Arithmetic Mean Questions

The Arithmetic Mean is a fundamental concept in statistics and mathematics, often referred to as the average. It represents the central tendency of a set of values and helps to simplify data analysis. In this blog, we’ll explore a variety of Arithmetic Mean Questions, provide detailed answers, offer PDFs for practice, and cover Arithmetic Mean Questions for grouped data.

1.0What is an Arithmetic Mean?

The Arithmetic Mean, commonly known as the average or simply the mean, is calculated by summing all the values in a dataset and dividing that sum by the total number of items in the set. When the numbers are evenly distributed, the arithmetic mean (AM) corresponds to the central value. Moreover, the method used to calculate the AM can vary based on the size of the dataset and how the data is distributed.

The Arithmetic Mean is found by taking the sum of all the values in a dataset and dividing it by the total number of values. For example, if you have values 5, 10, and 15, their Arithmetic Mean is:

$\text{ Mean }=\frac{5+10+15}{3}=\frac{30}{3}=10$

2.0Arithmetic Mean Formula

For a given set of n values x₁, x₂, x₃, ..., x_n, the Arithmetic Mean $\bar{x}$ is:

$\bar{x}=\frac{x_1+x_2+\dots +x_n}{n}$

Or

$\bar{x}=\frac{\text{ Sum of values }}{\text{ Number of values }}$

3.0Key Types of Arithmetic Mean Questions

1. Simple Arithmetic Mean Questions 

These involve basic datasets and simple calculations to find the average.

Example:

Find the arithmetic mean of the numbers 4, 8, 12, and 16.

Solution:

$\bar{x}=\frac{4+8+12+16}{4}=\frac{40}{4}=10$

2. Arithmetic Mean Questions for Grouped Data   

In these questions, data is given in the form of a frequency distribution. The formula for calculating the Arithmetic Mean in grouped data is:

$\bar{x}=\frac{\sum f_i{x}_i}{\sum f_i}$

Where f_i is the frequency and x_i is the midpoint of each group.

Example:

Find the mean of the following grouped data:

Solution: 

Calculate the midpoint for each class interval, multiply by the frequency, and divide the total by the sum of frequencies:

$\Rightarrow \bar{x}=\frac{(15\times 4)+(25\times 6)+(35\times 5)}{4+6+5}$

$\Rightarrow \bar{x}=\frac{60+150+175}{15}$

$\Rightarrow \bar{x}=\frac{385}{15}$

$\Rightarrow \bar{x}=25.67$

4.0Arithmetic Mean Questions and Answers

Example 1: Find the arithmetic mean of the following numbers: 15, 25, 35, 45, and 55.

Solution:  

Using the formula of Arithmetic Mean:

$\bar{x}=\frac{\text{ Sum of values }}{\text{ Number of values }}$

Add all the numbers and divide by the total number of values.

Sum of numbers = 15 + 25 + 35 + 45 + 55 = 175

$\text{ Arithmetic Mean }=\frac{175}{5}=35$

So, the arithmetic mean is 35.

Example 2: The arithmetic mean of 5 numbers is 64. Four of the numbers are 62, 58, 70, and 68. Find the fifth number.

Solution:  

Let the fifth number be x.

Using the formula of Arithmetic Mean:

$\bar{x}=\frac{\text{ Sum of values }}{\text{ Number of values }}$

$64=\frac{62+58++70+68+x}{5}$

64 × 5 = 62 + 58 + 70 + 68 + x

320 = 258 + x

x = 320 – 258

x = 62

Thus, the fifth number is 62.

Example 3: The arithmetic mean of 50 numbers is 60. If two numbers, 35 and 45, are added to the dataset, what will be the new arithmetic mean?

Solution:  

Formula of Arithmetic Mean 

$\bar{x}=\frac{x_1+x_2+\dots +x_n}{n}$

Therefore, the mean of these numbers is

$60=\frac{x_1+x_2+\dots +x_{50}}{50}$

60 × 50 = x₁ + x₂ + … + x₅₀ 

3000 = x₁ + x₂ + … + x₅₀ 

If two numbers, 35 and 45, are added to the dataset, then

New sum = 3000 + 35 + 45 = 3080

The total number of values is now 50 + 2 = 52. 

The new arithmetic mean is:

$\text{ New mean }=\frac{3080}{52}=59.23$

So, the new arithmetic mean is approximately 59.23.

Example 4: The arithmetic mean of 10 numbers is 25. If one of the numbers, say 40, is removed from the set, what will be the new arithmetic mean of the remaining 9 numbers?

Solution:  

Formula of Arithmetic Mean 

$\bar{x}=\frac{x_1+x_2+\dots +x_n}{n}$

Therefore, the mean of 10 numbers is

$25=\frac{x_1+x_2+\dots +x_{10}}{10}$

25 × 10 = x₁ + x₂ + … + x₁₀ 

250 = x₁ + x₂ + … + x₁₀ 

The sum of the original 10 numbers is 250.

If one number (40) is removed, the new sum becomes:

New sum = 250 – 40 = 210

The number of values is now 9, so the new arithmetic mean is:

$\text{ New }\text{ mean }=\frac{210}{9}=23.33$

So, the new arithmetic mean is approximately 23.33.

Example 5: The arithmetic mean of 7 numbers is 54. When one number is removed, the arithmetic mean of the remaining 6 numbers becomes 52. Find the value of the number removed.

Solution:  

Formula of Arithmetic Mean

$\bar{x}=\frac{x_1+x_2+\dots +x_n}{n}$ 

Therefore, the mean of 7 numbers is

$54=\frac{x_1+x_2+\dots +x_7}{7}$

The sum of the 7 numbers is:

54 × 7 = x₁ + x₂ + … + x₇

378 = x₁ + x₂ + … + x₇

So, the sum of 7 numbers is 378.

Now, the mean of 6 numbers be:

$52=\frac{x_1+x_2+\dots +x_6}{6}$

The sum of the 6 numbers is:

52 × 6 = x₁ + x₂ + … + x₆

312 = x₁ + x₂ + … + x₇

So, after removing one number, the sum of the remaining 6 numbers is 312.

The number removed is 378 – 312 = 66

So, the removed number is 66.

Example 6: The arithmetic mean of Class A’s 25 students is 68, and the arithmetic mean of Class B’s 30 students is 72. What is the overall arithmetic mean of all 55 students combined?

Solution:  

The sum of the scores for Class A is:

Sum for Class A = 68 × 25 = 1700

The sum of the scores for Class B is:

Sum for Class B = 72 × 30 = 2160

The total sum for both classes is:

Total sum = 1700 + 2160 = 3860

The overall arithmetic mean is:

$\text{ Overall mean }=\frac{3860}{55}=70.18$

So, the overall arithmetic mean is 70.18.

Example 7: The arithmetic mean of a set of numbers is 50. If each number in the set is increased by 10%, what will be the new arithmetic mean?

Solution:  

If each number is increased by 10%, the new mean will be:

New mean = 50 × 1.10 = 55

So, the new arithmetic mean is 55.

Example 8: The arithmetic mean of 40 numbers is 75. After removing 5 numbers, the arithmetic mean of the remaining 35 numbers becomes 78. Find the arithmetic mean of the 5 removed numbers.

Solution:  

The original sum of the 40 numbers is:

Original sum=75 × 40 = 3000

The sum of the remaining 35 numbers is:

Remaining sum=78 × 35 = 2730

Therefore, the sum of the 5 removed numbers is:

Sum of removed numbers = 3000 – 2730 = 270

The arithmetic mean of the removed numbers is:

$\text{ Mean of removed numbers }=\frac{270}{5}=54$

So, the mean of the removed numbers is 54.

Example 9: Find the arithmetic mean of the following grouped data:

Solution: 

Using the formula of Grouped Frequency:

$\Rightarrow \text{ Arithmetic Mean }=\frac{\sum f_i{x}_i}{\sum f_i}$

$\Rightarrow \text{ Arithmetic Mean }=\frac{310}{12}=25.83$

So, the arithmetic mean is 25.83.

5.0Arithmetic Mean Practice Questions

1. Find the arithmetic mean of 15, 25, 35, 45, and 55.
2.  A frequency distribution is given as follows:

Calculate the Arithmetic Mean.

3. The arithmetic mean of 6 numbers is 48. If five of the numbers are 50, 45, 55, 60, and 40, find the sixth number.
4. The arithmetic mean of 20 numbers is 70. When a new number is added to the dataset, the mean increases by 2. Find the value of the new number added.
5. In a class of 10 students, the average score is 85. The top 5 scorers are awarded bonus points, raising the class average to 88. If each of the top 5 students received an equal number of bonus points, how many bonus points were added to each of the top 5 students?

6.0Sample Questions on Arithmetic Mean Questions

1. How do you calculate the Arithmetic Mean?

Ans: The arithmetic mean is calculated using the formula:

$\text{ Arithmetic Mean }(AM)=\frac{\text{ Sum of all values }}{\text{ Number of values }}$

2. What is the Arithmetic Mean for grouped data?

Ans: To calculate the arithmetic mean for grouped data, the formula is:

$AM=\frac{\sum f_i{x}_i}{\sum f_i}$

Where:

• f_i is the frequency of each class interval
• x_i is the midpoint of each class interval

3. How do you solve arithmetic mean questions involving grouped data? 

Ans: To solve arithmetic mean questions for grouped data, use the formula:

$AM=\frac{\sum f_i{x}_i}{\sum f_i}$

Here, f_i represents the frequency of the class intervals, and x_i is the midpoint of each class interval. You multiply each class frequency by the midpoint, sum the results, and divide by the total frequency.