Student's t-distribution

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In probability and statistics, Student's t-distribution (or simply the t-distribution) is any member of a family of continuous probability distributions that arises when estimating the mean of a normally distributed population in situations where the sample size is small and population standard deviation is unknown. It was developed by William Sealy Gosset under the pseudonym Student.

The t-distribution plays a role in a number of widely used statistical analyses, including Student's t-test for assessing the statistical significance of the difference between two sample means, the construction of confidence intervals for the difference between two population means, and in linear regression analysis. The Student's t-distribution also arises in the Bayesian analysis of data from a normal family.

If we take a sample of n observations from a normal distribution, then the t-distribution with {\displaystyle \nu =n-1} degrees of freedom can be defined as the distribution of the location of the sample mean relative to the true mean, divided by the sample standard deviation, after multiplying by the tandardizing term {\displaystyle {\sqrt {n}}}.

⇒ 그러니깐 결국 t 는 샘플 사이즈를 고려할수 있는, 하는 분산인 것임.

In this way, the t-distribution can be used to say how confident you are that any given range would contain the true mean.

The t-distribution is symmetric and bell-shaped, like the normal distribution, but has heavier tails, meaning that it is more prone to producing values that fall far from its mean. This makes it useful for understanding the statistical behavior of certain types of ratios of random quantities, in which variation in the denominator is amplified and may produce outlying values when the denominator of the ratio falls close to zero. The Student's t-distribution is a special case of the generalised hyperbolic distribution.

T-distributions are slightly different from Gaussian, and vary depending on the size of the sample. (from https://en.wikipedia.org/wiki/Standard_error  )

Note: The Student's probability distribution is approximated well by the Gaussian distribution when the sample size is over 100. For such samples one can use the latter distribution, which is much simpler.