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My teacher explained the Central Limit Theorem and provided some examples. He told us that even if we don't have a normally distriuted variable, if we are working with sample means we can consider they are normally distributed (because of the CLT, and under the assumptions of: the samples have a large number of elements (typically larger than 30), the standard deviation and the mean of the population is finite).

But he applied it to a problem with the ages of people.

And I have a question. How can the distribution of the sample mean be normal if the age is always positive? The normal distribution is supposed have a domain from -inf to +inf.

skan
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    (+1) You might benefit from studying our popular thread on [intuitive explanations of the CLT](http://stats.stackexchange.com/questions/3734). – whuber May 13 '15 at 15:59

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You are right that the sample mean will never be normally distributed. But as an approximation it is still quite OK: The probability that the approximating normal distribution is negative, becomes very small even for moderate sample sizes. Just calculate it e.g. for mean age 12, standarddeviation 5 and sample size 40! You usually don't have to care about probabilities less than $10^{-6}$.

However, there are cases, where your notion makes more of a difference, so it is worth keeping it in mind. Namely, if relative frequencies are approximated as normally distributed. In this case, you need either really huge sample sizes or you have to keep your fingers crossed that the probability you want to estimate by the relative frequency, not too far from 0.5.

Horst Grünbusch
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