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Statistics & Probability

Summarise data honestly: mean, median, mode, variance and standard deviation, each computed in the open. Counting principles and combinations feed straight into probability.

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Introductory Sampling and data Populations and samples, kinds of data, how a sample can mislead. mean of 4, 8, 15, 16, 23, 42 Introductory Describing data with graphs Stemplots, histograms, box plots — and the quartiles behind them. median of 3, 1, 4, 1, 5, 9, 2, 6 Introductory Mean, median and mode Centre and spread: mean, median, mode, range, variance, standard deviation. stats 2, 4, 4, 4, 5, 5, 7, 9 Core Probability Events, complements, independence, the addition and multiplication rules. 1/6 * 1/6 Core Counting: permutations and combinations Factorials, ordered and unordered selections. 5 choose 2 Core Discrete random variables Probability distributions, expected value, and the binomial, geometric and Poisson families. 10 choose 3 * (1/2)^3 * (1/2)^7 Core Continuous random variables Density functions: probability is area, and the uniform and exponential distributions. integrate 1/10 dx from 2 to 5 Core The normal distribution The bell curve, z-scores, and the 68–95–99.7 rule. integrate e^(-x^2/2)/sqrt(2*pi) dx from -1 to 1 Core The central limit theorem Why averages are normal: the sampling distribution of the mean. 10/sqrt(25) Core Confidence intervals Estimate ± margin of error, and what “95% confident” actually claims. 50 + 1.96*10/sqrt(100) Core Hypothesis testing Null and alternative hypotheses, test statistics, p-values, and the two kinds of error. (52 - 50)/(10/sqrt(25)) Advanced Comparing two samples Differences of means and proportions, and paired samples. sqrt(4^2/20 + 3^2/25) Advanced Chi-square tests Goodness of fit, independence and homogeneity: comparing observed counts with expected ones. (20-25)^2/25 + (30-25)^2/25 Core Linear regression and correlation The least-squares line, the correlation coefficient, and prediction. line through (1,2) and (3,6) Advanced ANOVA and the F distribution Comparing several means at once by comparing variances. variance of 2, 4, 4, 4, 5, 5, 7, 9

Chapters from OpenStax Contemporary Mathematics

Every section of the book, condensed into a lesson with its own practice problems.

7. Probability

The Multiplication Rule for CountingPermutationsCombinationsTree Diagrams, Tables, and OutcomesBasic Concepts of ProbabilityProbability with Permutations and CombinationsWhat Are the Odds?The Addition Rule for ProbabilityConditional Probability and the Multiplication RuleThe Binomial DistributionExpected Value

8. Statistics

Gathering and Organizing DataVisualizing DataMean, Median and ModeRange and Standard DeviationPercentilesThe Normal DistributionApplications of the Normal DistributionScatter Plots, Correlation, and Regression Lines

Symbols used here

\pm
plus or minus
Both signs at once: x = 3 ± 2 means 5 and 1.
\approx
approximately equal
Equal to the precision shown, not exactly.
n!
factorial
n × (n−1) × … × 1; the number of orderings of n things. 0! = 1.
\binom{n}{k}
binomial coefficient, "n choose k"
Number of k-element subsets of n things: n!/(k!(n−k)!).
\sum_{k=1}^{n} a_k
summation
Add a_k for k = 1 up to n.
A \cup B,\ A \cap B,\ A \setminus B
union, intersection, difference
In either; in both; in A but not B.
\bar{x},\ \mu
sample mean, population mean
Average of the data; average of the whole population.
\sigma,\ s,\ \sigma^2
standard deviation, sample s.d., variance
Typical distance from the mean; its square.
P(A),\ P(A \mid B)
probability, conditional probability
Chance of A; chance of A given that B happened.
E[X],\ \operatorname{Var}(X)
expected value, variance
Probability-weighted average of X; its spread.
N(\mu, \sigma^2),\ z
normal distribution, z-score
The bell curve with mean μ and variance σ²; (x − μ)/σ.

Questions people ask

Mean or median — which should I use?

Median when the data have outliers or a long tail (incomes, house prices); mean when the data are roughly symmetric and you want every value to count. Report both if they disagree — the gap is itself information.

What does a p-value actually say?

The probability of seeing data at least this extreme if the null hypothesis were true. It is not the probability that the null hypothesis is true.

Why divide by n − 1 for the sample variance?

The sample mean sits closer to the sample than the true mean does, so squared deviations from it are slightly too small on average; dividing by n − 1 instead of n corrects the bias.

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