Understanding the Impact of Arithmetic Operations on the Mean

Have you ever encountered a problem involving arithmetic operations on a set of observations and wondered how it affects their mean? In this article, we will explore such a scenario. Specifically, we will dive into a problem where the mean of 10 observations is given, and we need to determine the new mean after adding 2 to each observation and then multiplying the result by 3. We will explore this step-by-step and discuss the general principles behind these operations.

Given Problem and Initial Setup

We start with the following information:

The mean of 10 observations is 5. We are asked to add 2 to each of the 10 observations and then multiply the result by 3.

To solve this, let's start with the basics. The formula for the mean (average) of a set of observations is:

Mean frac{Sum of all observations}{Number of observations}

Original Mean Calculation

Given the mean is 5 for 10 observations, the total sum of the observations can be calculated as:

Sum of observations 5 times 10 50

Adding 2 to Each Observation

When we add 2 to each of the 10 observations, the total sum will increase by 2 times 10 20. Therefore, the new sum of the observations is:

New Sum 50 20 70

Multiplying by 3

Next, we need to multiply the new sum by 3:

Final Sum 70 times 3 210

Calculating the New Mean

To find the new mean, we divide the final sum by the number of observations (which is still 10):

New Mean frac{210}{10} 21

General Principles

From this specific example, we can see that the mean of a set of observations is influenced by arithmetic operations on the individual observations. The core principles can be generalized as follows:

Adding a constant value to each observation increases the total sum by the product of that constant and the number of observations. The mean is thus adjusted by that constant value. Multiplying the observations by a constant value scales the total sum accordingly, and the mean reflects this scaling.

Algebraic Approach

Using algebraic methods, we can express the problem more compactly:

Lets denote the sum of the original observations as S. Given:

frac{S}{10} 5 Rightarrow S 50

1. Adding 2 to each observation increases the sum by 2 times 10 20 to get a new sum of:

New Sum S 2 times 10 50 20 70

2. Multiplying the new sum by 3:

Final Sum 70 times 3 210

3. The new mean, thus, is:

New Mean frac{210}{10} 21

Conclusion

Through this detailed analysis, we have demonstrated how to calculate the new mean after performing arithmetic operations on a set of observations. The key is to understand the effect of each operation on the sum and subsequently the mean. By practicing these steps and generalizing the principles, you can solve similar problems efficiently.