Question

On a test that has a normal distribution, a score of 54 falls two standard deviations
above the mean, and a score of 42 falls one standard deviation below the mean.
Determine the mean of this test.

Answer 1

Let μ be the mean of the distribution and let σ be its standard deviation.

We know that 54 falls two standard deviations above the mean, this can be express as:

$\mu+2\sigma=54$

We also know that 42 falls one standard deviation below the mean, this can be express as:

$\mu-\sigma=42$

Hence, we have the system of equations:

$\begin{gathered} \mu+2\sigma=54 \\ \mu-\sigma=42 \end{gathered}$

To find the mean we solve the second equation for the standard deviation:

$\sigma=\mu-42$

Now we plug this value in the first equation:

$\begin{gathered} \mu+2(\mu-42)=54 \\ \mu+2\mu-84=54 \\ 3\mu=138 \\ \mu=\frac{138}{3} \\ \mu=46 \end{gathered}$

Therefore, the mean of the distribution is 46