# statistics t test sample size

in the formula for t test, t = (mean1 - mean2)/sqrt[(var1 + var2)/N]

1. is N the total number of subjects in both groups being compared or the number of subjects in each group or condition?

2. df = N-2 is N the total number of subjects in both groups being compared or the number of subjects in each group or condition?

• Where did you get those formulas? The one for $t$ is not consistent with the one for $df$.
– whuber
Jun 11, 2014 at 22:12
• from a statistician, but the p value obtained through excel differs. what is the correct formula? Jun 11, 2014 at 23:41

There are two fairly common forms for the independent-samples two-sample t-test:

1. The assumed-equal-variance form:

$$t = \frac{\bar {X}_1 - \bar{X}_2}{s_p \cdot \sqrt{\frac{1}{n_1}+\frac{1}{n_2}}}\,,\quad$$ where $$s_p = \sqrt{\frac{(n_1-1)s_{1}^2+(n_2-1)s_{2}^2}{n_1+n_2-2}}\,$$, with df $$\nu=n_1+n_2-2$$.

2. The Welch (unequal variance) approximation:

$$t = {\overline{X}_1 - \overline{X}_2 \over s_d}\,,$$ where $$s_d = \sqrt{{s_1^2 \over n_1} + {s_2^2 \over n_2}}$$, with approximate df $$\nu= \frac{(s_1^2/n_1 + s_2^2/n_2)^2}{(s_1^2/n_1)^2/(n_1-1) + (s_2^2/n_2)^2/(n_2-1)}.$$.

The equation you have is for equal-n, equal-variance and is a simplified version of the first form.

The $$N$$ in your first formula is the sample size in each group, $$n=n_1=n_2$$. Your df is then $$2n-2$$.

The $$N$$ in your second equation is inconsistent with the $$N$$ in your first equation - they seem to be being used for different things. Specifically, in the second equation, it seems to represent the total sample size, $$n_1+n_2$$.