Standard Error for difference between two means

I'm aware that the formula for calculating the standard error of a difference between two means is

sqrt(SD1^2/N1 + SD2^2/N2)


Whereas for calculating the standard error of a mean is just sd/sqrt(n).

This question comes from Open Intro Statistics free online book:

5.37 Prison isolation experiment, Part I. Subjects from Central Prison in Raleigh, NC, volunteered for an experiment involving an “isolation” experience. The goal of the experiment was to find a treatment that reduces subjects’ psychopathic deviant T scores. This score measures a person’s need for control or their rebellion against control, and it is part of a commonly used mental health test called the Minnesota Multiphasic Personality Inventory (MMPI) test. The experiment had three treatment groups:

(1) Four hours of sensory restriction plus a 15 minute “therapeutic” tape advising that professional help is available.

(2) Four hours of sensory restriction plus a 15 minute “emotionally neutral” tape on training hunting dogs.

(3) Four hours of sensory restriction but no taped message.

Forty-two subjects were randomly assigned to these treatment groups, and an MMPI test was administered before and after the treatment. Distributions of the differences between pre and post treatment scores (pre - post) are shown below, along with some sample statistics. Use this information to independently test the effectiveness of each treatment. Make sure to clearly state your hypotheses, check conditions, and interpret results in the context of the data.

Using the TR1 as an example, the solution to the exercise, given at the end of the book, calculates the standard error as 3.287 for a T-score of 1.89. This therefore uses the formula SD/sqrt(n) = 12.3/sqrt(14) = 3.287

Given that the question relates to a difference between two means (6.21), why was the standard error calculated in this way? Because, I thought that the SE was calculated by the first formula in my question above sqrt(SD1^2/N1 + SD2^2/N2)?

I can see that with the information provided in the question it's not possible to calculate the SE using this method, since we would need the standard deviation of each group, not just that of the differences.

So, I'm confused about when which formula to use when calculating the standard error for a difference in means. I thought it was sqrt(SD1^2/N1 + SD2^2/N2) however the exercise solution uses SD/sqrt(n)

When to sue which and why not sqrt(SD1^2/N1 + SD2^2/N2)?

• The first formula is used when there independence between the two means. The "pre - post" difference implies that the pre and post were taken on the same individuals and therefore likely not independent. You might want to look at en.wikipedia.org/wiki/Student%27s_t-test. – JimB Feb 15 '18 at 4:24
• Ah so it's because it's the difference between means of paired data? – Doug Fir Feb 15 '18 at 4:29
• Yes. It is because of the paired data. – JimB Feb 15 '18 at 4:30