Updated July 26, 2023
Arithmetic Mean Formula (Table of Contents)
What is the Arithmetic Mean Formula?
The term “arithmetic mean” basically refers to the mathematical average of two or more numbers. However, the method to calculate the arithmetic mean can vary based on the frequency of each variable in the data set – simple average (equally weighted) or weighted average.
The formula for the arithmetic mean for equally weighted variables can be derived by summing up all the variables in the data set and then divide the result by the number of variables. Mathematically, it is represented as,
or
Where,
 x_{i} = i^{th} variable
 n = Number of Variables in the data set
In case of unequally weighted variables, the formula for the arithmetic mean can be derived by summing up the products of each variable and its frequency and then the result is divided by the sum of the frequencies. Mathematically, it is represented as,
or
Where
 x_{i} = i^{th} variable
 f_{i} = Frequency of the i^{th} variable
Examples of Arithmetic Mean Formula (With Excel Template)
Let’s take an example to understand the calculation of Arithmetic Mean in a better manner.
Arithmetic Mean Formula – Example #1
Let us take an example of a batsman who scored the following runs in his last 10 innings during last one year: 45, 65, 7, 10, 43, 35, 25, 17, 78, 91. Calculate the batsman’s average in his last 10 innings.
Solution:
Arithmetic Mean is calculated using the formula given below
Arithmetic Mean =∑ x_{i}/ n
 Arithmetic Mean = (45 + 65 + 7 + 10 + 43 + 35 + 25 + 17 + 78 + 91) / 10
 Arithmetic Mean = 41.60
Therefore, the batsman’s average remained 41.60 runs per innings in his last 10 innings.
Arithmetic Mean Formula – Example #2
Let us take an example of a class with 45 students. Recently, there was a weekly test conducted for science in which the students were evaluated on the scale of 1 to 10. Based on the following information calculate the average marks in the test.
Solution:
Arithmetic Mean is calculated using the formula given below
Arithmetic Mean = ∑ (f_{i }* x_{i}) / f_{i}
 Arithmetic Mean= ((3 * 3) + (4 * 9) + (6 * 18) + (7 * 12) + (9 * 3)) / 45
 Arithmetic Mean = 264 / 45
 Arithmetic Mean = 5.87
Therefore, the average score of the class in the science test was 5.87.
Arithmetic Mean Formula – Example #3
Let us take an example of two data sets with two different arithmetic means. The first data set has 10 variables with a mean of 45, while the second data set has 7 variables and a mean of 42. Determine the arithmetic mean of the two data sets combined.
Solution:
Arithmetic Means of the combined data set is calculated using the formula given below
Arithmetic Mean = ((m_{1}*n_{1}) + (m_{2}*n_{2})) / (n_{1}+n_{2})
 Arithmetic Mean= (45 * 10 + 42 * 7) / (10 + 7)
 Arithmetic Mean = 43.76
Therefore, the arithmetic means of the combined data set is 43.76.
Explanation
The formula for arithmetic mean can be calculated by using the following steps:
Step 1: Firstly, collect and sort out the variables for which the arithmetic mean has to be calculated. The variables are denoted by x_{i}.
Step 2: Next, determine the number of variables in the data set and it is denoted by n in case of equally weighted variables. Otherwise, figure out the frequency of each variable and they are denoted by f_{i} and the number of variables is the summation of the frequencies.
Step 3: Finally, the formula for the arithmetic mean for equally weighted variables can be derived by adding all the variables and then the result is divided by the number of variables in the data set as shown below.
Arithmetic Mean = ∑ x_{i}/ n
However, in case of weighted average, the formula for the arithmetic mean can be derived by summing up the products of each variable and its frequency and then the result is divided by the sum of the frequencies as shown below.
Arithmetic Mean = ∑ f_{i }* x_{i }/ f_{i}
Relevance and Uses of Arithmetic Mean Formula
The concept of arithmetic mean is very simple and elementary. But it is still very important as it is often used as a statistical indicator to assess the average outcome in a data set. In fact, it allows evaluating which of the variables are better or lower than the average of the group. It is also used as a measure to represent the average value across the entire data series. Further, arithmetic mean is used in cases where geometric mean or harmonic means are less useful, such as average grade, weight, etc.
Arithmetic Mean Formula Calculator
You can use the following Arithmetic Mean Calculator
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