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The average is also known as the arithmetic mean. It is calculated by adding up all the numbers and then dividing the sum by the total number of values in the set. This Average formula helps us find the average for any group of data. This article will discuss the use of the average formula, its examples and more.
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Average Symbol
The average is also known as the mean. Mean is a commonly used mathematical term. It is often represented by the symbol x̄, pronounced as “x bar”. Additionally, the Greek letter “μ” is sometimes used to denote the average, especially in statistics.
Average Formula
To calculate the average in mathematics, you use the following formula:
Average Formula = sum of observations total number of observations
If you have a set of numbers, say n numbers like x₁, x₂, x₃, …, xₙ, the average can be calculated using the formula:
Average Formula = x1 + x2 + x3 + x4 + . . . + xn n
Also Check: Area of a Circle
How to Calculate Average?
One must follow the steps given below to calculate the average of a set of numbers:
- List all the numbers (observations) and count how many there are (let’s call this number n).
- Add up all the numbers to find their total sum.
- Divide the sum you found in Step 2 by the total number of observations (n).
- Simplify the result to get the average.
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What is Average Used For?
The average is a useful way to summarise a large set of data with just one number, representing the central value of the data. Here are some practical applications of averages:
- By calculating the average time it takes to commute to work or school, you can better plan your daily schedule.
- In cricket, a player’s batting average helps assess their performance over time.
- Before purchasing a product, the average customer review score can guide your decision.
- Averages like household income, unemployment rate, or inflation rate are used to understand economic trends.
- Businesses calculate the average daily sales of a product to determine how much stock they need.
What is Mean?
In mathematics, the mean is a measure of central tendency, commonly known as the average or arithmetic mean. Mean is calculated by dividing the sum of all values by the total number of values.
Mean = sum of observations total number of observations
Also Check: Circumstance of Circle
Types of Mean
There are three types of means in mathematics:
- Arithmetic Mean: Arithmetic Mean is the most common type, calculated as described above.
- Geometric Mean: Geometric Mean is the nth root of the product of n values, often used for growth rates.
- Harmonic Mean: Harmonic Mean is the reciprocal of the average of the reciprocals, useful in certain types of averages like rates.
Arithmetic Mean
The arithmetic mean, often simply called the average, is calculated by adding up all the values and then dividing by the number of values. The formula to calculate the arithmetic mean for a set of values x1, x2, x3, . . . xn is
Arithmetic Mean (A. M.) = x1 + x2 + x3 + x4 + . . . + xn n
Geometric Mean
The geometric mean is a method to measure central tendency. It is calculated by multiplying all the values together and then taking the nth root of the product, where n is the total number of values. The formula to calculate the geometric mean for values x1, x2, x3, . . . xn is
Geometric Mean (G. M.) = nx1 . x2 . x3 . . . xn
Harmonic Mean
The harmonic mean is a type of Pythagorean mean, alongside the arithmetic and geometric means. It is calculated by dividing the number of values by the sum of the reciprocals of the values. The harmonic mean tends to be lower than both the arithmetic and geometric means. The formula for the harmonic mean for values x1, x2, x3, . . . xn is
Harmonic Mean (H. M.) =n(1x1) + (1x2) + . . . + (1xn)
Also Check: Fibonacci Sequence
Important Formulas for Calculating Averages
Below are some key formulas related to averages. These formulas are useful for board exams and competitive tests.
Average of the First n Natural Numbers
The sum of the first n natural numbers
sum = n(n + 1)2
Average of the first n natural numbers
average = n + 12
Average of the Squares of the First n Natural Numbers
The sum of the squares of the first n natural numbers
sum = n(n + 1)(2n + 1)6
Average of the squares of the first n natural numbers
average = (n + 1)(2n + 1)6
Average of the Cubes of the First n Natural Numbers
The sum of the cubes of the first n natural numbers:
Sum = (n(n + 1)2)2
Average of the cubes of the first n natural numbers:
Average = n(n + 1)24
Average of the First n Natural Odd Numbers
The sum of the first n natural odd numbers:
Sum = n2
Average of the first n natural odd numbers:
Average=n
Average of the First n Natural Even Numbers
The sum of the first n natural even numbers:
Sum=n(n+1)
Average of the first n natural even numbers:
Average=n+1
Solved Examples of Average
Example 1: Find the average of the first 10 natural numbers.
Solution: To find the average of the first 10 natural numbers we can use the formula for the average of the first n natural numbers:
average = n + 12
Here, n = 10
average = 10 + 12
average = 112 = 5.5
Answer: The average of the first 10 natural numbers is 5.5.
Example 2: Calculate the average of the squares of the first 5 natural numbers.
Solution: To calculate the average of the squares of the first n natural numbers, we use the below-given formula:
average = (n + 1)(2n + 1)6
Here, n=5
Therefore, average = (5 + 1)(10 + 1)6
average = (6)(11)6 = 11
Answer: The average of the squares of the first 5 natural numbers is 11.
Example 3: Find the average of the cubes of the first 4 natural numbers.
Solution: To calculate the average of the cubes of the first n natural numbers, we can use the below-given formula:
Average = n(n + 1)24
Here, n = 4
Average = 4 (4 + 1)24
Average = 25
Answer: The average of the cubes of the first 4 natural numbers is 25.
Example 4: Determine the average of the first 7 natural odd numbers.
Solution: To calculate the average of first n natural odd numbers, the average is simply n.
Average=n
Here, n = 7
Average=7
Answer: The average of the first 7 natural odd numbers is 7.
Practice Questions of Average
Question 1: Find the average of the first 15 natural numbers.
Question 2: Calculate the average of the squares of the first 6 natural numbers.
Question 3: Determine the average of the cubes of the first 5 natural numbers.
Question 4: What is the average of the first 10 natural odd numbers?
Question 5: Find the average of the first 8 natural even numbers.
Question 6: If the average of the cubes of the first 7 natural numbers is calculated, what will be the result?
FAQs on Average
What's the difference between average, mean, and arithmetic mean?
"Average," "mean," and "arithmetic mean" are often used interchangeably. "Mean" and "arithmetic mean" specifically refer to the sum of values divided by the number of values. "Average" is a broader term that can refer to different types of central tendencies.
How do you calculate the average of a set of numbers?
To calculate the average of a set of numbers, we need to add all the numbers together, and then divide by the total count of numbers.
What are the types of means used in statistics?
Arithmetic Mean: Arithmetic Mean is the sum of values divided by the count. Geometric Mean: Geometric Mean refers to the nth root of the product of values. Harmonic Mean: Harmonic Mean is the Reciprocal of the average of reciprocals.
