Calculate distributions: probability, quantile, mean

In brief

Choose a distribution, enter the parameters, ask the question: “at most (≤)”, “less than (<)”, “at least (≥)”, “more than (>)”, “between a and b” or “value for a probability”. The result appears as a decimal and in percent (0.0443 = 4.43%), as a sentence and as a picture with the coloured and hatched area or the hatched bars.

Discrete or continuous

• For the binomial, Poisson, hypergeometric and custom distributions X counts whole values. “At most 5” is then not the same as “less than 5”: the value 5 itself has its own probability. This is the most common school mistake, which is why Statable shows all four questions one below the other.
• For the normal, t, χ² and F distributions single values have probability 0. “At most” and “less than” are the same there, and so are “at least” and “more than”.

Quantile

For discrete distributions the quantile is the smallest value x with P(X ≤ x) ≥ p (the definition used by R). Statable shows the probability actually reached, because it is usually slightly above p, as only whole values exist. It is not the rejection region of a test: that is determined by the tile “Understand the test for a proportion” using the school rule.

Custom distribution

Enter values x and probabilities P(x) in a table, as decimals or fractions (1/6). Statable calculates E(X) = Σ x · P(x) and Var(X) = Σ x² · P(x) − E(X)². If the sum is not 1, nothing is adjusted silently: a question appears whether rounded values were used, with the choices “Scale to 1” and “Keep as is”. E(X) is the long-run mean, not the most frequent value.

Normal approximation of the binomial distribution

• The normal curve with μ = n · p and σ = √(n · p · (1 − p)) lies over the bars. P(a ≤ X ≤ b) is shown exactly next to the approximation without and with continuity correction. The correction calculates from a − ½ to b + ½, because bar k reaches from k − ½ to k + ½; it is more accurate, some schools calculate without it.
• Laplace condition: if σ is greater than 3, the approximation is considered usable; otherwise it is rough. The symbol ✓ or ! stands next to the text, colour alone carries no information.
• Sigma rules: the table shows 1σ, 2σ, 3σ and the school values 1.64σ, 1.96σ, 2.58σ with the approximation and the exact binomial probability of the whole-number limits. The school values are rounded, the exact ones are 1.6449, 1.9600 and 2.5758; so 1.64σ gives only about 89.9% rather than “90%”.

Mean, variance and standard deviation

Every distribution shows μ, σ² and σ. What does not exist is stated and not shown as “NaN”: the t distribution has no mean for at most 1 degree of freedom and no finite variance for at most 2; the F distribution has a mean from df₂ > 2 and a variance from df₂ > 4.

Hypergeometric

Statable labels N (elements in total), K (of them hits) and n (drawn without replacement), as taught in school. R writes phyper(q, m, n, k) with m = K hits, n = N − K misses and k = drawn; the names differ, the result does not. Example lottery “6 of 49”: N = 49, K = 6, n = 6.

For specialists

Densities after Loader (1999, as in R dbinom, dpois, dhyper), tails by summing from the mode outwards, beta and gamma functions with continued fractions, t and F distributions via the beta function. Quantiles of continuous distributions agree with R to a relative 1e-8, quantiles of discrete distributions exactly. Checked against R and against PSPP; printed examples: table of the binomial cumulative function n = 20, p = 0.3 (F(5) = 0.4164), standard normal table.

Sources

• KMK (Standing Conference of the German Ministers of Education): educational standards in mathematics for the general university entrance qualification (2012), guiding idea “data and chance” (binomial distribution, mean and standard deviation of discrete random variables, simulations).
• Lösungswege 8 (ÖBV, Austrian textbook), chapters 5 to 7: continuous random variables, normal distribution with approximation of the binomial distribution.
• Teacher training Baden-Württemberg: binomial and normal distribution, Laplace condition, continuity correction.
• Loader, C. (2000): Fast and accurate computation of binomial probabilities.