Average in Maths: Definition, Symbol, Formula & Solved Examples

All of us are somewhat familiar with the concept of Average. Questions related to the Average Quantitative Aptitude section is one of the easiest sections but sometimes the questions are framed in a tricky way so it is very necessary to be well aware of all the key concepts related to the Averages section. We are going to cover the key concepts of the Averages section along with the various types of questions, important formulas along with various tips and tricks. We have also added a few solved examples that candidates will find beneficial in their exam preparation. Read the article thoroughly to clear all the doubts regarding the same.

Average Definition

Average is the mean value which is equal to the ratio of the sum of the number of a given set of values to the total number of values present in the set. We apply an average in various areas of real life. So, Average is defined as the sum of the observations divided by the number of observations.

Average = Sum of observations / Number of observations

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Symbol

Averages can basically be defined as the mean value which can be expressed as x bar (x̄), it is also known as the Average Symbol. The average symbol can also be denoted by μ.

The method to calculate the average of given numbers is very simple. All we need to do is add up all the values and then divide the total by how many values there are. So, the formula for average in Maths can be written as:

Average = Sum of observations / Number of observations

For example, if we are given n numbers like x₁, x₂, x₃, ..., xₙ, then the average (also called the mean) of these numbers will be:

Average = (x₁ + x₂ + x₃ + ... + xₙ) / n

How to Calculate Average?

Finding the average of a set of values is quite easy. You just need to add up all the numbers and then divide the total by how many numbers there are.

You can calculate the average by following these three basic steps:

Step 1: Add the Numbers

Start by adding all the numbers in the list together to get the total sum.

Step 2: Count the Numbers

Next, count how many numbers are in the group — this is the number of observations.

Step 3: Divide to Find the Average

Finally, divide the total sum by the number of values to get the average.

Let’s See an Example:

Suppose we have a set of numbers: 20, 21, 23, 22, 21, 20, 23
We need to find their average.

Using the formula:
Average = (Sum of all values) ÷ (Number of values)

So,
Average = (20 + 21 + 23 + 22 + 21 + 20 + 23) ÷ 7
Average = 150 ÷ 7
Average = 21.42

Average of Negative Numbers

If there is a negative number present in a given list of numbers then candidates can calculate the average by using the same formula mentioned above

Average = Sum of the observations / Number of observations

Types of Averages in Simple Terms

1. Arithmetic Mean:

This is the most common type of average. To calculate it, you add up all the numbers in a group and then divide the total by how many numbers there are.
For example, if you have 5, 10, and 15:
Arithmetic Mean = (5 + 10 + 15) / 3 = 30 / 3 = 10

2. Geometric Mean:

This average is used when you multiply numbers instead of adding them. You take the product (multiply all the numbers) and then find the nth root (where n is how many numbers there are).
For example, if you have 2, 4, and 8:
Geometric Mean = ³√(2 × 4 × 8) = ³√64 = 4

3. Harmonic Mean:

This type of mean is used when the numbers are in the form of rates or ratios. To find it, you divide the number of values by the sum of the reciprocals (1 divided by each number).
For example, for 2, 3, and 4:
Harmonic Mean = 3 / [(1/2) + (1/3) + (1/4)] = 3 / (13/12) = 2.77 (approx)

Difference Between Mean and Average

The main difference between mean and average are given below:

Average	Mean
A general term for a central value of a data set. Includes mean, median, and mode. Common in everyday language. Less precise, can refer to different measures of central tendency. "Average score" could be mean, median, or mode.	A specific type of average, typically the arithmetic mean. Refers only to the arithmetic average (unless otherwise specified) Used in mathematics and statistics more formally. More precise, usually indicates a specific calculation. "Mean score" always means the sum divided by the number of values

To get better clarity on this, read and understand Mean, Median, Mode.

Tips and Tricks to Solve Questions Based on Averages

Students can find different tips and tricks for solving the questions related to Averages from below:

Tip 1: Average = Sum of the observations / Number of observations

Tip 2: The average of any consecutive series is the middle term (median).

⇒ For example 8, 10, 12 in this series middle term is 10 which is also the average of the series.

Required percentage = [(12 – 8)/8] × 100 = 50%

Average Formula in Maths

Students can find the important average formulas to solve questions based on Averages:

• Average of first n natural numbers = (n+1) / 2
• Average of squares of first n natural numbers = (n+1) (2n+1) / 6
• Average of cubes of first n natural numbers = n (n+1)2 / 4
• Average of first n even numbers = n + 1
• Average of squares of first n even numbers = 2 (n+1) (2n+1) / 3
• Average of cube of first n even numbers = 2n (n+1)2
• Average of first n odd numbers = n
• Average of squares of first n odd numbers = (2n+1) (2n-1) / 3
• Average of cube of first n odd numbers = n(2n2 – 1)

Solved Examples on Averages

Example 1: The average age of a cricket team of eleven players is 27 years. If two more players are included in the team the average becomes 26 years, then the

average age (in years) of the two included players is∶

Solution: Average age of eleven players is = 27 years

⇒ Sum of age of 11 players = 27 × 11 = 297 years

⇒ If two players are included in the team, then average age of 13 players = 26 years

⇒ Sum of age of 13 players = 13 × 26 = 338 years

⇒ Sum of age of two players (included) = 338 – 297 = 41 years

Hence, Average age of two players (included) = 41/2 = 20.5 years

Example 2: In a group of 25 people, the average expenditure per person is Rs. x. If one person spends Rs. 48 and another person spends Rs. x (same as the average), and the remaining 23 people each spend Rs. 30, find the average expenditure x and the total money spent by all 25 people.

Solution: Total expenditure = 23 × 30 + x + 48

Average expenditure = Total expenditure / 25

So,
⇒ 25x = 23 × 30 + x + 48
⇒ 25x = 690 + x + 48
⇒ 25x - x = 738
⇒ 24x = 738
⇒ x = 32

Total money spent by all = 25 × 32 = Rs. 800

Example 3: The average of 3 consecutive even numbers is 10, then the third number is by what percent more than the first number?

Solution: Let the first number be x.

So, the second and the third number will be (x + 2) and (x + 4) respectively. According to the question

⇒ x + x + 2 + x + 4 = 10 × 3

⇒ x = 8

The first number is 8.

⇒ The second number be = 8 + 2 = 10

⇒ The third number be = 8 + 4 = 12

Hence, Required percentage = [(12 – 8)/8] × 100 = 50%

Example 4: The averages of the first 15 odd numbers are by what percent less/more than the average of the first 15 even numbers?

Solution: As we know,

⇒ Average of the first n odd numbers = n

⇒ Average of the first n even numbers = n + 1

⇒ Average of the first 15 odd numbers = 15

⇒ Average of the first 15 even numbers = 15 + 1 = 16

Hence, Required percentage = [(16 – 15)/16] × 100 = 6.25%

Example 5: Find the averages of cubes of the first 10 odd numbers.

Solution: The average of cubes of first n odd numbers = n (2n2 – 1)

Hence, The average of cubes of first 10 odd numbers = 10(2 × (10)2 – 1) = 10 × 199 = 1990

Example 6: A cricketer had a certain average of runs for his 43 innings. In his 44th innings, he is bowled out for no score on his part. This brings down his average by three runs. Find his new average of runs.

Solution: Concept:

Total runs = Number of wickets × Average

Averages of bowler = Total runs / Number of wickets

Let the average of runs be x

Total runs = Number of innings × Average

⇒ 43 × x = 44 × (x – 3)

⇒ x = 132

New average = x – 3 = 132 – 3 = 129

Example 7: The average age of a family of 6 members is 25 years. If the age of the youngest member of the family is 8 years, then find the average age of the members of the family just before the birth of the youngest member.

Solution: Total age of a family of 6 members = 6 × 25 = 150 years

⇒ Total age of a family of 6 members before 8 years = 150 – (8 × 6) = 102 years

Hence, The average age of the family just before the birth of the youngest = 102/(6 – 1) = 20.4 years

Example 8: In a hostel, 52 students are living. If 18 students are joined in this hostel then average expenditure will be 3 rupees less whenever total expenditure 510 rupees will be increased. Find the total expenditure initially.

Solution: Let the averages expenditure of initially students be x,

⇒ 52x + 510 = (52 + 18) × (x – 3)

⇒ 18x = 720

⇒ x = 40

Hence, Total expenditure of 52 students = 52 × 40 = 2080

If you are checking Averages article, check related maths articles:
Harmonic Mean Formula	Number System
Trigonometry	Statistics
Geometric Mean	Partnership
Harmonic Mean	Mean Median Mode

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