1 - Class interval,  Column 2 - Frequencies (fi), Column 3 - Class marks xi (corresponding) and Column 4- xifi (corresponding product of column 2 and column 3)

Step 2: Calculate mean by the formula ∑xifi/∑fi

Here, class mark xi is the middle value of the interval. i.e., xi = (upper limit) + (lower limit) / 2.

Solved Example

Question: Calculate the mean.

Solution:

Thus, Mean = ∑fi xi/∑fi 

= 2840/60

= 47.33 

Thus, the mean weight of the given data is 47.34 Kg.

Find Mean Using Assumed Mean Method

Step 1: Create a table containing five columns: Column 1 - Class interval,  Column 2 - Frequencies (fi) Column 3 - Class marks xi, Column 4 - Corresponding deviation di = xi - Assumed mean a i.e and Column 5 - xifi (corresponding product of column 2 and column 3)

Step 2: Calculate mean by the formula ∑xifi/∑fi

Here, assumed mean A is the central value from class marks.

Solved Example

Question: Calculate the mean using the Assumed Mean Method.

Solution:

Let assumed mean A be 50.

Thus, Mean = A + (∑fi di)/∑fi 

= 50 + (-160)/60 = 47.34

Thus, mean weight of the given data using assumed mean method is 47.34 Kg.

Properties of Mean

The mean, in statistics, has several important properties:

1. The mean is sensitive to all values in the dataset.
2. The mean is the balance point of the dataset i.e, if the data values were placed on a number line, the mean would be the point at which the data distribution is balanced.
3. Mean is the average of a dataset obtained by summing up the values in a dataset and dividing by the total number of values.
4. If each observation in a dataset is increased by a uniform amount, the mean of new observations will also increase by the same amount.
5. If each observation in a dataset is decreased by a uniform amount, the mean of new observations will also increase by the same amount.

Practice problems

1. Calculate the mean of the first 10 prime numbers.
2. Calculate the mean of the first five multiples of 5.
3. Calculate the mean weight for the following bodybuilders weight data using a direct method.

4. Calculate the mean scores for the following student report data using assumed mean method.

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