T-Distribution Calculator
Free T-Distribution Calculator. Find probabilities, critical values, and percentiles for Student's t-distribution with any degrees of freedom.
Calculate Probability from T-Value
Understanding the T-Distribution
What is the T-Distribution?
The Student's t-distribution is a probability distribution that is used to estimate population parameters when the sample size is small and the population standard deviation is unknown.
Unlike the normal distribution, the t-distribution has heavier tails and its shape depends on the degrees of freedom (df). As df increases, the t-distribution approaches the standard normal distribution.
Key Properties
- •Symmetric: Centered around zero, bell-shaped
- •Heavier tails: More probability in the tails than normal distribution
- •Degrees of freedom: Shape depends on df (df = n-1 for one sample)
- •Convergence: Approaches normal distribution as df → ∞
T-Distribution vs Normal Distribution
T-Distribution
- • Used when population standard deviation is unknown
- • Heavier tails than normal distribution
- • Shape depends on degrees of freedom
- • More conservative for small samples
- • Critical values are larger than z-scores
Normal Distribution
- • Used when population standard deviation is known
- • Standard bell curve shape
- • Fixed shape regardless of sample size
- • Less conservative for small samples
- • Critical values are z-scores
Understanding Degrees of Freedom
Degrees of freedom (df) represent the number of independent pieces of information available to estimate a parameter. In t-distributions, df affects the shape and spread of the distribution.
One Sample T-Test
df = n - 1
where n = sample size
Two Sample T-Test
df = n₁ + n₂ - 2
where n₁, n₂ = sample sizes
Paired T-Test
df = n - 1
where n = number of pairs
Common Applications
Statistical Testing
- • One-sample t-tests for population mean
- • Two-sample t-tests for comparing means
- • Paired t-tests for before/after studies
- • Confidence intervals for small samples
Real-World Examples
- • Medical research with small patient groups
- • Quality control in manufacturing
- • Educational assessment studies
- • Market research with limited data
Related Statistical Tools
T-distribution calculations are part of a broader statistical toolkit. Explore these related tools:
Z-Score Calculator
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Try Z-Score Calculator →T-Test Calculator
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Estimate population parameters with confidence intervals.
Try Confidence Interval Calculator →