The Impact of Adding a Constant to Each Number in a Set: A Detailed Analysis

When working with data in statistics, one common operation involves adding a constant to each number in a set. This process can have a significant impact on the average of the set. We will explore this concept through a detailed example to understand the changes in the arithmetic mean and the sum of the numbers.

Understanding the Initial Set

Let's start by understanding the initial set of 15 numbers where the average is 5. The sum of these 15 numbers is calculated as follows:

Given: The average of 15 numbers is 5 Calculating the total sum of the original numbers: Total sum Average × Number of values 5 × 15 75

This means the sum of the 15 numbers is 75.

Adding a Constant to Each Number

Now, we add 5 to each of the 15 numbers in the set. To find the new sum, we follow these steps:

Since there are 15 numbers, adding 5 to each will increase the total sum by: 5 × 15 75 Calculating the new total sum: New total sum Original total sum Increase 75 75 150

With the new total sum, we can now find the new average.

Calculating the New Average

The new average is calculated as follows:

Divide the new total sum by the number of values: New average New total sum ÷ Number of values 150 ÷ 15 10

Hence, the new average of the 15 numbers is 10. This demonstrates that adding a constant to each number in a set increases the average by that same constant.

Alternative Methods

We can approach this problem using both a long and a short method. Let's consider both:

Long Method

Given: The average of 15 numbers is 5 Total sum of the 15 numbers 5 × 15 75 Adding 5 to each number increases the total sum by 75 (5 × 15) New total sum 75 75 150 New average 150 ÷ 15 10

Short Method

Assuming each of the original 15 numbers is 5, and adding 5 to each number results in 10:

New average 10

This succinctly shows that the new average is directly the sum of the original number and the constant added.

In conclusion, adding a constant to each number in a set not only increases the total sum but also raises the arithmetic mean (average) of the set by the same constant. This principle is fundamental in statistical analysis and can be applied to various data sets for better understanding and interpretation.