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Understanding means and averages is fundamental in statistics. The mean, often called the average, is a measure of the center of a data set. To calculate the mean:

• Add up all the numbers in the data set.
• Divide the sum by the number of items in the data set.

For Data Set I, you calculate the mean by summing up 4, 8, 15, 9, and 11. This results in 47. Dividing by 5 (the number of values) gives you 9.4.
For Data Set II, the values 8, 16, 30, 18, and 22 sum up to 94. Dividing 94 by 5 yields a mean of 18.8.
These means help summarize each data set with a single representative value.

Comparing data sets allows us to find meaningful differences and similarities. In this exercise, we see two data sets with values related by a multiplicative factor.
Data Set II is constructed by multiplying each value from Data Set I by 2.
This comparison shows that transformations affect the overall properties of the data, such as the mean. Understanding such relationships helps in identifying patterns and making predictions based on data.
By comparing these data sets’ means, you quickly see how the transformation impacts the central tendency—doubling each value doubles the mean.

A multiplicative transformation involves multiplying each element of a data set by a constant factor. In our example, Data Set I is transformed into Data Set II by multiplying each value by 2.

• This transformation scales all values identically.
• It alters the spread as well as the location (mean) of the data.

When you multiply each element by a constant, the mean also gets multiplied by the same factor. Thus, while the original mean of Data Set I is 9.4, the mean of the transformed Data Set II naturally becomes 18.8, which is also double 9.4.

Performing statistical analysis systematically is crucial to obtaining accurate results. Here's a recommended approach based on the exercise:

Step 1: Compute the Individual Means

Calculate the sum of the data set values and divide by the total number of values. This gives a snapshot of the typical value in your data.

Step 2: Transform and Analyze

If dealing with a transformation, analyze how each data point changes. This requires understanding the effect of these changes on the mean and other statistical parameters.

Step 3: Compare and Conclude

Compare the results from different data sets to draw conclusions. Check how transformations like scaling affect the figures.
This structured approach helps in efficiently handling data, making sense of it, and drawing meaningful conclusions.