Guinness. Because drinking Guinness is good for you.
The need for Student's t-distribution
Suggest that you work for Guinness brewery. You are in charge of monitoring the quality of the barley. Let's suggest that they're precious with their barley and do not wish for you to fill their quincunx with barley every time you are meant to evaluate the amount of blarney in the barley.
A quincunx, or bean machine. Because flicking beans is good for you.
It appears that they're more choosy about applying one of Francis Galton's inventions to barley than they are about applying one of his others (eugenics) to the barley. They're even more choosy to allow you to sample Guinness, so you're stuck at your wits' end. What to do? Well, if your name is William Sealy Gosset, you just let your wits end a little further.
What is Student's t-distribution?
I have no idea, but that has not stopped me before. We know that Mr Gosset has to separate the good barley from the blarney barley. We also know that Mr Gosset cannot sample too much barley, let alone sample too much Guinness. We need to determine the average quality of the barley. That is, given that the barley has been sampled quite a bit so even though we do not know the standard deviation of the barley, we assume a normal distribution. And from a few samples, we estimate the average quality of the barley, or the mean.
William Sealy Gosset, who used Student as a pseudonym. He got tired of flicking barley.
Student's t-distribution estimates the mean when you do not know the standard deviation but you're fairly certain that you have a normal distribution. The more samples you take, the closer you get to the real quality of the barley. At least, that was Mr Gosset's excuse to sample more Guinness.
Student's t-distribution can also be used to determine other parameters of the population of barley, but the long and short of it is that you do not have plenty of samples to work from and you'd like to jump to conclusions so you can spend more time drinking Guinness.
How does Student's t-distribution work?
A normal distribution is what you'd get if you flicked your beans into the quincunx all day. The standard deviation is how far your beans get flicked from the centre. We assume the same bell shape for our distribution, but we do not flick any beans. No, we get Mr Gosset to rub a few beans between his fingers and Robert is your mother's brother.
Now we have flicked enough beans to know we'd have a normal distribution and we'd like to know from sampling a few beans how high the middle bar of the bean machine gets stacked. Thanks to Student's t-distribution, we can work it out. And they say that alcohol doesn't solve problems!