Statistical Distributions & Critical Values Calculator
Compute density, cumulative and tail probabilities, quantiles and critical values for normal, Student’s t, chi-square and F distributions.
Exact distribution algorithms · valid df · no symmetric χ²/F tails
Results
Review or adjust your inputs, then calculate to see your results and interpretation.
Understand this calculation
Curve height is probability density; area is probability. The CDF is area to the left, a survival probability is area to the right, and a quantile finds a location from cumulative probability. α fixes the planned rejection region; the observed statistic determines the p-value.
See calculation method and assumptions
Formula
Standard normal density
Student's t density
χ² density, x≥0
F density, x≥0
Cumulative, survival and quantile
Variables, rates and periods
- f(x), F(x), S(x)
- Density, cumulative and survival probability; density is not point probability.
- ν, d₁, d₂
- Positive degrees of freedom; F needs numerator and denominator.
- Γ, B
- Gamma and beta normalization functions.
- xq, q, α
- Quantile, quantile probability and rejection area; χ²/F do not use symmetric two tails.
The roles of four distributions
- The standard normal has mean zero and SD one;
- t appears in mean inference with estimated SD.
- Chi-square describes squared statistics and changes skewness with df.
- F is a ratio of appropriately scaled chi-square quantities with numerator and denominator df, and supplies the ANOVA reference.
p and α are different areas
- α sets a fixed rejection region.
- The p-value accumulates probability beyond the observed statistic in the chosen alternative's direction.
- Comparing p with α gives a threshold decision, but neither measures effect size or the probability that the study is true.
The drawing is not the probability domain
- The chart draws a finite readable range;
- probability can remain beyond its edges.
- Numerical results use the full distribution core, not pixels or coarse chart integration.
- Low df can have long tails, so off-chart values remain available as text.
Compute density, cumulative and tail probabilities, quantiles and critical values for normal, Student’s t, chi-square and F distributions.
Worked example
- Choose a distribution and its required df; F needs two degrees of freedom.
- Inspect a statistic and tail probability, or enter a cumulative probability to find a quantile.
- Compare the shaded area, critical value and α. Chi-square and F do not have a symmetric two-tail shortcut.
Frequently asked questions
Why does df change the t distribution?
Estimating population SD from a sample adds uncertainty, producing heavier tails at lower df. The t approaches the normal as df increases, but the formal calculation still uses t.
Why are χ² and F tests usually right-tailed?
The corresponding test statistics are squared-discrepancy sums or variation ratios, for which departure often appears as a large value. The distributions are generally asymmetric; doubling the right tail is not a generic two-sided test.
Are extremely small p-values shown as zero?
Numerical underflow is different from an exact zero. Underflowed tails receive a bound or scientific-notation indication rather than an assertion of exact zero; failed convergence is reported as an error.
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