What happens to the standard deviation if 7 is added to all measurements within a sample?

What happens to the standard deviation if 7 is added to all measurements within a sample?

• A. The mean remains the same.
• B. The standard deviation remains the same.
• C. Both a and b are true.
• D. Both a and b are false.

Answer: (B)

When a constant value, such as 7, is added to all measurements within a sample, it affects the dataset in a specific way. The mean of the dataset will increase by the same constant amount that has been added, which means it will no longer remain the same. However, the standard deviation measures the amount of variation or dispersion of a set of values relative to the mean. Since the addition of a constant does not change the relative distances between the data points, the standard deviation remains unchanged. Thus, the correct assertion is that the standard deviation remains the same when a constant is added to all values in the dataset. This is a fundamental property of standard deviation: it is invariant under linear transformations that involve additions or subtractions. In this case, the analysis correctly identifies that while the mean changes, the standard deviation does not, validating the choice that states the standard deviation remains the same.

When a constant value, such as 7, is added to all measurements within a sample, it affects the dataset in a specific way. The mean of the dataset will increase by the same constant amount that has been added, which means it will no longer remain the same. However, the standard deviation measures the amount of variation or dispersion of a set of values relative to the mean. Since the addition of a constant does not change the relative distances between the data points, the standard deviation remains unchanged.

Thus, the correct assertion is that the standard deviation remains the same when a constant is added to all values in the dataset. This is a fundamental property of standard deviation: it is invariant under linear transformations that involve additions or subtractions. In this case, the analysis correctly identifies that while the mean changes, the standard deviation does not, validating the choice that states the standard deviation remains the same.