Subject
Statistics
Describe data with summary statistics, reason about probability, and work with the normal distribution and regression.
Begin at I if you’re new to Statistics.
Mean, Median, Mode
Three ways to describe the middle of a data set. Mean averages, median picks the middle, mode finds what repeats most.
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Range and IQR
Two simple measures of spread. Range covers the full extent; IQR covers the middle 50%, robust against outliers.
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Standard Deviation
Variance measures the average squared deviation from the mean; standard deviation is its square root, in the original units.
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Five-Number Summary
Describe a data set with five values: min, Q1, median, Q3, max. The foundation of the box plot.
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Graphical Displays
Read histograms, box plots, dot plots, and stem-and-leaf displays. Each makes a different feature of the data obvious.
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Distribution Shape
Classify a distribution as symmetric, skewed right, skewed left, or uniform — from a picture or from mean/median.
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Sampling Methods
Identify the sampling design: simple random, stratified, systematic, cluster, or convenience.
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Observational vs Experimental Studies
If the researcher assigned the treatment, it's an experiment. If they just measured, it's observational. Only experiments can show cause.
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Probability Fundamentals
When outcomes are equally likely, probability is a counting problem: favorable outcomes divided by total outcomes. Plus the complement rule.
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Independent Events
When two events don't affect each other, the probability they both happen is the product of their probabilities. The multiplication rule.
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Conditional Probability
Probability of A given B already happened. Restrict the sample space to just the B outcomes, then count favorable A within that.
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Two-Way Tables
Marginal, joint, and conditional probabilities — all read directly from the cells of a two-way table.
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Permutations and Combinations
Count arrangements (order matters) and selections (order doesn't). Factorial, P(n,k), and C(n,k) are the three core tools.
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Discrete vs Continuous Variables
Counting? It's discrete. Measuring on a real scale? It's continuous. The litmus test: could you have 4.7 of it?
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Expected Value
The long-run average of a discrete random variable. Multiply each value by its probability, then add. The key tool for analyzing games and decisions.
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Normal Distribution and z-scores
The bell curve, described by its mean and standard deviation. A z-score converts any raw value to standard-deviations-from-the-mean.
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Empirical Rule (68-95-99.7)
Quick percentages for normal data: 68% within 1 SD, 95% within 2 SD, 99.7% within 3 SD. Combined with symmetry, you can read off almost any region.
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Binomial Probability
Probability of exactly k successes in n independent trials with success probability p. P(X = k) = C(n, k) · p^k · (1-p)^(n-k).
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Correlation Coefficient
r measures the strength and direction of a linear relationship between two variables, from -1 to 1. r² is the fraction of variance explained.
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Linear Regression
The line of best fit y = mx + b. Compute slope and intercept from data; use the line to predict y from any x.
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Scatter Plots and Residuals
Use the regression line to predict y. Residual = actual − predicted. A pattern in residuals signals a poor linear fit.
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Confidence Intervals and Hypothesis Tests
Turn an estimate into a range with a confidence interval. Test a claim with a z-statistic. The framework underlying all modern inference.
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