if we have the following conditions:

• the underlying distribution is Normal
• the population standard deviation (𝜎) is unknown, we cannot transform 𝑋̅ to standard normal
• the sample size is too small to apply Central Limit Theorem (CLT)

However, we can estimate 𝜎 using the sample standard deviation (𝑠) and transform 𝑋̅ to a variable with a similar distribution, the t distribution. There are actually many t distributions, indexed by degrees of freedom (𝑑𝑓).

t-distribution does the following:

• adjusts for the additional uncertainty around 𝑠
• the smaller the sample size:
  • the more uncertain we are about 𝑠
  • t-distribution becomes less like z-distribution (i.e. becomes flatter and wider)
• the larger the sample size:
  • the more certain we are about 𝑠
  • t-distribution becomes more like z-distribution

T-Distribution Probability Density Function

where:

• 𝑣 - degrees of freedom
• 𝚪 - gamma function

T-Distribution vs Z-Distribution (i.e. Standard-Normal-Distribution)

a 𝑡_𝑛−1-distribution looks like a z-distribution 𝑁𝑜𝑟𝑚𝑎𝑙(0,1) but it has heavier tails. A heavier tail accounts for the fact that there is more uncertainty when sample standard deviation (𝑠) is used in place of population standard deviation (𝜎)

T-Distribution vs Degrees of Freedom

as the degrees-of-freedom increase, the t-distribution becomes more like the z-distribution (standard normal distribution)

Resources

• tutorial - https://www.youtube.com/watch?v=UetYS3PaHIo
• t-distribution calculator - https://stattrek.com/online-calculator/t-distribution.aspx