Distributions & Critical Values
Compute density, cumulative and tail probabilities, quantiles and critical values for normal, Student’s t, chi-square and F distributions.
Question, try, then observe
The same location: the same tail probability?
Switch among four genuine reference curves while retaining the statistic. Degrees of freedom change the shape. α moves the rejection region; it does not change p for a fixed statistic and distribution.
What did you just see?
Treat the statistic as a position on a curve. Moving farther into the tested tail usually leaves less area, but one location can have different tail probabilities under different distributions or df.
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.
Another 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.
Show formal definition and formula
A reference distribution describes possible statistic values and probabilities under a model. Quantiles and tail areas translate t, ANOVA, chi-square and regression statistics into comparable evidence.
Standard normal density
Student's t density
χ² density, x≥0
F density, x≥0
Cumulative, survival and quantile
- 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.
Common mistakes
Density height is not a point probability: one exact value has probability zero in a continuous distribution. A p-value is not P(H₀ true), and α=0.05 does not mean a conclusion is 95% likely to be correct. A reference curve does not validate assumptions.
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.
You can now explore…
- Read probability curves and tail areas
- Compare t, normal, χ² and F critical values