What is Average ? How to Calculate the Average of Numbers Using Average Formula?

In this article, you will learn what average maths is and how it is calculated. You will also learn the average formula.

What is Average?

When a single value represents a set of numbers or objects, it is called average. In mathematics, the average refers to the middle value of a set of data obtained by dividing the sum of all values of the set by the total number of values.

It is useful when there are changing values in a set of numbers because it helps represent the data set with just one single value. Thus, the mean of the average is to find the mean value of a data set. Therefore, sometimes average is also named as mean.

Average Formula

The average of any data set of numbers is calculated by adding all values and dividing the result by the number of values. The formula of average can be written as:

For example in the data set {2,4,5,7,8,9}, the average can be calculated as:

$$ Average \;=\; \frac{2+4+5+7+8+9}{6} $$ $$ Average \;=\; 356 $$ $$ Average \;=\; 5.83 $$

Which is the middle value of the given set of numbers.

How do you Calculate the Average Value?

Consider the group of n numbers x_n=x₁,x₂,x₃,…,x_n-1,xn the average of xn can be calculate by following formula:

$$ Average \;=\; \frac{x_1,x_2,x_3,…,x_{n-1},x_n}{n} $$

In more generalised form,

$$ Average \;=\; \sum \frac{x_n}{n} $$

Where,
xn = is the sum of all values.
n= is the number of values.

Let’s see the following examples to understand how to calculate average.

So the average marks of the students are 55.

Related Formulas

The following formulas are related to average formula,

1. Mean
2. Median
3. Arithmetic mean

Mean

Generally the mean is calculated by dividing the sum of all values to the total number of values. The mean formula is given by;

$$ Mean \;=\; \frac{ \sum x_n }{n} $$

The same formula is used for average that’s why it is related to average.

Median

The median of a data set can be calculated by ordering the number from smallest to largest and selecting the middle value. There are two cases in the median formula.

1. When there is the odd number of data values, the median formula is: $$ Median \;=\; {\frac{n+1}{2}}^{th} term $$
2. When there is even number of data values, $$ Median \;=\; \frac{ (\frac{n}{2})^{th}term \;+\; (\frac{n}{2} + 1)^{th}term}{2}$$

Arithmetic Mean

The arithmetic mean is the ratio of all observations to the total number of observations in a sample data. In statistics, it refers to the average of the observations. The arithmetic mean formula can be written as:

$$ \bar X \;=\; \sum \frac{f_i}{N} $$ WhereX is the mean value of the given data set, f is the frequency of the individual data and N is the sum of all frequencies.

FAQ’s

What is the Formula to Find an Average?

The formula to find the average of any data set is:

$$ Average \;=\; \frac{Sum \; of \; all \; Values}{Total \; Number \; of \; Values} $$

What is the Average Value Formula in Calculus?

In calculus, the average value formula is used to find the average of a function. It is obtained by taking the integral of the function over the given interval and dividing by the length of the interval. The formula of average of a function over the interval [a,b] can be written as:

$$ f_{avg} \;=\; \frac{1}{b-a} \int_a^b f(x)dx $$

Where
b-a = is the length of the interval.