Average

Average in Maths is a number expressing the central value in a set of data it is calculated by dividing the sum of the values in the set by their number.  The basic formula for the average of n numbers x₁, x₂,……x_n is 

(Average)A = (x₁ + x₂ + …….. + x_n)/n

Average is the another name of mean and average in mathematics is used for finding and defining various values. In this article, we will learn about the average definition, average formula, average solved examples, and others in detail.

What is Average in Maths?

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 by the number of the values. For n terms, its average is given by first taking the sum of n numbers and then dividing them by n.

Average Definition

Average is defined as the value obtained by the dividing the sum of data by the given number of data.

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

Average Symbol

In mathematics average is another name for average and it denoted by taking bar on x, i.e. x̄. We also use Greek letter(μ) to denote the average.

Average Formula in Maths

The average in mathematics is calculated using the formula sum of values divided by 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 observation and find total number of observation(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 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 Mean?

Mean in mathematics is defined as the average of numbers. It is other name of average and it is calculated using the formula, mean of numbers is sum of values divided by total numbers of values. We have three types of mean in mathematics that are,

• Arithmatic Mean
• Geometric Mean
• Harmonic Mean

Now let’s learn about them in detail.

Arithmetic Mean

Arithmetic mean is another name of the average it is the sum of values divided by the number of values. The formula to calculate the arithmetic mean for n values x_1, x_{2, …,} x_n is,

A.M. = (n₁ + n₂ + n₃ + n₄ + … + n_n)/n

Geometric Mean

Geomrtric Mean is one of the measure of the central tendency. It is calculated by taking the n th root of the product of the all the given number. The formula to calculate the geometric mean for n values x₁, x₂, …, x_n is,

G.M. = ⁿ√(x₁.x₂…x_n)

Harmonic Mean

Harmonic mean is one of the Pythagorean means other than Arithmetic Mean and Geometric Mean. It is calculated by dividing the number of the reciprocal by the sum of the reciprocal values. Harmonic mean is always lower as compared to the geometric and arithmetic mean.

The formula to calculate the harmonic mean for n values x₁, x₂, …, x_n is,

H.M. = n/{(1/x₁) + (1/x₂) + … + (1/x_n)}

Average of Negative Numbers

Average of the negative number is simply calculated by taking the sum of the observation divided by the number of the observations. Negative numbers have no effect in finding 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

Average of two number is simply the sum of two numbers divided by 2. Suppose we are given two numbers ‘a’ and ‘b’ then its 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 Points on Average

Sum of the important tips and tricks on average has been discussed below:

Average of first n natural numbers,

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

Average of first n natural number squares,

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

Average of first n natural number cubes,

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

Average of first n natural odd numbers,

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

Average of first n natural even numbers,

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

Read More,

• Arithmetic Progression
• Geometric Progression
• Weighted Average

Examples on Average

Example 1: Find the average of the square of 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+1)(2n+1)/6 

Average = (16+1)(2×16+1)/6 

= 17 x 33 /6 

= 187/2 

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 x 9 = 783 

Average of first 5 observations = 79 

Sum of first 5 observations = 79 x 5 = 395 

Sum of 6th,7th and 8th = 92 x 3 = 276 

9th number = 783 – 395 – 276 = 112 

Example 3: Five years ago the average 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:

H + W = 25 

Sum of ages of both 5 years before = 25×2 = 50 

Today, sum of their ages is = 50 + 5 + 5 = 60 

Today avg. of H + W + C = 21 

Sum of ages of H , W and C = 21×3 = 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 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. 

Acc. 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 are 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, 61 + 34 = 95 

16 + 43 = 59 

Diff = 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

Acc. 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 with 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

Example 8: A biker travels at speed of 60 km/hr from A to B and returns with 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 x 2a /10a 

= 240/5 

= 48 

Hence, the average speed is 48 km/hr.

Practice Questions on Average

Q1. Average temp. of Monday, Tuesday, Wednesday and Thursday are 31°, and the average temp. of Tuesday, Wednesday, Thursday, and Friday are 29.5°. If the temp of Friday is 4/5 times of Monday. Find the temp of Monday.

Q2. Average age of boys in school is 13 years and of girls is 12 years. If the total number of boys is 240, then find the number of girls if the average of school is 12 years 8 months.

Q3. If the runs scored by a batsman in 5 matches are 56, 102, 23, 45 and 78. Find the average run scored by him.

FAQs on Average

1. What is Average?

Average in mathematics is defined as the mean of two numbers. It is used to find the central value of the data set. It is calculated by taking the ratio of all the observation by number of observations.

2. What is Average Formula?

The formula to calculate the average of numbers is,

Average = (Sum of Terms)/ (Number of Terms)

3. What is Average of First n numbers?

The average of first n natural is calculated below,

Sum of n natural number = n(n + 1)/2

• Average of n natural number = n(n + 1)/2n = (n + 1)/2

4. Is Average and Mean the Same?

Yes, average in mathematics is similar to the mean in mathematics. It is calculated by dividing the sum of vales by number values.

5. What is Average of Two Numbers?

Average of two numbers is found using the formula. Suppose we have two numbers ‘a’ and ‘b’. Average of two numbers,

Average = (a+b)/2

6. What is Weighted Average?

Weighted Average is average of the dataset that takes the weight of each data in a dataset

7. How to Calculate Weighted Average?

We can calculate weighted average by taking sum of product of each data with their weight and then dividing the total weight by total number of data