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Statistics & Probability

Averages, spread, counting and 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

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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Probability

The Multiplication Rule for Counting Permutations Combinations Tree Diagrams, Tables, and Outcomes Basic Concepts of Probability Probability with Permutations and Combinations What Are the Odds?The Addition Rule for Probability Conditional Probability and the Multiplication Rule The Binomial Distribution Expected Value

Statistics

Gathering and Organizing Data Visualizing Data Mean, Median and Mode Range and Standard Deviation Percentiles The Normal Distribution Applications of the Normal Distribution Scatter Plots, Correlation, and Regression Lines

Describing data

Statistics Mean Median Mode (statistics)Standard deviation Variance Quartile Box plot Histogram Skewness Outlier Correlation

Inference

Statistical inference Sampling (statistics)Confidence interval Statistical hypothesis test P-value Type I and type II errors Student's t-test Analysis of variance Chi-squared test Regression analysis Simple linear regression Statistical significance Bayesian inference

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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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