The t-distribution and the normal distribution are both continuous probability distributions, but they differ in several key ways:

1. Shape:

   • The normal distribution is symmetric and bell-shaped, with a fixed width determined by the standard deviation.
   • The t-distribution is also symmetric and bell-shaped, but it has heavier tails, meaning it is more prone to producing values that fall far from its mean.

2. Degrees of Freedom (df):

   • The normal distribution does not depend on the sample size; it has a fixed shape.
   • The t-distribution depends on the degrees of freedom (typically related to sample size). As the degrees of freedom increase, the t-distribution approaches the shape of the normal distribution. For small sample sizes (low degrees of freedom), the t-distribution has thicker tails, reflecting greater uncertainty in estimating the population parameters.

3. Use Case:

   • The normal distribution is used when the population standard deviation is known, or when dealing with large sample sizes (typically $n > 30$).
   • The t-distribution is used when the sample size is small and the population standard deviation is unknown. It compensates for the added variability in small samples by having heavier tails.

4. Formula:

   • The probability density function of the normal distribution is: $f(x) = \frac{1}{\sqrt{2\pi}\sigma} e^{-\frac{(x-\mu)^2}{2\sigma^2}}$ where $\mu$ is the mean, and $\sigma$ is the standard deviation.
   • The t-distribution is more complex, depending on the degrees of freedom $\nu$, but its formula is: $f(x) = \frac{\Gamma\left(\frac{\nu+1}{2}\right)}{\sqrt{\nu\pi} \, \Gamma\left(\frac{\nu}{2}\right)} \left( 1 + \frac{x^2}{\nu} \right)^{-\frac{\nu+1}{2}}$ where $\Gamma$ is the gamma function, and $\nu$ is the degrees of freedom.

5. Convergence:

   • As the sample size increases (degrees of freedom increase), the t-distribution converges to the normal distribution. For large samples, the difference between the two becomes negligible.

Summary

• Normal distribution: Known population standard deviation, large sample sizes.
• t-distribution: Unknown population standard deviation, small sample sizes, heavier tails for more variability.

Would you like more details or have any questions?

Here are 5 relative questions:

1. How does the Central Limit Theorem relate to the normal distribution?
2. What are the practical applications of the t-distribution in statistical tests?
3. How do you calculate degrees of freedom in a t-test?
4. Why does the t-distribution have heavier tails than the normal distribution?
5. In what scenarios would you prefer the normal distribution over the t-distribution?

Tip: Use the t-distribution when you're working with small samples and need to account for additional uncertainty in your estimates.