Probability Distributions

Historical Background of Students t Distribution

Students t distribution was published in 1908 by a statistician named William Sealy Gosset. His employer, Guinness Brewery, forbade employees from publishing research, so he used the pen name “Student.” It literally means “Student’s t-distribution,” not a distribution for students!

Students t Distribution Introduction

Suppose, you are interested in finding the average height of women in a certain. Measuring height of every single women is impossible. So, one can take a small sample, say 10 women, and compute the average height of that sample.

Let us suppose, you that your sample average is (probably) not true average for all women. So, you took another sample of 10 and you get a slightly different average. Therefore, there is uncertainty in sample average given that sample is small. This is where the Students t distribution comes in.

Normal and Students t Distribution

In various posts related to normal probability distribution, either confidence interval and hypothesis tests were performed for small sample because the test variable follows normal distribution. However, it is necessary for confidence intervals and hypotheses tests that the population standard deviation ($\sigma$) is known either for small or large samples. This situation is usually unlikely in any real-life situation that the population standard deviation is known and yet try to estimate $\mu$.

A good solution in this case is to use the sample standard deviation $s$ (instead of population standard deviation $\sigma$) from small sample to estimate $\sigma$. However, the estimation of $\sigma$ from a small sample is generally not accurate enough for use in the normal distribution test. We know that if $x$ is normally distributed, then the distribution of

$$z = \frac{\overline{x} – \mu} {\frac{\sigma}{\sqrt{n}}}$$

is the standard normal distribution for any sample size $n$, but the distribution of

$$t = \frac{\overline{x} – \mu}{\frac{s}{\sqrt{n}}}$$

is not. The distribution of random variable is the Students t distribution with $n-1$ degrees of freedom.

Dealing with Small Sample Uncertainty

The t-distribution is used to make inferences (such as confidence intervals and hypothesis tests) about a population average when you have a small sample size and you don’t know the population’s true standard deviation. One can use it whenever ones have to estimate the variation from the sample itself.

Properties of Students t distribution

• The students t distributions are a family of probability density functions. For each possible degrees of freedom, there is a unique students t distribution with that degree of freedom.
• Like the standard normal distribution, the Students t distributions are symmetric, bell-shaped probability density functions with a mean of 0. However, the t distribution is a wider bell curve with thicker tails than the standard normal curve.
• As the degree of freedom increases, the t distributions become closer to a normal distribution. Thus, for large sample sizes ($n\ge 30$), one can use $s$ in palace of $\sigma$ and then use the standard normal distribution.

Applications in the Real World

The following are some real world applications that make use of Students t distribution.

1. Medical Trials: Testing if a new drug lowers blood pressure using a group of 15 patients. You use the t-distribution to see if the observed effect is real or just random chance in your small group.
2. Psychology/Social Science Research: A researcher surveys 20 people to see if a new therapy reduces anxiety. With such a small “$n$” (sample size), they use t-tests (based on the t-distribution) to analyze their results.
3. A/B Testing for a Startup: A small website with low traffic runs an A/B test on its checkout button. Only 100 visitors saw each version. To decide which button is truly better, use the t-distribution to account for the high uncertainty from the low visitor count.
4. Quality Control in a Small Factory: A factory gets a new batch of 8 ball bearings. They measure the diameter and use the t-distribution to create a range (“We are 95% confident the true average diameter of this batch is between $X$ and $Y$ mm”).
5. Any Student’s Science Fair Project: A high school student tests which fertilizer makes plants grow taller, using 5 plants per fertilizer group. The statistical analysis in their poster will likely be based on the t-distribution.

R Frequently Asked Questions