# The standard normal distribution vs the t-distribution

Given an IID normally distributed sample $X_1,...,X_n$ for $n$ small with mean $\mu$, standard deviation $\sigma$, sample mean $\overline{X}$ and sample standard deviation $s$ (the unbiased estimator form). I understand that

$$\frac{\overline{X} - \mu}{\frac{\sigma}{\sqrt{n}}} \sim N(0,1),$$

but I'm having trouble reconciling this with the fact that

$$\frac{\overline{X} - \mu}{\frac{s}{\sqrt{n}}} \sim t_{n-1}.$$

Since the $t$-distribution is like the standard normal distribution but with a higher variance (smaller peak and fatter tails), this would seem to suggest that the sample standard deviation $s$ systematically underestimates the population standard deviation, making it in fact a biased estimator.

• $s^2$ is unbiased for $\sigma^2,$ so, yes, $s$ is biased for $\sigma.$ Jul 22, 2014 at 1:51

Actually $$s$$ doesn't need to systematically underestimate $$\sigma$$; this could happen even if that weren't true.

As it is, $$s$$ is biased for $$\sigma$$ (the fact that $$s^2$$ is unbiased for $$\sigma^2$$ means that $$s$$ will be biased for $$\sigma$$, due to Jensen's inequality*, but that's not the central thing going on there.

* Jensen's inequality

If $$g$$ is a convex function, $$g\left(\text{E}[X]\right) \leq \text{E}\left[g(X)\right]$$ with equality only if $$X$$ is constant or $$g$$ is linear.

Now $$g(X)=-\sqrt{X}$$ is convex,

so $$-\sqrt{\text{E}[X]} < \text{E}(-\sqrt{X})$$, i.e. $$\sqrt{\text{E}[X]} > \text{E}(\sqrt{X})\,$$, implying $$\sigma>E(s)$$ if the random variable $$s$$ is not a fixed constant.

Edit: a simpler demonstration not invoking Jensen --

Assume that the distribution of the underlying variable has $$\sigma>0$$.

Note that $$\text{Var}(s) = E(s^2)-E(s)^2$$ this variance will always be positive for $$\sigma>0$$.

Hence $$E(s)^2 = E(s^2)-\text{Var}(s) < \sigma^2$$, so $$E(s)<\sigma$$.

So what is the main issue?

Let $$Z=\frac{\overline{X} - \mu}{\frac{\sigma}{\sqrt{n}}}$$

Note that you're dealing with $$t=Z\cdot\frac{\sigma}{s}$$.

That inversion of $$s$$ is important. So the effect on the variance it's not whether $$s$$ is smaller than $$\sigma$$ on average (though it is, very slightly), but whether $$1/s$$ is larger than $$1/\sigma$$ on average (and those two things are NOT the same thing).

And it is larger, to a greater extent than its inverse is smaller.

Which is to say $$E(1/X)\neq 1/E(X)$$; in fact, from Jensen's inequality:

$$g(X) = 1/x$$ is convex, so if $$X$$ is not constant,

$$1/\left(\text{E}[X]\right) < \text{E}\left[1/X\right]$$

So consider, for example, normal samples of size 10; $$s$$ is about 2.7% smaller than $$\sigma$$ on average, but $$1/s$$ is about 9.4% larger than $$1/\sigma$$ on average. So even if at n=10 we made our estimate of $$\sigma$$ 2.7-something percent larger** so that $$E(\widehat\sigma)=\sigma$$, the corresponding $$t=Z\cdot\frac{\sigma}{\widehat\sigma}$$ would not have unit variance - it would still be a fair bit larger than 1.

**(at other $$n$$ the adjustment would be different of course)

Since the t-distribution is like the standard normal distribution but with a higher variance (smaller peak and fatter tails)

If you adjust for the difference in spread, the peak is higher.

Why does the t-distribution become more normal as sample size increases?

The standard normal distribution vs the t-distribution