# Question about t statistic. Of these two, which formula is correct? t(x) = frac(bar(X) - mu )(sqrt(frac(1)(n)sum left ( X_(i) - bar(X)right )^(2)))sqrt(n)

Question about t statistic. Of these two, which formula is correct?
$t\left(x\right)=\frac{\overline{X}-\mu }{\sqrt{\frac{1}{n}\sum {\left({X}_{i}-\overline{X}\right)}^{2}}}\sqrt{n-1}$
$t\left(x\right)=\frac{\overline{X}-\mu }{\sqrt{\frac{1}{n-1}\sum {\left({X}_{i}-\overline{X}\right)}^{2}}}\sqrt{n}$
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Franklin Frey
They are mathematically equivalent, because
$\frac{\sqrt{n-1}}{\sqrt{1/n}}=\sqrt{n}\sqrt{n-1}=\frac{\sqrt{n}}{\sqrt{1/\left(n-1\right)}}.$
That said, the second formula is a better reflection of how the t statistic is calculated, because it is obtained by estimating the standard deviation of the population from the sample when it is unknown. Thus,
${s}^{2}=\frac{1}{n-1}\sum _{i=1}^{n}\left({X}_{i}-\overline{X}{\right)}^{2}$
is the unbiased estimator for the variance ${\sigma }^{2}$ when ${X}_{i}$ are iid normal random variables with unknown mean μ and unknown variance ${\sigma }^{2}$. Then the statistic
$\frac{X-\mu }{s/\sqrt{n}}$
is Student $t$ distributed where $s/\sqrt{n}$ is the standard error of the mean.