Average in Maths

An average is the middle value of a group of numbers. The average is calculated by adding up all the numbers in a group and dividing the total by how many of numbers there are.

The image below shows three rows of apples with 6, 11, and 7 apples, and if we take the average of all three rows, we get 8 apples in each row.

The average in mathematics is defined as the central value of the given data set. It is the ratio of the sum of all the values to the number of values. For n terms, its average is given by first taking the sum of n numbers and then dividing them by n.

Average = Sum of Values/Number of Values

In this article, we'll explore what average is in maths, including its symbol, average formula in maths, and how to calculate the average. We'll also cover step-by-step instructions for calculating the average and several detailed examples.

Average in Maths Definition

An average is a single number expressing a set of data. It is calculated by dividing the sum of the values in the set by their number, also called the arithmetic mean. The basic formula for the average of n numbers x₁, x₂,......x_n is 

(Average) = (x₁ + x₂ +........ + x_n)/n

it is denoted x̄ (read as "x bar"). We also use the Greek letter "μ" to denote the average.

Average Formula

The average in mathematics is calculated using the formula sum of values divided by the number of values. Hence, the average formula is given as

Average = Sum of Values/Number of Values

For given n numbers x₁, x₂, x₃ ,….., x_n the average is given by the formula,

Average = (x₁+ x₂ + ... + x_n)/n

How to Calculate Average?

Study the following steps to find the average of various numbers

• Step 1: Note all the observations and find the total number of observations (say n)
• Step 2: Find the sum of all observations.
• Step 3: Divide the sum obtained in Step 2 by the number of observations (n)
• Step 4: Simplify to obtain the required value of the Average.

Example: Find the average of 3, 4, 7, 8, 10, and 12.

Solution:

Given values,

3, 4, 7, 8, 10, 12

Number of Observations = 6
Sum of Observations = 3 + 4 + 7 + 8 + 10 + 12 = 34
Average = 34/6 = 5.67

What is Average Used For?

The average is used to represent a large amount of data with a single number. It helps us find the central value of a data set.

Some practical applications of averages are:

1. Calculating the average time of commute to work or school can help you plan your schedule.
2. Calculating the batting average in cricket helps in assessing a batsman's performance.
3. Calculating average customer reviews before buying new things.
4. Calculating average household income, average unemployment rate, or average inflation rate to understand economic trends.
5. Calculating average daily sales of a product to stock the right amount.

Average Calculator

This tool allows users to quickly and easily calculate the average of a set of numbers. By entering a list of values, the calculator will automatically compute the mean, helping you to better understand and analyze your data.

What is Mean?

Mean in mathematics is the measure of central tendency. It is also called the average or arithmetic mean. It is calculated by dividing the sum of values by the total number of values.

There are three types of mean in mathematics:,

Types of Mean

Three types of means are:

Difference Between Mean and Average

This table provides a structured comparison of mean and average :

Average of Negative Numbers

The average of the negative numbers is simply calculated by taking the sum of the observations divided by the number of observations. Negative numbers do not affect the finding of the average of the negative numbers. This is explained by the example,

Example: Find the average of -8, -4, 0, 4, 8

Solution:

Given, -8, -4, 0, 4, 8

Number of Observations = 5
Sum of Observations = (-8) + (-4) + 0 + 4 + 8 = 0
Average = 0/5 = 0

Average of Two Numbers

The average of two numbers is simply the sum of the two numbers divided by 2. Suppose we are given two numbers 'a' and 'b', then their average is calculated as,

Average = (a+b)/2

Example: Find the average value of 80 and 100

Solution:

Given,

• a = 80
• b = 100

Average = (a + b)/2
= (80 + 100)/2 = 180/2
= 90

Important Formulas on Average

Some of the important tips and tricks to solve average questions are mentioned below. These formulas will help students and will be useful in boards and competitive exams.

Average of the first n natural numbers:

• Sum of first n natural numbers = n(n + 1)/2
• Average of first n natural numbers = (n + 1)/2

Average of the first n natural number squares:

• Sum of square of first n natural numbers = n(n + 1)(2n + 1)/6
• Average of square of first n natural numbers = (n + 1)(2n + 1)/6n

Average of the first n natural number cubes:

• Sum of cube of first n natural numbers = [n(n + 1)/2]²
• Average of cube of first n natural numbers = n(n + 1)²/4

Average of the first n natural odd numbers:

• Sum of first n natural odd numbers = n²
• Average of first n natural odd number = n

Average of the first n natural even numbers:

• Sum of first n natural even numbers = n(n + 1)
• Average of first n natural even numbers = n + 1

Also Check:

• Important Average Formulas
• Arithmetic Progression
• Geometric Progression
• Weighted Average

Average Formula in Excel

To calculate the average without using the AVERAGE function, we can sum all numeric values and divide by the count of numeric values. We can use SUM and COUNT functions like this:

= SUM(A1:A5)/COUNT(A1:A5) ( Average Calculation )

Here:

• SUM function is used to add multiple numeric values within different cells and
• COUNT function to count the total number of cells containing only numbers.

Also Check:

How to Calculate Average in Excel ( Formula & Examples)

Solved Examples on Average in Maths

Here are some numerical examples on average with solutions. These solved examples will help students understand and practice the concept of average.

Also Check:

Tricks to Solve Average Questions

Example 1: Find the average of the squares of the first 16 natural numbers.

Solution:

Sum of square of first n natural number = n(n+1)(2n+1)/6 
Avg. of square of first n natural number = n(n+1)(2n+1)/6n

Average = 16(16+1)(2x16+1)/6 x 16 
= 16 x 17 × 33 /96 
= 8976/96 = 93.5

Example 2: The average of 9 observations is 87. If the average of the first five observations is 79, and the average of the next three is 92. Find the 9th observation. 

Solution:

Average of 9 observations = 87 
Sum of 9 observations = 87 × 9 = 783 

Average of first 5 observations = 79 
Sum of first 5 observations = 79 × 5 = 395 

Sum of 6th,7th and 8th = 92 × 3 = 276 
9th observation = 783 - 395 - 276 = 112 

Example 3: Five years ago, the average age of the Husband and wife was 25 years; today, the average age of the Husband, wife, and child is 21 years. How old is the child? 

Solution:

Five years ago, the average age of the Husband and Wife was 25 years
Sum of their ages five years ago = 25x2 = 50 

Today, current total age = 50 + 5 + 5 = 60 
The average age of the Husband, Wife, and Child today si 21 years,
Total sum of ages = 21 x 3 = 63

Age of child = 63 - 60 = 3 years

Example 4: There are 42 students in a hostel. If the number of students increased by 14. The expense of the mess increased by Rs 28 per day. While the average expenditure per head decreased by Rs 2. Find the original expenditure. 

Solution:

Total students after increment = 42 + 14 = 56 
Let the expenditure of students is A Rs/day. 
Increase in expenditure Rs 28/day. 

According to question 

42A + 28 = 56(A - 2) 
42A + 28 = 56A - 112 
14A = 140 
A = 10 

Hence, the original expenditure of the student was Rs 10/day. 

Example 5:The average of 200 numbers is 96, but it was found that 2 numbers, 16 and 43, were mistakenly calculated as 61 and 34. Find his correct average. It was also found that the total number is only 190. 

Solution:

Average of 200 numbers = 96 
Sum of 200 numbers = 96 x 200 = 19200 

Two numbers mistakenly calculated as 61 and 34 instead of 16 and 43. 
So,
Incorrect sum added = 61 + 34 = 95 
Correct sum added = 16 + 43 = 59 
Diffrence = 95 - 59 = 36 

So, Actual sum of 200 numbers = 19200 - 36 = 19164 
Total numbers are also 190 instead of 200

So, correct average = 19164/190 = 100.86

Example 6: A batsman scored 120 runs in his 16th innings; due to this, his average increased by 5 runs. Find his current average. 

Solution:

Let the average of 15 innings is A
New average = A + 5
Total runs in 16 innnings = 15A + 120

According to question 

15A + 120 = 16(A + 5) 
15A + 120 = 16A + 80 
A = 40 

Hence, current average of the batsman is (40 + 5) = 45 

Example 7: There are three natural numbers if the average of any two numbers is added to the third number 48,40, and 36 will be obtained. Find all the natural numbers. 

Solution:

Let a, b and c are the numbers

Given 
(a+b)/2 + c = 48

=> a + b + 2c = 96 .........(1) 
(b+c)/2 + a = 40 

=> 2a + b + c = 80 ..........(2) 
(c+a)/2 + b = 36 

=> a + 2b + c = 72 ..........(3) 
Add (1)(2)(3), we get 
4(a + b + c) = 248 
a + b + c = 62 

From 1, 2, and 3
(a+b+c) + c = 96 
62 + c = 96 
c = 34

a + (a+b+c) = 80 
a + 62 = 80 
a = 18

b + (a+b+c) = 72 
b + 62 = 72 
b = 10

Therefore value of a = 18, b = 10 and c = 34

Example 8: A biker travels at a speed of 60 km/hr from A to B and returns at a speed of 40 km/hr. What is the average speed of the total journey? 

Solution:

Let a is the distance between A and B
Total distance travel in journey = 2a 

Time to travel from A to B = Distance/speed = a/60 
Time to travel from B to A = Distance/speed = a/40 
Total time of journey = a/60 + a/40 

Average speed = Total distance/Total time 
=2a / (a/60 + a/40) 
=240 × 2a /10a 
= 240/5 
= 48 

Hence, the average speed is 48 km/hr.

Practice Questions on Average in Maths

Question 1: The Average temp on Monday, Tuesday, Wednesday, and Thursday is 31°, and the average temperature on Tuesday, Wednesday, Thursday, and Friday is 29.5°. If the temperature on Friday is 4/5 times that of Monday. Find the temperature for Monday.

Question 2: The average age of boys in school is 13 years, and that of girls is 12 years. If the total number of boys is 240, then find the number of girls if the average school age is 12 years 8 months.

Question 3: If the runs scored by a batsman in 5 matches are 56, 102, 23, 45, and 78. Find the average run scored by him.

Question 4: The average score of a class of 50 students is 72. If the average score of 30 boys in the class is 78, what is the average score of the girls?

Question 5: The average height of 8 men is 5.8 feet. If a new person joins the group, the average height becomes 5.9 feet. What is the height of the new person?

Question 6: A batsman has an average score of 48 runs in 20 innings. How many runs must he score in the next inning to raise his average to 50 runs?

Question 7: The average of 5 consecutive odd numbers is 41. Find the smallest of these numbers.

Question 8: The average weight of a group of 15 boys is 60 kg. If 5 boys leave the group and the average weight of the remaining boys increases by 2 kg, what was the average weight of the boys who left?

Question 9: A car travels 150 km in the first hour, 120 km in the second hour, and 180 km in the third hour. What is the average speed of the car over the 3 hours?

Question 10: The average salary of 5 employees in a company is $4000. If one new employee joins and the average salary increases by $500, what is the salary of the new employee?

Summary - Average in Maths

The average, also known as the arithmetic mean in mathematics, is a measure used to represent the central tendency of a set of numerical data. It is calculated by summing up all the values in the dataset and then dividing the sum by the total number of values. This provides a single value that is representative of the entire dataset, offering insights into its typical or central value.