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I'm newcomer in statistic so I'll ask probably stupid question: if normally distributed observation required to use t-test, why couldn't be used z-statistic and subsequent calculation of probability?

I mean, our observations came from $N(0, 1)$, so why it must use some strange terrific t-distribution, when using normal distribution fits naturally. What is intuitive behind $T = \frac{\overline{X} - \mu}{\frac{s}{\sqrt{n}}}$ came not from $N(0, 1)$, but from $t(n - 1)$.

And I have another question: Is there situations, when data isn't distributed normally? I mean, the CLT says: if there are enough of our observations, then their sum is subject to a normal distribution.

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  • $\begingroup$ the first question isn't clear $\endgroup$
    – Aksakal
    Commented Aug 4, 2023 at 22:56
  • $\begingroup$ I've edited it. $\endgroup$
    – Maxim
    Commented Aug 4, 2023 at 23:46

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answer to the second question: CLT says what it says assuming certain conditions. Depending on the version of CLT and the dataset the conditions may or not be present. So, the answer is that in many fields most data isn't normally distributed.

answer to the first question: look up the story of life of Gosset, a brewer from Guinness. He came up with Student t distribution. The problem is that you rarely know the true variance of the population. So, you have to estimate it from the sample. Sample variance is a function of the sample, i.e. a random variable itself. So, when you plug a random variable into the Gaussian formula instead of true variance, what you get is not Gaussian but Student t distribution.

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  • $\begingroup$ I know CLT with finite expectation and variance conditions. And only one distribution with such properties - Cauchy distribution, but I f it faced it extremely rarely. So can you explain in more detail? $\endgroup$
    – Maxim
    Commented Aug 4, 2023 at 23:52
  • $\begingroup$ variance is only one aspect, dependence is another. in many cases outside engineering and physics it is easy to see how noises that affect the variable are not independent, which makes application of CLT problematic. as for examples, it is very hard to find stock return series that are normal $\endgroup$
    – Aksakal
    Commented Aug 5, 2023 at 2:31
  • $\begingroup$ @Maxim, the Cauchy distribution is not that exotic - for example, the ratio of two standard normals has a standard Cauchy distribution. And the CLT requires that you (theoretically) add up infinitely many independent random variables, but the real world is not that infinite. OK, you most often get very good approximation with already about n=10 - so good that in the old days normal distributed variables were simulated by adding up twelve uniform random variables. $\endgroup$
    – Ute
    Commented Aug 5, 2023 at 5:10
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For your second question, in addition to @aksakal 's post, most variables are not "added together" and many can't be.

Many, many variables are not normally distributed. Any categorical variable. But also all sorts of numeric ones. Income is a famous example, but almost any variable that involves money. Even some physical variables (weight of adult humans, for instance).

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