Arithmetic Mean and Range 

Arithmetic mean and range are fundamental concepts in statistics and data analysis. The arithmetic mean represents the average value of a data set, providing insight into its central tendency. The range, on the other hand, measures how spread out the values are by calculating the difference between the highest and lowest numbers. Understanding both helps in summarizing and interpreting data effectively. In this blog, you'll learn how to find the range and arithmetic mean, their formulas, and solve real-world examples.

1.0What is Arithmetic Mean?

The arithmetic mean (often simply called the "mean" or "average") is the sum of all numbers in a data set divided by the number of elements.

Arithmetic Mean Formula

$\text{Arithmetic Mean}=\frac{Sumofallvalues}{Numberofvalues}$

Arithmetic Mean Between Two Numbers

If you are asked to find the arithmetic mean between two numbers a and b, simply use:

$\text{Arithmetic Mean}=\frac{a+b}{2}$

2.0What is Range?

The range of a data set is the difference between the highest and lowest values. It gives a basic idea of how spread out the data is.

Range Formula

$\text{Range}=\text{Maximum Value}-\text{Minimum Value}$ 

Also Read: Domain and Range of a Relation

3.0Arithmetic Mean and Range Examples

Example 1: Find the arithmetic mean and range of the data: 10, 12, 15, 18, 20

Solution:

• Mean = $\frac{10+12+15+18+20}{5}=\frac{75}{5}=15$
• Range = 20 - 10 = 10

Example 2: Find the arithmetic mean between 28 and 52.

Solution: $\text{Mean}=\frac{28+52}{2}=\frac{80}{2}=40$

Example 3: A data set has the numbers: 3, 7, 10, 5, 8. Find the arithmetic mean and range.

Solution:

• Mean = $\frac{3+7+10+5+8}{5}=\frac{33}{5}=6.6$
• Range = 10 - 3 = 7

Example 4: The arithmetic mean of the numbers x - 2, x, x + 2, x + 4 is 10. Find the value of x and the range.

Solution:

Given numbers: x - 2, x, x + 2, x + 4

• Mean = $\frac{(x-2)+x+(x+2)+(x+4)}{4}=10$

Simplify:

$\frac{4x+4}{4}=10$

$\Rightarrow x+1=10$

$\Rightarrow x=9$

So the numbers are: 7, 9, 11, 13

• Range = 13 - 7 = 6

Example 5: An arithmetic progression has five terms: a - 2d, a - d, a, a + d, a + 2d. Show that the arithmetic mean of these numbers is a, and find the range in terms of d.

Solution:

• Mean = $\frac{(a-2d)+(a-d)+a+(a+d)+(a+2d)}{5}=\frac{5a}{5}=a$
• Max = a + 2d, Min = a - 2d 

⇒ Range = (a + 2d) - (a - 2d) = 4d

Example 6: The marks obtained by a student in five subjects are: Math: 90, Physics: 85, Chemistry: 80, English: 75, Computer: 95. Find the arithmetic mean and range.

Solution:

• Mean = $\frac{90+85+80+75+95}{5}=\frac{425}{5}=85$
• Range = 95 - 75 = 20

Example 7: The mean of 6 numbers is 20. Five of them are: 18, 22, 19, 20, 21. Find the sixth number and the range.

Solution:

• Total sum = 6 × 20 = 120
• Sum of 5 numbers = 18 + 22 + 19 + 20 + 21 = 100
• Sixth number = 120 - 100 = 20

Full data: 18, 19, 20, 20, 21, 22

• Range = 22 - 18 = 4

Example 8: A student's average marks in 4 tests is 78. If he wants to raise his average to 80 after 5 tests, what should be his score in the 5th test? Also, find the range if the first four scores were 75, 80, 77, and 80.

Solution:

• Total after 4 tests = 4 × 78 = 312
• Total required after 5 tests = 5 × 80 = 400
• 5th test score = 400 - 312 = 88

New data: 75, 80, 77, 80, 88

• Range = 88 - 75 = 13

Example 9: Find the mean and range of the data set $\frac{1}{2},\frac{3}{4},1,\frac{5}{4},\frac{3}{2}$:

Solution:

Convert all to like denominators: $\frac{2}{4},\frac{3}{4},\frac{4}{4},\frac{5}{4},\frac{6}{4}$

Sum = $\frac{2+3+4+5+6}{4}=\frac{20}{4}=5$

$\Rightarrow Mean=\frac{5}{5}=1$

Range = $\frac{3}{2}-\frac{1}{2}=1$

Also Read: Solved Questions on Arithmetic Mean

4.0Why Learn Arithmetic Mean and Range?

They are commonly used in statistics, economics, science, and competitive exams like JEE, NTSE, and school assessments.

5.0Practice Questions on Arithmetic Mean and Range

1. Find the arithmetic mean and range of: 45, 55, 60, 70, 75.
2. What is the arithmetic mean between 17 and 33?
3. A data set has numbers: 5, 9, 12, 14, 18. Find its mean and range.
4. The mean of five numbers is 30. If four of them are 28, 32, 29, and 31, find the fifth number.
5. The range of a data set is 24 and the lowest value is 56. What is the highest value?