The Students t-distribution is a statistical probability distribution that is used to analyze small sample sizes. It is similar to the normal distribution, but is better suited for smaller samples where the population standard deviation is unknown.
The Students t-distribution is used because it provides more accurate results when working with small sample sizes. It takes into account the uncertainty of the population standard deviation and allows for a wider range of possible outcomes.
The Students t-distribution is derived from the standard normal distribution by dividing the difference between the sample mean and the population mean by the standard error of the mean. This results in a distribution with fatter tails, allowing for a better representation of the uncertainty in small sample sizes.
The main assumptions of the Students t-distribution are that the data follows a normal distribution, the observations are independent of each other, and the sample is a random sample from the population. Additionally, the sample size should be small (usually less than 30) and the population standard deviation should be unknown.
The main difference between the Students t-distribution and the normal distribution is that the t-distribution has fatter tails, meaning it allows for a wider range of possible outcomes. This is due to the uncertainty of the population standard deviation in small sample sizes. Additionally, the t-distribution has a parameter called degrees of freedom, which affects the shape of the distribution and is not present in the normal distribution.