How to calculate standard deviation of change in a data set? I'm writing a review in which I need to find the change in mean score of a treatment group before and after treatment. The standard deviation of the mean is known for pre and post treatment seperately. Is it possible to calculate the standard deviation for the change in score?
Example data:
Number of participants = 29
Pre-treatment mean and SD = 68.07, 25.43
Post-treatment mean and SD = 58.31, 21.94
Mean change in score = 68.07 - 58.31 = 9.76
P value = 0.001
What is the standard deviation of this change?
 A: It would be best if you had access to the raw data, but it is possible to back-calculate what you want given the mean difference, the number of observations, and the p-value. This experiment is a within-subjects design, so the p-value should have been calculated with a paired t-test.
First, we can use the p-value and the number of observations to determine the value of the t-statistic. Because this is a paired design, there are 28 degrees of freedom. I'm going to assume the p-value you have is two-tailed, so we can use a t-distribution calculator to figure out the t-value corresponding the the 99.95th percentile of the distribution, which is $ t = 3.674 $.
Remember that the t-statistic is just an expression of the mean difference divided by the standard error of the mean difference:
$$ t = \dfrac{\bar{X} - \bar{Y}}{S.E.(\bar{X} - \bar{Y})} $$
In the case of the paired t-test, this is really just:
$$ t = \dfrac{(\bar{X} - \bar{Y}) * \sqrt{n}}{S.D.(\bar{X} - \bar{Y})} $$
You know t, n, and the mean difference, so you can solve this equation to find that the standard deviation of the mean difference is:
$$ S.D.(\bar{X} - \bar{Y}) = 14.3 $$
