Question

A teacher gives a test in which the average of the test scores is 68 and the SD is 14. Realizing that the test was probably too difficult, the teacher decides to scale the tests by adding 10 points to all scores. What is the mean and SD of the scaled test scores?

SD = standard deviation

Answer 1

Answer:

mean = 78

Standard deviation = 14

Step-by-step explanation:

Let's denote the original mean as $\sf \mu$ and the original standard deviation as $\sf \sigma$.

For the original test scores:

• $\sf \textsf{Original Mean }\sf ( \mu ) = 68$
• $\sf \textsf{Original Standard Deviation} ( \sigma) = 14$

Now, the teacher scales the test scores by adding 10 points to all scores.

The new mean ($\sf \mu'$) and the new standard deviation ($\sf \sigma'$) can be calculated using the following relationships:

$\sf \textsf{New Mean ( \sf \mu' )} = \textsf{Original Mean ( 3\sf \mu )} + \textsf{Constant Added}$

$\sf \textsf{New Standard Deviation ( \sigma' )} = \textsf{Original Standard Deviation (\sf \sigma) }$

In this case, the constant added is 10:

$\sf \textsf{New Mean ( \sf \mu' } = 68 + 10 = 78$

$\sf \textsf{New Standard Deviation (\sf \sigma' )} = 14$

Therefore, the mean of the scaled test scores is 78, and the standard deviation remains unchanged at 14.

Key Point:

Adding or subtracting a constant value to all scores only affects the mean, not the standard deviation.

Answer 2

Answer:

Mean = 78

Standard deviation = 14

Step-by-step explanation:

When a constant value is added to each data point in a set, the mean of the set increases by that constant, but the standard deviation remains unchanged.

Given that the original mean (μ) is 68 and the standard deviation (σ) is 14, and the teacher adds 10 points to all scores, the new mean (μ') of the scaled test scores will be:

$\mu' = \mu + \text{constant}$

$\mu' = 68 + 10$

$\mu' =78$

The standard deviation (σ') remains the same as the original standard deviation (σ):

$\sigma' = 14$

Therefore, the mean of the scaled test scores is 78, and the standard deviation is 14.