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AP Statistics review: key concepts

AP Statistics flashcards: describing data, regression, sampling, experiments, probability, binomial models, intervals and tests, with worked numbers.

38 cards6 topicsAbout 15 minutes a roundAP Statistics style course content (one-term introductory university statistics)

Independent study material written by StudyBrix. AP and SAT are trademarks of College Board and ACT is a trademark of ACT, Inc.; neither organisation is involved with or has reviewed this deck.

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About this deck

This deck is for students in an AP Statistics style course and for anyone preparing for a first university statistics class. It covers exploring one- and two-variable data, least-squares regression, collecting data through samples and experiments, probability rules and random variables, binomial and geometric models, sampling distributions, confidence intervals and significance tests for proportions and means, and chi-square tests.

You need comfortable algebra and the ability to use a calculator for square roots and normal or t probabilities. Every numerical example was checked, so work it yourself and compare. Where a card gives a critical value such as z* = 1.96 or t* = 2.131, practise finding it from a table or calculator too.

Statistics is graded heavily on communication. Each time you finish a calculation, write a one-sentence conclusion in context that names the parameter and the population. For tests, always state hypotheses, check conditions (random, 10% rule, large counts or normality), compute, and conclude. Cards in the inference groups are designed to drill that order.

Common mistakes: interpreting a 95% confidence interval as a 95% chance the parameter is inside this particular interval; saying a P-value is the probability the null hypothesis is true; and concluding cause and effect from an observational study. Another frequent error is accepting the null hypothesis; we can only fail to reject it.

Before you start: Grade 11 algebra; comfort with fractions, percentages, square roots and reading graphs.

All 38 cards

Cover the definitions with a sheet of paper and test yourself, or use the study modes above.

Exploring data

Five-number summary and IQR: Minimum, Q1, median, Q3 and maximum; the interquartile range Q3 - Q1 measures the spread of the middle half.Example: 2, 4, 4, 5, 7, 8, 9, 10, 12: median 7, Q1 4, Q3 9.5, IQR 5.5.
1.5 x IQR outlier rule: A value is flagged as an outlier if it lies more than 1.5 IQR below Q1 or above Q3.Example: Q1 = 4, Q3 = 9.5, IQR = 5.5: fences at -4.25 and 17.75, so 12 is not an outlier.
z-score: Number of standard deviations a value lies from the mean: z = (x - mean)/SD.Example: A score of 85 when the mean is 70 and SD is 10 has z = 1.5.
Skewed distributions and centre: In a right-skewed distribution the mean is pulled above the median; in a left-skewed one it falls below. The median is resistant to extreme values.Example: Incomes are right skewed, so the median income is usually reported.
Linear transformation of data: Adding a constant shifts measures of centre but not spread; multiplying by b multiplies both centre and spread (SD by |b|).Example: Temperatures with mean 20 C and SD 5 C in Fahrenheit (x 1.8 + 32): mean 68 F, SD 9 F.

Regression

Correlation r: Measures the direction and strength of a linear relationship between two quantitative variables; unitless, between -1 and 1, and sensitive to outliers.Example: r = 0.92 between study hours and score is strong and positive, but still not proof of cause.
Least-squares regression line: The line y-hat = a + bx that minimizes the sum of squared residuals; b = r (sy/sx) and it passes through (x-bar, y-bar).Example: r = 0.8, sy = 10, sx = 5, x-bar = 20, y-bar = 50: b = 1.6, a = 50 - 32 = 18.
Residual: Observed y minus predicted y; positive means the point is above the line.Example: With y-hat = 18 + 1.6x, the point (25, 60) has residual 60 - 58 = 2.
Coefficient of determination: r^2: the proportion of variation in y explained by the linear model with x.Example: r = 0.8 gives r^2 = 0.64: 64% of the variation in y is accounted for by x.
Residual plot: Graph of residuals against x or predicted values; a random scatter supports a linear model, while a curve suggests a nonlinear pattern.Example: A U-shaped residual plot means a straight line is not the right model.
Extrapolation: Using a regression line to predict far outside the range of x values in the data; such predictions are unreliable.Example: A growth line for ages 2 to 10 predicts silly heights at age 40.

Collecting data

Stratified random sample: Divide the population into similar groups (strata) and take a simple random sample from each one.Example: Sample 50 students at random from each grade level to compare opinions.
Cluster sample: Divide the population into groups that each resemble the whole, then randomly select entire groups.Example: Randomly pick 5 homerooms and survey every student in them.
Sources of sampling bias: Voluntary response (self-selected people), undercoverage (some groups left out) and nonresponse (chosen people do not answer) all distort results.Example: An online poll that anyone can answer gives a voluntary response sample.
Experiment versus observational study: An experiment imposes a treatment on subjects; an observational study only records existing conditions. Only well-designed experiments support cause and effect.Example: Comparing grades of students who choose to drink coffee is observational.
Blocking: Grouping similar experimental units and randomizing treatments within each group to reduce variability from a known factor.Example: Splitting subjects by gender before randomly assigning a diet, then comparing within each block.

Probability and random variables

General addition rule: P(A or B) = P(A) + P(B) - P(A and B), subtracting the overlap so it is not counted twice.Example: P(A) = 0.5, P(B) = 0.4, P(A and B) = 0.2: P(A or B) = 0.7.
Conditional probability: The probability of A given that B occurred: P(A | B) = P(A and B)/P(B).Example: With the numbers above, P(A | B) = 0.2/0.4 = 0.5.
Independent events: Knowing one event happened does not change the other's probability: P(A | B) = P(A), or P(A and B) = P(A)P(B).Example: Since P(A | B) = 0.5 = P(A), A and B above are independent.
Expected value and SD of a discrete random variable: Mean = sum of x P(x); variance = sum of (x - mean)^2 P(x), and SD is its square root.Example: X = 0, 1, 2 with P 0.5, 0.3, 0.2: mean 0.7, variance 0.61, SD 0.78.
Combining independent random variables: Means add or subtract directly; variances always add (never subtract), so take the square root of the summed variances.Example: Independent SDs 3 and 4: SD of the sum or difference is sqrt(9 + 16) = 5.
Binomial distribution: Counts successes in n independent trials with the same success probability p: P(X = k) = nCk p^k (1 - p)^(n - k).Example: n = 5, p = 0.3: P(X = 2) = 10 x 0.09 x 0.343 = 0.3087.
Binomial mean and standard deviation: Mean = np and SD = sqrt(np(1 - p)).Example: n = 100, p = 0.2: mean 20, SD sqrt(16) = 4.
Geometric distribution: Counts trials until the first success: P(X = k) = (1 - p)^(k - 1) p.Example: p = 0.3: P(first success on trial 3) = 0.7^2 x 0.3 = 0.147.

Sampling distributions and intervals

Sampling distribution of a sample proportion: Centre p, SD sqrt(p(1 - p)/n), approximately normal when np and n(1 - p) are both at least 10.Example: p = 0.4, n = 150: SD = sqrt(0.24/150) = 0.04.
Central limit theorem: For large n, the sampling distribution of x-bar is approximately normal whatever the population shape, with SD sigma/sqrt(n).Example: sigma = 12 and n = 36: the SD of x-bar is 2.
One-proportion z interval: p-hat plus or minus z* sqrt(p-hat(1 - p-hat)/n), given random sampling, the 10% condition and large counts.Example: p-hat = 0.55, n = 400, 95%: SE = 0.0249, margin 0.049, interval about (0.501, 0.599).
Interpreting a confidence level: If we took many samples and built an interval from each, about that percentage of the intervals would capture the true parameter.Example: 95% describes the method's long-run success rate, not one interval's chance.
Sample size for a margin of error: Solve ME = z* sqrt(p(1 - p)/n) for n, using p = 0.5 when no estimate exists, and round up.Example: 95% with ME 0.03: n = (1.96/0.03)^2 x 0.25 = 1067.1, so sample 1068.
One-sample t interval for a mean: x-bar plus or minus t* (s/sqrt(n)) with df = n - 1, used when sigma is unknown.Example: x-bar = 72, s = 8, n = 16: SE 2, t* = 2.131 (df 15), interval 72 plus or minus 4.26.

Significance tests

Null and alternative hypotheses: H0 states no effect or a specific parameter value; Ha states what we suspect, using <, > or not equal. Both refer to the parameter.Example: H0: p = 0.5 and Ha: p > 0.5 for whether a coin favours heads.
P-value: Assuming H0 is true, the probability of a statistic at least as extreme as the one observed; small values are evidence against H0.Example: P-value 0.03 < alpha 0.05, so reject H0.
One-proportion z test: z = (p-hat - p0)/sqrt(p0(1 - p0)/n), using the null value in the SD.Example: p0 = 0.5, p-hat = 0.56, n = 200: z = 1.70, one-sided P-value about 0.045.
Type I and Type II errors: Type I: rejecting a true H0 (probability alpha). Type II: failing to reject a false H0.Example: Type I: approving a drug that does not work; Type II: missing one that does.
Power of a test: The probability of correctly rejecting a false H0; it rises with larger samples, larger alpha and larger true effects.Example: Doubling n raises power to detect the same true difference.
Paired t procedures: Used when observations come in natural pairs; analyse the differences as one sample with df = pairs - 1.Example: Before and after scores for 12 students: test the mean of the 12 differences.
Chi-square goodness of fit test: Compares observed category counts with expected counts from a claimed distribution; df = categories - 1.Example: Die rolled 60 times: counts 8, 12, 10, 9, 11, 10 vs 10 each give chi-square 1.0 with df 5.
Expected count in a two-way table: For independence or homogeneity tests, expected = (row total x column total)/grand total.Example: Row total 40, column total 50, grand total 200: expected count 10.