# When to use t-distribution instead of normal distribution?

According to the Student's t-distribution wiki article the t-distribution is used instead of the Normal distribution "when estimating the mean of a normally distributed population in situations where the sample size is small and the population standard deviation is unknown".

An unknown population standard deviation implies that it would have to be estimated from the samples itself which is inaccurate with small sample sizes. According to the Z-test wiki article a sample size >= 30 implies the use of a normal distribution, a sample size < 30 implies the use of the t-distribution. (t-test for reference.) Is this assumption common best practice? How does this relate to sample size determination (estimation of a mean)?

• When you cannot influence sample size and you have a small sample, t is more robust than normal. This is widely accepted, but I would need to go back to the book to provide a proof. The „estimation of mean“ link you provide uses a normal distribution. So poor for small samples. – Peter Nov 10 '19 at 19:51

## 2 Answers

Student t-distribution by definition is a distribution of mean estimates from samples taken from the normally distributed population.

T-distribution has thicker tails and it gets thinner with increase of degrees of freedom, which in turn depends on sample distribution. So, at some point it closely resembles normal distribution and can be substituted by it.

As far as I remember (although, not 100% sure) this sample size of 30 threshold is valid for the t-test with α=0.05 (widely accepted Type I error level). Although, of you have much smaller α (e.g. 0.0001) you need to go much further to the tails of the distributions where the difference between t-distribution and normal distribution will be much more evident and thus you are better off using t rather than normal for larger sample size.

Another issue is standard deviation (or rather here we are talking about standard error). Normal distribution (and thus z-test) requires knowledge of population standard deviation. If you do not know it, you need to estimate it from the sample and it is obviously will be only an estimation of the population standard deviation. Student t-distribution handles estimated standard deviation better because using it with normal distribution (which should have only population standard deviation) will create extra errors (you will incorrectly estimate your type I (and Ii) errors).

So, the answer is, if you do not know population mean (which is almost always the case in the real world), use t-distribution. If you know population mean, be careful with 30 sample threshold. Depends on your application it can be significantly higher.

So we use t-distribution over normal distribution when the sample size is small because the answers are more accurate. T-distribution is generally used for smaller sample sizes so yes to answer your question, its a good practice. Because as the sample size increases, the t distribution curve starts resembling a normal distribution curve anyways. And when the population distribution of a given set of data is normal, we use the normal distribution anyways. Whereas the T statistic = (Sample mean – hypothesised mean)/sample standard error

• You made a copy-paste of this ResearchGate comment, at least quote your sources. – Leevo Apr 3 '20 at 13:30
• My bad. Rewrote the answer in my own words and further simplified it – Sam Apr 3 '20 at 13:59