The Mean of New Observations After Multiplying and Adding

When dealing with data, it often becomes necessary to perform operations such as multiplying and adding constants to the original observations. This article will guide you through the process of finding the new mean of the observations after these operations have been applied. We'll start with an initial mean, multiply each observation by a constant, and then add a constant to each result. Let's walk through the steps with a clear example.

Understanding the Problem

Given the mean of a dataset is 10. If each observation is multiplied by 5 and then 1 is added to each result, we need to determine the mean of the new observations. This involves two main steps: multiplying by a constant and adding a constant to each observation.

Step-by-Step Solution

To find the mean of the new observations, we can follow these steps:

Step 1: Original Mean

The given mean of the original data is 10.

Step 2: Multiplying Each Observation by 5

When you multiply each observation by a constant (in this case, 5), the mean of the new observations is also multiplied by that constant.

Let's compute the new mean after this operation:

New Mean after multiplication: 5 times; 10 50

Step 3: Adding 1 to Each Result

When you add a constant to each observation (in this case, 1), the mean of the new observations increases by that constant. Therefore, the new mean after this addition is:

Final New Mean: 50 1 51

Thus, the mean of the new observations is 51.

Addressing Ambiguity: Understanding the Phrase “Each Result”

The phrase "1 is added to each result" can be ambiguous. Assuming "result" here means each observation, the sum of all observations increases by the number of observations (n). Let's consider a more detailed solution:

Alternative Solution

Multiplying each observation by 3, the sum of 9n gets multiplied by 3 where n is the number of data points. Assuming "result" to mean observation (i.e., each member of data), the sum increases by n:

Final Sum: 3(9n) n 28n

Thus, the new average is:

New Average: 28n/n 28

Generalizing the Concept

The mean is not independent of changes in origin and scale. Therefore, if we multiply all data by any number k, the mean is multiplied by the same number k. Similarly, if we add any number to all data, the mean is also added by the same number.

Thus, if we multiply the original mean (10) by the constant 5 and then add 1:

Mean of new observations: 5 times; 10 1 51

Let's consider the case with n data points:

Total of original observations: 10n

When each term is multiplied by 5, the total increases by a factor of 5:

Increased total: 10 times; 5 50n

When 1 is added to each term, the total increases by n:

New total: 50n n 51n

Thus, the new mean is:

New mean: 51n / n 51

Conclusion

In summary, after multiplying each observation by a constant and then adding a constant, the new mean can be computed easily using the steps outlined above. Understanding these operations is crucial for data manipulation and analysis.

Would you like to explore more scenarios or need further assistance with data manipulation? Feel free to ask!