Solve the following problems. a. After 6 points have been...

Solve the following problems. a. After 6 points have been added to every score in a sample, the mean is found to be $M=70$ and the standard deviation is $s=13 .$ What were the values for the mean and standard deviation for the original sample? b. After every score in a sample is multiplied by $3,$ the mean is found to be $M=48$ and the standard deviation is $s=18 .$ What were the values for the mean and standard deviation for the original sample?

Solve the following problems.
a. After 6 points have been added to every score in a sample, the mean is found to be $M=70$ and the standard deviation is $s=13 .$ What were the values for the mean and standard deviation for the original sample?
b. After every score in a sample is multiplied by $3,$ the mean is found to be $M=48$ and the standard deviation is $s=18 .$ What were the values for the mean and standard deviation for the original sample?

Key Concepts

Linear Transformations

Linear transformations in statistics involve applying operations such as adding a constant or multiplying by a constant to every data point in a dataset. These operations have systematic effects on the mean and standard deviation: adding a constant shifts the mean without changing the standard deviation, while multiplying by a constant scales both the mean and the standard deviation by that constant.

Arithmetic Mean

The arithmetic mean is a measure of central tendency that is calculated by adding all the values in a dataset and dividing by the number of values. It provides a summary statistic that indicates the average outcome of the data, and it is sensitive to every value found in the dataset.

Standard Deviation

Standard deviation is a measure of dispersion that quantifies the amount of variation or spread of a set of data values. It shows how much the individual data points typically differ from the mean of the dataset, giving insight into the consistency or variability within the data.

Effects on Statistical Measures

Understanding the effects of operations on statistical measures is crucial in data analysis. When a constant is added to every observation, the mean increases by that constant but the variability, as measured by the standard deviation, remains the same. Conversely, multiplying every observation by a constant will multiply the mean and the standard deviation by that constant, altering both central tendency and dispersion.

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