Average Definition

Calculating an average is a fundamental mathematical operation used in a wide range of applications, from determining the average temperature for a city to calculating an athlete's average speed during a race and calculating average marks scored by a student. It is a simple yet essential concept that can provide valuable insights into large data sets. This article will explore how to calculate an average and solve some questions.

What is an Average?

An average, also known as the Mean, is the total of a group of given elements divided by the number of elements in the set. For example, if we have a set of ten numbers, 1, 2, 3, 4, 5, 6, 7, 8, 9, and 10, the average would be calculated as follows:

(10 + 20 + 30 + 40 + 50 + 60 + 70 + 80 + 90 + 100) / 10 = 55

Therefore, the average of these ten numbers is 55.

How to Calculate an Average?

Calculating an average is a simple process involving a few small steps. The following steps outline how to calculate an average:

Step 1: Calculate the sum of all the elements in the given set.

Step 2: Note the total number of values in the given set.

Step 3: Divide the obtained sum of the numbers by the total number of values.

Step 4: The number obtained by the above calculation is our average.

Let us take the example of calculating the average of the following numbers: 10, 20, 30, 40, and 50.

Step 1: Calculate the sum of all the elements in the given set.

10 + 20 + 30 + 40 + 50 = 150

Step 2: Note the total number of values in the given set.

There are five values in the set.

Step 3: Divide the obtained sum of the numbers by the total number of values.

150 / 5 = 30

Step 4: The number obtained by the above calculation is our average.

Therefore, the average of the set of numbers 10, 20, 30, 40, and 50 is 30.

Types of Averages

Different types of averages can be calculated depending on the conditions. The most commonly used types of averages are the Arithmetic Mean, Geometric Mean, and Harmonic Mean.

1. Arithmetic Mean

The arithmetic Mean is the most commonly used type of average. It is calculated as adding all the elements in a set divided by the total number of values present. It is used to find the average of numerical data, and it is the type of average that we have been discussing in this article. Arithmetic Mean, also known as the average or the Mean, is a statistical measure used to find the central tendency of a set of numbers. Addition of all the elements in a set divided by the total number of values present in the set.

The arithmetic Mean is frequently used to depict an average value or a broad trend in a data set. It is utilized in many disciplines, including finance, economics, science, and engineering, to analyze and interpret data. It is simple to calculate the arithmetic Mean and offers a helpful summary of the data set. The formula for calculating the arithmetic Mean is:

Arithmetic Mean = (Sum of all the numbers) / (Number of numbers in the set)

For example, if we have a set of numbers 3, 5, 7, and 9, we can find the arithmetic mean as follows:

Arithmetic Mean = (3+5+7+9) / 4

Arithmetic Mean = 24/4

Arithmetic Mean = 6

Therefore, the arithmetic mean of numbers 3, 5, 7, and 9 is 6.

The arithmetic Mean has some limitations, and it may only sometimes correctly represent the data set. For example, the arithmetic Mean may be skewed if the data set has extreme values or outliers.

Some Examples of Calculating the Arithmetic Mean

Example 1: Suppose a student takes five tests in a course and receives the following scores: 85, 92, 78, 89, and 90. What is the student's average score?

Solution:

To calculate the arithmetic Mean, we add up all the scores and divide by the number of tests taken:

Arithmetic Mean = (85 + 92 + 78 + 89 + 90) / 5

Arithmetic Mean = 86.8

Therefore, the student's average score is 86.8.

Example 2: A small shopkeeper sells ten items and notes the following sales for each product: $120, $200, $300, $75, $150, $225, $250, $100, $175, and $125. What is the average sale price of the items?

Solution:

To calculate the arithmetic Mean, we add up all the sales and divide by the number of items sold:

The total number of items sold is 10.

Arithmetic Mean = (120 + 200 + 300 + 75 + 150 + 225 + 250 + 100 + 175 + 125) / 10

Arithmetic Mean = 172

Therefore, the average sale price of the items is $172.

Example 3: A company has four employees with annual salaries of $50,000, $75,000, $100,000, and $125,000. Find out the average salary of the employees.

Solution:

To calculate the arithmetic Mean, we add up all the salaries and divide by the number of employees:

In this case, the number of employees is 4.

Arithmetic Mean = (50,000 + 75,000 + 100,000 + 125,000) / 4

Arithmetic Mean = 87,500

Therefore, the average salary of the employees is $87,500.

2. Geometric Mean

Finding the predominant trend of a group of numbers that have been multiplied together is done using an average known as the geometric Mean. The average growth or change rate of a group of values, such as population growth in an area or investment returns, is determined using the geometric Mean. The mathematical formula for calculating the geometric Mean is:

Geometric Mean = (x1 * x2 * x3 * ... * xn) ^ (1/n)

Where x1, x2, x3, ... xn is the n numbers in the set.

In the case of two numbers, a and b, the geometric Mean is calculated using the formula (a * b) ^ (1 / 2)

For example, if two numbers are 4 and 9, the geometric Mean is calculated as (36) ^ (1 / 2) = 6.

For example, if we want to find the geometric Mean of the following numbers: 2, 4, 8, 16, and 32.

Geometric Mean = (2 * 4 * 8 * 16 * 32) ^ (1/5)

Geometric Mean = 11.31

Therefore, the geometric Mean of the set of numbers 2, 4, 8, 16, and 32 is 11.31.

The geometric Mean helps calculate the median increment rate of a group of values over time. For example, if we want to calculate the average increment rate of a company's savings over the past ten years, we can use the geometric Mean to find the average rate of change. However, it is important to consider other measures of central tendency in certain situations to obtain a correct representation of the data.

Some Examples of Calculating the Geometric Mean

Example 1: Suppose a stock has the following annual returns over the past five years: 10%, 15%, 8%, 12%, and 20%. What is the geometric mean return of the stock?

Solution:

To find the geometric mean return of the stock, we can use the formula:

Geometric mean return = (1 + r1) x (1 + r2) x ... x (1 + rn) ^ (1/n) - 1

Where r1, r2, ..., rn are the stock's annual returns over n years.

Putting the values in the formula from the problem, we get:

Geometric mean return = [(1 + 0.10) x (1 + 0.15) x (1 + 0.08) x (1 + 0.12) x (1 + 0.20)] ^ (1/5) - 1

= (1.10 x 1.15 x 1.08 x 1.12 x 1.20) ^ (1/5) - 1

= 1.1127 - 1

= 0.1278 or 12.78%

Therefore, the geometric mean return of the stock over the past five years is 11.27%.

Example 2: A population of rabbits increases by 5%, 10%, 12%, and 8% in four consecutive years. What is the mean yearly growth rate of the population?

Solution:

To calculate the geometric Mean, we take out the product of all the growth rates together and then take the nth root of the product, where n depicts the number of growth rates:

In this case, n is 4.

Geometric Mean = (1.05 * 1.10 * 1.12 * 1.08) ^ (1/4) - 1

Geometric Mean = 1.089 -1

= 0.089

Therefore, the average annual growth rate of the population is 1.089 -1 = 0.089 or 8.9 %

Example 3: Suppose a bond has the following annual coupon rates over the next three years: 4%, 5%, and 6%. What is the geometric Mean of the coupon rates?

Solution:

To calculate the geometric Mean, we take out the product of all the coupon rates together and then take the nth root of the product, where n depicts the number of coupon rates

In this case, n is 3.

Geometric Mean = (1.04 * 1.05 * 1.06) ^ (1/3)

Geometric Mean = 1.05

Therefore, the geometric Mean of the coupon rates is 1.05 - 1 = 0.05 or 5%.

3. Harmonic Mean

The harmonic Mean is another type of average user to find the central tendency of a set of numbers. The harmonic Mean calculates rates and ratios, such as speed or distance.

The formula for calculating the harmonic Mean is:

Harmonic Mean = n / (1/x1 + 1/x2 + 1/x3 + ... + 1/xn)

Where n is the number of values in the set, and x1, x2, x3, ... xn are the values in the set.

For example, if we want to find the harmonic Mean of the following numbers: 2, 4, and 8.

Harmonic Mean = 3 / (1/2 + 1/4 + 1/8)

Harmonic Mean = 3 / 0.875

Harmonic Mean = 3.43

Therefore, the harmonic Mean of numbers 2, 4, and 8 is 3.43.

The harmonic Mean is useful for calculating ratios because it considers the common values in the set. For example, if we want to find the average speed of a car that travels at 60 km/h for half the distance and 80 km/h for the other half, we can use the harmonic Mean to calculate the average speed.

Some Examples of Calculating the Harmonic Mean

Example 1: Suppose a car travels 60 miles at a speed of 40 mph and then travels another 60 miles at a rate of 60 mph. What is the average speed of the car over the entire trip?

Solution:

To calculate the harmonic Mean, we first find the reciprocal of each speed, then take the arithmetic mean of the reciprocals, and finally, take the reciprocal of the arithmetic Mean:

Harmonic Mean = 2 / [(1/40) + (1/60)]

Harmonic Mean = 48 mph

Therefore, the average car speed over the entire trip is 48 mph.

Example 2: Suppose a company has three factories that produce 200 units, 300 units, and 500 units, respectively. What is the average production rate of the factories?

Solution:

Finding the inverse of each production rate is the first step in calculating the harmonic Mean. Next, the arithmetic mean of the inverses is calculated, and finally, the inverse of the arithmetic Mean is calculated:

Harmonic Mean = 3 / [(1/200) + (1/300) + (1/500)]

Harmonic Mean = 290.41 units

Therefore, the average production rate of the factories is 290.41 units.

Example 3: Suppose a runner runs a 5-mile race in 30 minutes and then a 10-mile race in 60 minutes. What is the average pace of the runner over the entire distance?

Solution:

To calculate the harmonic Mean, we first find the reciprocal of each pace (minutes per mile), then take the arithmetic mean of the reciprocals, and finally, take the reciprocal of the arithmetic Mean:

Harmonic Mean = 2 / [(1/30) + (1/60)]

Harmonic Mean = 40 minutes per mile

Therefore, the average pace of the runner over the entire distance is 40 minutes per mile.

Conclusion

In this article, we learnt that calculating an average is a fundamental mathematical operation used in a wide range of applications, from determining the average temperature for a city to calculating an athlete's average speed during a race and calculating average marks scored by a student. It is a simple yet essential concept that can provide valuable insights into large data sets. Calculating an average is a basic mathematical procedure that has many practical applications. By following the simple steps in this article, you can calculate an average for any group of values. Remember to consider the average appropriate for your situation to get the most accurate result. The arithmetic Mean is a fundamental statistical measure used to find the central tendency of a set of numbers. However, it is advisable to look into other measures of central tendency in certain conditions to obtain a correct representation of the data. The geometric Mean is a useful measure of central tendency used to find the mean growth rate of a group of values. The harmonic Mean calculates rates and ratios, such as speed or distance.