Student's T Distribution

It is called "Student's" t-distribution because its creator, William Sealy Gosset, published his groundbreaking 1908 paper under the pen name "Student."

This is a continuous probability distribution that looks very similar to the standard normal (z) distribution. It is symmetric, bell-shaped, and centered at zero. However, it is shorter, flatter, and has heavier tails than a normal distribution.

This distribution is used to see how confident a sample mean to the population mean.

This distribution was invented to work with smaller sample sizes. When the sample is small, the normal distribution approximation adds more uncertainty. The student's T distribution corrects this uncertainty.

Core Properties

• Symmetry & Mean: Like the normal distribution, it is perfectly symmetric around a mean of $0$.
• Heavier Tails: Because sample standard deviations fluctuate more than population standard deviations, extreme values are more common in a t-distribution.
• Degrees of Freedom ($df$): The t-distribution isn't just one curve; it is a family of curves defined by its degrees of freedom, which typically equals your sample size minus one ($df = n - 1$).
• Asymptotic to the Normal: As your sample size (and thus degrees of freedom) increases, the t-distribution morph into the standard normal distribution. Usually, once sample sizes exceed $n = 30$, the differences are negligible.

T-Distribution vs. Normal (Z) Distribution

Use-cases

1. The population standard deviation is unknown.
2. The sample size is small (traditionally $n < 30$).

$t$ - statistic

This is very similar to how $Z$ statistic is defined in normal distribution. Here since the true population mean and deviation is unknown, we use a hypothesized population mean and sample std. deviation.

$$T = \frac{\bar{X} - \mu}{\frac{s}{\sqrt{n}}}$$

Where:

• $\bar{X}$ is your sample mean
• $\mu$ is the hypothesized population mean
• $s$ is your sample standard deviation
• $n$ is your sample size