Introduction

In order to find the constant that needs to be added to each observation so that the mean becomes k times the original mean, we need to understand the concept of mean and how it is affected by adding a constant.

Mean

The mean, also known as the average, is a measure of central tendency that represents the typical or average value of a set of observations. It is calculated by summing up all the observations and dividing the sum by the total number of observations.

Effect of Adding a Constant

When a constant is added to each observation in a data set, it does not change the relative positions of the observations but shifts the entire distribution by the same amount. This means that the mean will also be shifted by the same constant.

Constant for k times the Original Mean

To find the constant that needs to be added to each observation so that the mean becomes k times the original mean, we can use the following formula:

Constant = (k * original mean) - original mean

This formula ensures that when the constant is added to each observation, the resulting mean will be k times the original mean.

Explanation

Let's consider an example to understand this concept. Suppose we have a data set with the following observations:

10, 15, 20, 25, 30

The original mean of this data set can be calculated by summing up all the observations and dividing the sum by the total number of observations:

Original mean = (10 + 15 + 20 + 25 + 30) / 5 = 20

Now, let's say we want the mean to be 2 times the original mean. Using the formula mentioned earlier, we can calculate the constant:

Constant = (2 * 20) - 20 = 40 - 20 = 20

This means that if we add 20 to each observation, the resulting mean will be 2 times the original mean. Let's verify this:

New observations = 10 + 20, 15 + 20, 20 + 20, 25 + 20, 30 + 20 = 30, 35, 40, 45, 50

New mean = (30 + 35 + 40 + 45 + 50) / 5 = 200 / 5 = 40

As we can see, the new mean is indeed 2 times the original mean.

Conclusion

To find the constant that needs to be added to each observation so that the mean becomes k times the original mean, we can use the formula (k * original mean) - original mean. Adding this constant to each observation will shift the distribution by the same amount, resulting in the desired mean.