What is the difference between the normal distribution and the t distribution?

I have been studying statistics for a few years now, and one thing that I have learned is that there is always something new to learn.

I have been studying normal distributions, which are bell curves, and also t distributions. The normal distribution and t distribution are both important probability distributions in statistics.

Normal distribution

1. The shape of the normal distribution is a bell curve that is centred around the mean.
2. The two parameters of the normal distribution are the mean (mu) and standard deviation (sigma).
3. The normal distribution is often used when the sample is large and the standard deviation is known.
4. The tails of the normal distribution are known, so extreme values far from the mean are less likely.
5. The normal distribution is not dependent on degrees of freedom.
6. According to the central limit theorem, the distribution of the sample mean approaches a normal distribution as the sample size increases, regardless of the population distribution.

T distribution

1. The shape of the t distribution is symmetric and bell shaped, but the tails are heavier.
2. The parameters of the t distribution are the degrees of freedom (df), which is calculated as the sample size minus one.
3. The t distribution is used when the sample size is small and the population standard deviation is unknown.
4. The t distribution has heavier tails than the normal distribution, so extreme values are more likely as compared to the normal distribution.
5. The shape of the t distribution depends on the degrees of freedom. As the degrees of freedom increase, the more the t distribution begins to resemble the normal distribution.

The illustration below depicts a t distribution with only 5 degrees of freedom overlaid on the normal distribution. The two distributions can be clearly differentiated on the graph:-

In the illustration below, the degrees of freedom have been increased to 30 and it is evident that the t distribution is now very close to the normal distribution that it is overlaid on:-

The differences between the t distribution and normal distribution are:-

1. The t distribution is more appropriate for small sample sizes, whereas the normal distribution is used for large sample sizes.
2. The t distribution is used when the population standard deviation is unknown, and the normal distribution is used when the population standard deviation is known.
3. The t distribution has heavier tails, providing a more conservative estimate for small samples.