boxplot() — Box plot

boxplot(x)

Key idea

A boxplot summarises data with five numbers: minimum, Q1, median, Q3, and maximum.
The box covers Q1 to Q3 (the quarter and three-quarter marks), holding the middle 50% of values. (e.g. Q1=4 to Q3=10)
The line inside the box marks the median, the middle value. (e.g. median = 7)
The box width is the interquartile range (IQR), a measure of spread. (e.g. IQR = 10 - 4 = 6)
Whiskers stretch to the rest of the data, and far-off dots are flagged as outliers. (e.g. a point at 30)

Definition

$whisker fences = Q_{1} - 1.5 IQR and Q_{3} + 1.5 IQR$

$Q_{1}$: the 25th percentile (lower quartile)
$Q_{3}$: the 75th percentile (upper quartile)
$IQR$: Q_3 minus Q_1, the middle-half width

Where you'd use it

Three classes sat the same test. Drawing a boxplot for each side by side, you can instantly see that class B’s box sits higher (a median around 80) and is shorter, meaning B scored both higher and more consistently than the others.
You have 50 delivery times mostly between 20 and 40 minutes, plus one that took 120. The boxplot draws that lone slow delivery as a separate dot beyond the whisker, flagging it as an outlier.
For a set of house prices, the boxplot shows the median line pushed toward the bottom of the box with a long upper whisker — a quick visual that prices are skewed high by a few expensive homes.

Frequently asked questions

Does the box show the mean?: No — the line is the median. The box spans Q1–Q3 (the IQR).
What are the dots beyond the whiskers?: Points past about 1.5×IQR, flagged as possible outliers.

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.