statistics

The normal density $N(\mu, \sigma^2)$, drawn across four standard deviations either side of the mean.

The standard normal $N(0, 1)$ — the density every other normal is a shift and a stretch of. It is normal with its parameters fixed, called the way you would call it yourself: inside the pack its own constructs are named without the pack prefix.

A normal density with the probability between two points shaded, and the bounds marked on the axis. The curve it shades is named inside the template, so two uses in one graph never collide (D36) and the caller supplies no id.

A vertical line at the mean, labelled $\mu$.

Vertical lines one standard deviation either side of the mean.

A point sitting on a normal curve, labelled with its z-score. x and z are both supplied because a template cannot compute one from the other: substitution splices text and the value folds parse literals rather than evaluating arithmetic. Writing $z$ out is the honest form until the language grows expression abstraction.

The continuous uniform density on $[a, b]$.

The exponential density with rate $\lambda$.

The Laplace density with location $\mu$ and scale $b$.

The logistic density with location $\mu$ and scale $s$.

The Cauchy density with location $x_0$ and scale $\gamma$ — heavy enough in the tails that it has no mean, which is why it is drawn ten scales wide.

Student's $t$ with $\nu$ degrees of freedom — the normal's heavier-tailed cousin, which is what a small sample's mean actually follows.

The chi-squared density with $k$ degrees of freedom. Drawn from just above zero: at $x = 0$ the density is infinite for $k = 1$ and the curve would have nowhere to start.

The $F$ density with $d_1$ and $d_2$ degrees of freedom — the ratio of two scaled chi-squareds, and so what a variance ratio follows.

The gamma density with shape $\alpha$ and rate $\beta$, $\beta^\alpha x^{\alpha - 1} e^{-\beta x} / \Gamma(\alpha)$ — the exponential is shape $1$, and the chi-squared shape $k/2$ at rate $\tfrac{1}{2}$. Drawn from just above zero, as chi-squared is: below shape $1$ the density is infinite at $x = 0$.

The beta density on $[0, 1]$, $x^{a - 1} (1 - x)^{b - 1} \, \Gamma(a + b) / (\Gamma(a) \Gamma(b))$ — the conjugate prior for a probability, and the posterior it updates to. Drawn from just inside each end: below $a = 1$ or $b = 1$ the density is infinite there.

The Weibull distribution with shape $k$ and scale $\lambda$, as its density, its hazard $h(x) = (k/\lambda)(x/\lambda)^{k - 1}$ or its survival function $S(x) = e^{-(x/\lambda)^k}$. The shape decides the hazard's direction: falling below $1$, constant at $1$ (the exponential), rising above it. Drawn from just above zero: below shape $1$ the density and the hazard are infinite at $x = 0$.

The Gompertz law of mortality, a hazard growing exponentially with age, $h(x) = B c^x$: drawn as its density, its hazard or its survival function $S(x) = \exp(-B (c^x - 1) / \ln c)$. The defaults are a human lifetime in years.

The normal's cumulative distribution — the probability of landing at or below $x$, which is the shaded area of normal read off as a height.

The exponential's cumulative distribution, $1 - e^{-\lambda x}$.

The uniform's cumulative distribution — the straight line every other CDF is measured against.

The logistic's cumulative distribution — the sigmoid, which is this and nothing more.

The Cauchy's cumulative distribution, in terms of $\arctan$.

One tail of a normal beyond a critical value — the rejection region of a one-sided test, shaded and marked. side says which: upper shades above the critical value, lower below it.

The Bernoulli mass function — one trial, drawn as the two outcomes it has. Like every discrete construct here it is a lollipop rather than bars: a bar chart stands on a categorical axis, and the support of a distribution is a run of numbers, so the mass sits on a numeric axis like everything else on the plane. A stem claims nothing but the value it stands at, which is what $P(X = k)$ says.

The binomial mass function $\binom{n}{k} p^k (1-p)^{n-k}$ — one stem for each $k$ from $0$ to $n$, generated from the parameters rather than written out. One lollipop per $k$ rather than one holding every $k$: a series is text the compiler reads in one piece, and @for generates ELEMENTS, so there is no way to fold a loop into an attribute. Each mass is written once either way, which is the part that matters. A stem in highlight and one outside it are two @ifs, one colour each.

The Poisson mass function $e^{-\lambda} \lambda^k / k!$. The support is infinite, so to says where to stop drawing — far enough out that the tail is invisible, not so far that the stems vanish.

The geometric mass function $p(1-p)^{k-1}$ — the trial on which the first success arrives, counted from $1$.

The hypergeometric mass function $\binom{K}{k} \binom{N - K}{n - k} / \binom{N}{n}$ — the successes in $n$ draws without replacement from $N$ items, $K$ of them successes. A stem for every $k$ from $0$ to $n$: choose is $0$ outside its range, so a $k$ the population cannot supply stands at zero rather than needing bounds the template could not compute.

The negative binomial mass function $\binom{k - 1}{r - 1} p^r (1-p)^{k-r}$ — the trial on which the $r$th success arrives, counted as geometric counts it. The geometric is $r = 1$.

A strip of effects by group: one dot per comparison, a row per outcome, each row swarmed so a dense row reads as dense, and every dot coloured by which side of the baseline it fell on. The baseline is ruled across the plot and each side is named in a band behind the dots. The picture of a set of paired studies — organic against conventional, treatment against control — where what matters is how many land on each side and how far. values and groups are two columns of one long-format dataset: a row per comparison, its effect in one column and what it measured in the other. The groups go up the y axis as names, so the graph needs no y format: line.