splines() — Regression splines

Joins smooth cubic pieces to bend flexibly through data.

splines(knots, x, y)

Key idea

A knot is a chosen cut point on the x-axis. Fit a gentle cubic curve between each pair of knots, joined smoothly with no kinks. (e.g. 4 knots -> 5 cubic pieces)
More knots = more freedom to follow local wiggles.
Splines avoid the wild edge swings of a single high-power formula.
A smoothing spline puts a knot everywhere but adds a roughness penalty to stay smooth.

Definition

$f (x) = β_{0} + β_{1} x + β_{2} x^{2} + β_{3} x^{3} + k \sum θ_{k} (x - ξ_{k})_{+}^{3}$

$ξ_{k}$: the knots (join points)
$(x - ξ_{k})_{+}^{3}$: a cubic that switches on past the knot
$θ_{k}$: how much the curve bends at knot k

In plain words

Rather than one stiff curve for the whole range, bend several smooth pieces and glue them together seamlessly.

Where you'd use it

Modelling a dose–response curve that bends in a non-straight way.
Smoothing a noisy time trend without assuming a formula.
The building block of GAMs — one flexible spline per feature.

Frequently asked questions

What are splines for?: Flexible curve-fitting using piecewise polynomials joined smoothly at knots.

Statistics Playground

An interactive playground for probability and statistics. Pick a function, edit its parameters, and see the distribution curve, the computed value, and a plain-English interpretation — all computed in your browser (no server, no install). The functions mirror familiar R syntax (for example pnorm, dbinom, t.test) and the numbers come from numpy running under Pyodide.

The normal distribution

pnorm(q, mean = 0, sd = 1) — Cumulative probability for the normal distribution.
dnorm(x, mean = 0, sd = 1) — Height of the bell curve at a point (the density).
qnorm(p, mean = 0, sd = 1) — The inverse of pnorm; turns a probability back into a value.
rnorm(n, mean = 0, sd = 1) — Draws random values from a normal distribution.
scale(x) — Re-expresses values as z-scores (mean 0, sd 1).

Binomial & Poisson distributions

dbinom(x, size, prob) — Probability of exactly x successes in n trials.
pbinom(q, size, prob) — Probability of q or fewer successes.
dpois(x, lambda) — Probability of exactly x events at rate λ.
ppois(q, lambda) — Probability of q or fewer events at rate λ.

The chi-square distribution & hypothesis testing

pchisq(q, df) — Cumulative probability for the χ² distribution.
t.test(x, mu = 0) — One-sample t-test against a hypothesised mean.

Correlation & regression

cor(x, y) — Strength of a linear relationship, −1 to +1.
lm(y ~ x) — Fits a least-squares straight line through the points.

Descriptive statistics

mean(x) — The arithmetic average.
median(x) — The middle value of the data.
sd(x) — Standard deviation; typical distance of a value from the mean.
var(x) — Variance; the mean squared deviation from the mean.
IQR(x) — Span of the middle 50% of the data.
range(x) — The minimum and maximum values.
quantile(x) — Sample quantiles that split the data into equal slices.
summary(x) — The five-number summary plus the mean.
weighted.mean(x, w) — An average where each value carries a weight.

Combinatorics, sampling & visualisation

factorial(x) — x!, the ways to arrange x distinct items in order.
choose(n, k) — Ways to choose k items from n, order ignored.
sample(x, size, replace = FALSE) — Draw a random sample.
hist(x, breaks) — A frequency histogram of the data.
boxplot(x) — A box-and-whisker summary of the spread.