Question

A population has a mean ($\mu$) of 35 and a standard deviation ($\sigma$) of After 3 points are added to every score in the population, what are the new values for the mean and standard deviation?

• A. $\mu=35$ and $\sigma=5$
• B. $\mu=35$ and $\sigma=8$
• C. $\mu=38$ and $\sigma=5$
• D. $\mu=38$ and $\sigma=8$

Hint:

Adding or subtracting a constant changes the mean but has no effect on the standard deviation or variance.
Multiplying or dividing by a constant scales both the mean and the standard deviation.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
In statistics, it is important to understand how linear transformations—such as adding a constant to every score—affect the measures of central tendency (like the mean) and measures of dispersion (like the standard deviation).

Step 2: Detailed Explanation:
Let us analyze the mathematical effects of adding a constant (\(c\)) to every value (\(X\)) in a dataset:
Effect on the Mean (\(\mu\)):
The mean is a measure of the center of the distribution.
If we add a constant \(c\) to every value, the entire distribution shifts along the axis by that distance.
Mathematically, the new mean (\(\mu_{\text{new}}\)) is:
\[ \mu_{\text{new}} = \mu_{\text{old}} + c \]
Given \(\mu_{\text{old}} = 35\) and \(c = 3\):
\[ \mu_{\text{new}} = 35 + 3 = 38 \]
Effect on the Standard Deviation (\(\sigma\)):
The standard deviation measures the spread or dispersion of the data around the mean.
It is calculated based on the differences between individual scores and the mean:
\[ \sigma = \sqrt{\frac{\sum (X_i - \mu)^2}{N}} \]
If we add a constant \(c\) to both \(X_i\) and \(\mu\), the difference between them remains unchanged:
\[ (X_i + c) - (\mu + c) = X_i - \mu \]
Because the distance between the data points does not change, shifting the entire distribution does not affect its spread.
Therefore, the standard deviation remains the same:
\[ \sigma_{\text{new}} = \sigma_{\text{old}} = 5 \]

Step 3: Final Answer:
The new values are \(\mu = 38\) and \(\sigma = 5\).