Class 7 - Arithmetic Mean : Formula, Steps and Solved Examples

The arithmetic mean is a fundamental concept in Class 7 mathematics that represents the central value of a given set of numbers. It is commonly referred to as the “average”. The arithmetic mean describes a single value that best represents a group of data. This concept is widely used in real-life situations such as calculating marks, temperatures, daily expenses, etc. In this guide, you will learn the concept of arithmetic mean along with its definition, formula, and step-by-step method of calculation, along with examples for quick and easy understanding of the concept.

Table of Contents

• What Is Arithmetic Mean?

• Arithmetic Mean Formula

• Properties of Arithmetic Mean

• Solved Examples on Arithmetic Mean

• Arithmetic Mean vs Median vs Mode

• Advantages and Limitations of Arithmetic Mean

• Practice Questions on Arithmetic Mean

• Practice Worksheets on Arithmetic Mean

 

What Is Arithmetic Mean?

The arithmetic mean is the most commonly used measure of central tendency. In simple terms, it is what most people mean when they say "average". The mean of a data set is the sum of the data divided by the number of data values. The arithmetic mean is denoted by x̄ (read as "x bar") or sometimes by μ (mu) when referring to a population in statistics. The mean of data always lies between the greatest and the smallest observations.

Arithmetic Mean Formula

The basic formula for arithmetic mean is:
Mean = (Sum of all observations) / (Number of observations)
x̄ = (Sum of all observations) / (Number of observations)
In mathematical notation:
x̄=(x1+x2+x3+...+xn)/n=Σxi/n
Where:

• x1,x2,x3,…,xn are the individual observations

• n is the total number of observations

• Σ (sigma) represents the sum of all values

How to find the mean?

1. Find the sum of all the data values.

2. Divide the total by the number of data values.

3. Mean = (Sum of all observations) / (Number of observations)

Properties of Arithmetic Mean

• Property 1: If all observations are equal, the mean is equal to that value. If your data are 7, 7, 7, 7, 7, then the mean is 7.

• Property 2: The sum of deviations from the mean is always zero. If you take each value of the observation, subtract the mean from it, and add all those differences up, you always get 0.

• Property 3: Adding or subtracting a constant shifts the mean by the same constant. For example, if every value in a dataset increases by 5, the mean also increases by 5. If x̄ is the mean of x1,x2,x3,…,xn then the mean of (x1+k),(x2+k),...,(xn+k) is x̄ + k.

• Property 4: Multiplying or dividing by a constant scales the mean accordingly. For example, if every value is multiplied by 3, the mean is also multiplied by 3.

• Property 5: The mean of combined data can be calculated from group means. If Group 1 has n1 observations with mean x̄1 and Group 2 has n_{2} observations with mean x̄2, then the combined mean is x̄combined=(n1x̄1+n2x̄2)/(n1+n2).

Solved Examples of Arithmetic Mean

Example 1: Find the arithmetic mean of the first 8 natural numbers.
Solution: The first 8 natural numbers are 1, 2, 3, 4, 5, 6, 7, 8
Sum = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 = 36
n = 8
Mean = (Sum of all observations) / (Number of observations)
x̄ = 36 / 8 = 4.5
Therefore, the arithmetic mean of the first 8 natural numbers is 4.5.

Example 2: The daily temperatures recorded in a city over a week (in °C) were: 28, 31, 27, 33, 30, 29, 32. Find the arithmetic mean of the daily temperatures recorded in the city over the week.
Solution: Sum = 28 + 31 + 27 + 33 + 30 + 29 + 32 = 210
n = 7
Mean = (Sum of all observations) / (Number of observations)
x̄ = 210 / 7 = 30°C
Therefore, the arithmetic mean of the daily temperatures recorded in the city over the week is 30°C.

Example 3: The arithmetic mean of five numbers, 12, 18, x, 24, and 30, is 22. Find x.
Solution: Given mean = 22
Mean = (Sum of all observations) / (Number of observations) = 22
x̄ = (12 + 18 + x + 24 + 30) / 5 = 22
(84 + x) / 5 = 22
84 + x = 22 × 5 = 110
x = 26

Example 4: Class A has 30 students with a mean score of 70 in a test. Class B has 20 students with a mean score of 80 in the same test. Find the combined mean.
Solution: Given, the mean of class A = x̄1 = 70
Number of students in class A = n1 = 30
Mean of class B = x̄2 = 80
Number of students in class B = n2 = 20
Combined mean = (n1x̄1+n2x̄2)/(n1+n2)
x̄combined = (30 × 70 + 20 × 80) / (30 + 20)
= (2100 + 1600) / 50
= 3700 / 50
= 74
Therefore, the combined mean of class A and class B is 74

Example 5: Chaya scored the following marks in her first 4 maths tests: 87, 93, 91 and 88. What must be her score in the 5th test to have an average of 90 in the class?
Solution: Given, marks in first 4 tests = 87, 93, 91, 88
Required average for 5 tests = 90
Let x be the score of the fifth test
Mean = (Sum of all observations) / (Number of observations) = 90
90 = (87 + 93 + 91 + 88 + x)/5
90 × 5 = 87 + 93 + 91 + 88 + x = 359 + x
450 = 359 + x
x = 450 - 359 = 91
Therefore, her score in the 5th test to have an average of 90 is 91.

Arithmetic Mean vs Median vs Mode 

Mean, median, and mode are measures of central tendency used to represent a dataset with a single value. While the mean gives the average, the median shows the middle value, and the mode identifies the most frequently occurring number.

 

Advantages and Limitations of Arithmetic Mean

Advantages of Arithmetic Mean:

• It uses every single value in the dataset.

• It's uniquely defined; there's only one arithmetic mean for any dataset.

• It's algebraically tractable. It can be used in other calculations like variance and standard deviation.

• It's easy to compute and understand.

• It's the basis of many statistical formulas, including standard deviation and regression.

Limitations of Arithmetic Mean:

• It is heavily influenced by extreme values. One unusually large or small number can distort the mean significantly.

• It may produce a value that doesn't actually exist in the dataset. For example, the mean number of children per family being 1.8 is mathematically correct, but no family actually has 1.8 children.

• It is not useful for data that is not numerical. It's not always meaningful for qualitative or categorical data.

Practice Questions on Arithmetic Mean

1. If the mean of 5, 6, 8, 4, x, 8, 6, 6, 5 and 7 is 6, find the value of x.

2. Find the mean of the first ten even natural numbers.

3. The following are the heights of 6 students in a class. 156.4 cm, 155.4 cm, 151.5 cm, 153.1 cm, 149.3 cm and 151 cm. Find the mean height.

4. The following are the number of times a video was watched during a week on YouTube. 121, 145, 167, 190, 119, 108 and 144. Find the average number of times the video was watched during the week.

5. Marks obtained by 8 students are: 12, 15, 18, 20, 14, 16, 19, 17. Find the arithmetic mean.

6. Find the mean of: 25, 30, 35, 40, 45, 50

7. The temperatures (in °C) for 7 days are: 32, 34, 31, 33, 35, 36, 32. Find the average temperature.

8. Weights (in kg) of 5 students are: 42, 38, 40, 45, x. If the mean weight is 41 kg, find the value of x.

9. Find the arithmetic mean of the first 15 natural numbers.

10. Number of books read by 6 students are: 3, 5, 7, 4, 6, 5. Find the mean number of books read.

Practice Worksheets on Arithmetic Mean

Refer to the PDF or worksheet provided to solve additional practice problems and strengthen your understanding of the arithmetic mean.

Easy Level Worksheets

Medium Level Worksheets

Advanced Level Worksheets