## Average Calculator

#### Result Summary:

 Type Result Given set of numbers 0 Total numbers provided 0 Sum of Numbers 0 Average Value 0

### What is Average?

In mathematics and statistics, the average of a set of numbers is defined as the central value among the numbers. In other words, if we were to have a data set of some numbers, the average would help to determine a number which lies in the centre of the data set. The average is the ratio of the sum of all the values of the data set to the number of values in the data set.

The average is calculated by first adding up all the values of the data set, and then dividing by the total number of values. The average may or may not be equal to a value in the data set. For example, if we were to find the average of the numbers 4, 5 and 6, we would first find the sum = 4+5+6 = 15. Then, we divide the sum by the number of values, which is 3. Hence, the average =15/3 =5. Hence, 5 is the average of these three numbers and it lies in the centre of the data set.

Another example would be 4, 5, 6 and 7. Here, the average = (4+5+6+7)/4 = 5.5, which does not belong to the data set, but it is the central value.

The average is also known as the “mean” or “arithmetic mean”. It is a measure of central tendency. The average helps to determine the best representative number among the values in the data set, or the number which best describes all the numbers in the data set.

The average is usually denoted by $$“μ” \;or \;“\bar x ̅”.$$

### Properties of Average

Given below are some important properties of the average.

Data set – The values for which the average is to be calculated.

Sum – The total obtained after adding up all the values in the data set.

Average – This is the value obtained by dividing the sum by the total number of values in the set.

Mean/Arithmetic mean – This is the same as the average. It is a measure of central tendency in statistics.

Weighted average – When the values in the data set make unequal contributions to the data set, owing to some characteristics of the data, we compute the weighted average. In this case, each value is assigned a fixed weight, depending on how much they influence the data. To find the weighted average, all the values are multiplied with their respective weights and then added up. Finally, this sum is divided by the sum of all the weights. In the case of average (or arithmetic mean), each value in the data set has the same weight.

### Average Formula

The formula for average is: $$average \; = \; {sum \,of \,all \,the \,values \over number \,of \,values}$$

This can also be written as : $$\bar x \; = \; {1\over n} \sum_{i=1}^n x_i$$

In this formula, each xi indicates one value of the data set, and n denotes the total number of values.

The average of a data set may be positive or negative integers or real numbers.

### Characteristics of Average

• In the given data set, if the difference between each value is the same (that is, the values are equally spaced), and the number of values is an odd number, then the average is equal to the middle term of the data set. The middle term can be easily obtained if we first sort the data set in ascending/descending order before computing the average.
• If the data set is an arithmetic progression, then the average is, average=(first value+last value)/2
• For example, consider the data set 3, 6, 9, 12, 15, 18.

The average = (3 + 6 + 9 + 12 + 15 + 18) /6 = 63/6 =10.5

Since the data set is an arithmetic progression, the average can be computed faster using the above formula. Average = (first value + last value)/2 = (3+18)/2 = 21/2 = 10.5

• If each value of the data set is multiplied (or divided) by a number x, then the average also gets multiplied (or divided) by x.
• If a number x is added to (or subtracted from) each value of the data set, then the average increases (or decreases) by x.

• ### Areas of application

The average formula has numerous real-world applications. It is important for analysing mathematical or statistical data, but it also finds applications in economics and sociology. One example is the per capita income, which measures the average income of the population of a nation. If we were to find the average height of an adult Indian woman, we would need to sum up the heights of all Indian women and divide by the total number of adult women residing in India.

Given below is an example making use of the average formula.

Question: In a mathematics class, the average age of 20 boys is 19 years, and the average age of 15 girls is 17 years. What is the average age of the entire class?

Answer: First, we need to find the total age of all the boys in the class. The average age is 19, and the total number of boys is 20.

Hence, the total age of the boys = number of boys × average age = 20 × 19 = 380.

In the same way, we can find the total age of the girls, when it is given that the number of girls is 15 and their average age is 17.

Total age of the girls = number of girls × average age = 15 × 17 = 255.

We now need the average age of the entire class.

Number of students in the class (boys and girls combined) = 20+15 = 35

Total age of all the students = total age of the boys + total age of the girls = 380+255= 635.

Therefore, average age of the class = total age of all the students/ number of students = 635/35 = 18.14 ≈ 18 years.