How good is an assessment with these statistics? We are looking at an assessment for some high school students to measure "growth" in a subject.  The seller showed us their data on the assessment with the starting mean and standard deviation at the beginning of the course, along with the ending mean  and standard deviation, for a large number of students:
$$
\begin{array}{c|c|c|c}
 \style{font-family:inherit}{\text{Starting mean}} & \style{font-family:inherit}{\text{Starting stdev}}
& \style{font-family:inherit}{\text{Ending mean}} & \style{font-family:inherit}{\text{Ending stdev}}\\\hline
 47.56                                       & 7.608    &  48.56   & 7.612 \\ 
\end{array}
$$
This seems too small of a change and too noisy, but beyond highlighting these statistics, are there any standard or useful ways to quantify what can or cannot be inferred with such an assessment?

 A: There are two kinds of issues here---both important:
(1) Statistical Significance. In order to do a statistical test you need to know the
number of subjects. A difference that is unremarkable for
a small sample might be very impressive with a larger sample.
(Simple example: Tossing a coin 10 times and getting 70% Heads
is not evidence of a biased coin. Tossing a coin 1000 times
and getting 69% Heads is very strong evidence of bias.)
Also, the data are not presented in 'paired' format, so you
have no idea how many students improved. The increase in final
scores might result from (a) small bits of progress for almost
everyone or (b) huge gains from a small percentage of carefully
coached students. So again here, you have not been given data
in a format you can use to do a confirmatory test.
(2) Practical Importance. The difference between 'statistically significant' difference
and difference of 'practical importance' is an important one.
Even if you had data in a format that would allow you to
test for yourself and if you found that the one-point difference
is 'statistically significant' (likely to be real), one has
to wonder whether an increase of one point out of about 48 is
the kind of course-long improvement you'd be hoping for.
Putting it together:

*

*You have no way to know whether the one-point
difference is real or just due to random testing variation
and


*Even if the improvement is real there might be
a question whether it is of practical importance.
