In mathematics, the average is a measure that represents the central tendency of a set of numbers. It is commonly referred to as the arithmetic mean and is calculated by summing up all the numbers in a set and dividing the sum by the total count of numbers.
The concept of average dates back to ancient times. The ancient Egyptians used a form of average to distribute the workload among workers. The Greeks also had a notion of average, which they called "meson," meaning middle. However, the modern concept of average as the arithmetic mean was developed in the 17th century by mathematicians such as Christiaan Huygens and Gottfried Wilhelm Leibniz.
The concept of average is introduced in elementary school and is taught throughout middle school and high school. It is a fundamental concept in mathematics and is applicable in various fields, including statistics, economics, and science.
To understand the concept of average, one needs to have a basic understanding of addition, division, and counting. The step-by-step process to calculate the average is as follows:
For example, let's find the average of the numbers 5, 8, 12, and 15:
Therefore, the average of the given set is 10.
There are different types of averages used in mathematics, depending on the context and the nature of the data. The most commonly used types of averages are:
The average possesses several properties that make it a useful measure:
To find or calculate the average, follow these steps:
The formula for calculating the average is:
Average = Sum / Count
Where "Average" represents the average value, "Sum" represents the sum of all the numbers, and "Count" represents the total count of numbers in the set.
To apply the average formula, substitute the values of the sum and count into the formula and perform the division to obtain the average value.
For example, if the sum is 50 and the count is 5, the average can be calculated as:
Average = 50 / 5 = 10
The symbol commonly used to represent average is "x̄" (pronounced x-bar).
There are various methods for calculating the average, including:
Example 1: Find the average of the numbers 10, 15, 20, and 25. Solution: Sum = 10 + 15 + 20 + 25 = 70 Count = 4 Average = 70 / 4 = 17.5
Example 2: The average of five numbers is 12. If four of the numbers are 10, 12, 14, and 16, what is the fifth number? Solution: Sum = 10 + 12 + 14 + 16 = 52 Count = 4 Average = 12 Fifth number = (Average * Count) - Sum = (12 * 5) - 52 = 60 - 52 = 8
Example 3: The average of a set of numbers is 25. If one number is removed from the set, the average becomes 30. What is the removed number? Solution: Sum = Average * Count = 25 * Count New Sum = Average * (Count - 1) = 30 * (Count - 1) Removed number = Sum - New Sum = (25 * Count) - (30 * (Count - 1))
Question: What is the difference between mean and average? Answer: In mathematics, mean and average are often used interchangeably to refer to the arithmetic mean. However, in some contexts, mean can also refer to other types of averages, such as the geometric mean or harmonic mean.