# alternative formulas for standard error of a difference between two means?

I have a question that seems basic but I found two alternative answers online so I thought that I should ask for advice. I have an experiment where each subject makes decisions about target words in two conditions: the word can be either preceded by a related word (the related condition) or it can be preceded by an unrelated word (the unrelated condition). We measure their response times to the target words. My goal is to calculate the standard error associated to the difference between these two condition means. As each subject performs the same number of decisions in the related and unrelated conditions, the observations are paired.

Which formula is appropriate to use to calculate the standard error of the difference between the two condition means? When looking for this online, I found two alternatives, which I paste below. Are they expressing the same computation or are there reasons to choose one over the other? Also, are these formulas appropriate for when the observations are paired? Also, should I calculate mean and SE for each subject separately (and then average across subjects) or can I just treat all subjects' together?

• Have you put some numbers through and observed whether the results are different for the two formulas? – John May 14 '14 at 15:03
• @John: thanks for the comment. yes, i have, and the results are the same. however, i wanted to get a more conceptual explanation (also to know if there is a preferable way to express the SE), and also, the other two questions in my post are still open. – Sol May 14 '14 at 15:09
• You can write LaTeX formulas in questions, btw. – ziggystar May 14 '14 at 15:36

The formulas are equivalent:

$$\sqrt{\frac{\sigma_1^2}{n_1} + \frac{\sigma_2^2}{n_2}} = \sqrt{\frac{\sigma_1^2}{\sqrt{n_1}^2} + \frac{\sigma_2^2}{\sqrt{n_2}^2}} = \sqrt{\left(\frac{\sigma_1}{\sqrt{n_1}}\right)^2 + \left(\frac{\sigma_2}{\sqrt{n_2}}\right)^2} = \sqrt{SE_1^2 + SE_2^2}$$

1. First equality should be clear.
2. Second equality is just pulling the square outside the fraction.
3. Third equality is the formula for standard error of the mean. (also given in the question)

They're so close to the same there isn't really any substantial conceptual difference displayed through the different representations.

Equation 2 would typically be used in the interest of brevity and for an audience or manuscript focused solely on the SE. Equation 1 would be more useful in most cases of establishing derivation or relationships to underlying principles of variance.