Let two independent random samples of sizes $n_1$ and $n_2$ respectively have been drawn from the same normal population. Let $\overline{X_1}$ and $\overline{X_2}$ be the means and let $s_1$ and $s_2$ be their standard deviations. In order to test whether the the two sample means $\overline{X_1}$ and $\overline{X_2}$ differ significantly or not, the $t$-test statistic is given by

The goal is to test whether two sample means $\bar{x}_1$ and $\bar{x}_2$ are significantly different when both samples come from the same normal population.

We want to check if the difference between the two sample means is large enough to be real, or if it could just be due to random chance. The $t$-statistic measures exactly this — it compares the difference to how much difference we'd expect from random sampling alone.

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The $t$-test statistic always follows this pattern:

$t = \dfrac{\text{Difference in means}}{\text{Standard error of the difference}}$

This gives us:

$t = \dfrac{\bar{x}_1 - \bar{x}_2}{s\sqrt{\dfrac{1}{n_1} + \dfrac{1}{n_2}}}$

where:

$\bar{x}_1 - \bar{x}_2$ is the observed difference between the two sample means

$s$ is the pooled standard deviation (explained in the next step)

$\sqrt{\dfrac{1}{n_1} + \dfrac{1}{n_2}}$ is the adjustment factor that accounts for sample sizes

Note: the term inside the square root is $\dfrac{1}{n_1} + \dfrac{1}{n_2}$ and not $\dfrac{1}{n_1^2} + \dfrac{1}{n_2^2}$. This is because when you subtract two independent quantities, their variances add, and the variance of a sample mean is proportional to $\dfrac{1}{n}$, not $\dfrac{1}{n^2}$.

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Since both samples come from the same population, they share the same true variance. So instead of using $s_1$ and $s_2$ separately, we combine (pool) them into one better estimate:

$s = \sqrt{\dfrac{n_1 s_1^2 + n_2 s_2^2}{n_1 + n_2 - 2}}$

The numerator $n_1 s_1^2 + n_2 s_2^2$ is the total sum of squared deviations from both samples combined. Larger samples contribute more, which makes sense.

The denominator $n_1 + n_2 - 2$ is the total degrees of freedom. We subtract $2$ because we used up one degree of freedom estimating each sample mean — one for $\bar{x}_1$ and one for $\bar{x}_2$.

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$t = \dfrac{\bar{x}_1 - \bar{x}_2}{s\sqrt{\dfrac{1}{n_1} + \dfrac{1}{n_2}}}$ where $s = \sqrt{\dfrac{n_1 s_1^2 + n_2 s_2^2}{n_1 + n_2 - 2}}$

This $t$-statistic follows a $t$-distribution with $n_1 + n_2 - 2$ degrees of freedom.