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Also known as: Student’s t-Distribution
A distribution with heavier tails useful for small samples.
DESCRIPTION
The Student’s t-distribution arises when estimating a normal mean with unknown variance from small samples. It has heavier tails than the Normal, controlled by degrees of freedom (ν). As ν increases, it approaches the Normal distribution. It is used to construct confidence intervals and perform t-tests for means. Its robustness to outliers makes it useful in regression and modeling errors.
ALTERNATIVE DEFINITIONS
Wikipedia
In probability and statistics, Student’s t-distribution is any member of a family of continuous probability distributions that arise when estimating the mean of a normally distributed population in situations where the sample size is small and population standard deviation is unknown.
Highlights small-sample use and unknown variance scenario.
Wolfram MathWorld
The Student’s t-distribution is a distribution that arises when a normal random variable is divided by an estimate of its standard deviation.
Focuses on construction from standardized normal with estimated variance.
RELATED TERMS
Degrees of Freedom Confidence Interval t-test Robust Statistics Heavy Tails Variance Estimation Central Limit Theorem Hypothesis Testing
COMPETING TERMS
KNOWLEDGE AREAS
TARGET AUDIENCE
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EXTERNAL SOURCES
- Wikipedia - Student’s t-distribution
- Statlect - Definition, PDF, and properties
- Penn State STAT 414 - Course notes on t-distribution
- NIST - NIST coverage of t-based analysis
TOP QUESTIONS
When should I use the t-Distribution instead of Normal?
Use t when the sample size is small and variance is unknown. For large samples, the t approximates Normal.
Category: How-To
How do degrees of freedom affect the t-Distribution?
Lower degrees of freedom produce heavier tails. As ν→∞, the distribution converges to a Normal.
Category: Explanation
What is a common misuse of the t-Distribution?
Applying it without approximate normality of the underlying data. Consider nonparametric tests if normality is doubtful.
Category: Mistakes
Why is the t-Distribution robust to outliers?
Heavier tails assign more probability to extreme values, reducing the impact on inference compared to the Normal.
Category: Explanation
What are competing error models?
Normal errors are common, but t-errors can be preferable with outliers. Laplace errors are another robust option.
Category: Alternatives
Where is the t-Distribution applied?
t-tests, regression with t-errors, robust modeling, and Bayesian hierarchical models.
Category: Example
How are t critical values used?
They define rejection regions and confidence intervals when variance is estimated, via t-tables or software functions.
Category: How-To
What’s the origin of the name ‘Student’?
It’s the pen name of William Sealy Gosset, who introduced the distribution while working at Guinness.
Category: Origin
Is t-Distribution appropriate for heavy tails in finance?
It’s commonly used for returns modeling due to heavy tails; calibration of ν is crucial.
Category: Relevancy