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AP Statistics Flashcards

Central Limit Theorem

For a large enough sample size, the sampling distribution of the sample mean $\bar{x}$ is approximately normal regardless of the population's shape (rule of thumb $n \ge 30$).

Type I error

Rejecting a true null hypothesis — a false positive. Its probability equals the significance level $\alpha$.

Type II error

Failing to reject a false null hypothesis — a false negative. Its probability is $\beta$.

p-value

The probability of getting a result at least as extreme as the one observed, assuming the null hypothesis is true.

z-score

How many standard deviations a value is from the mean: $z = \dfrac{x - \mu}{\sigma}$.

Empirical rule (68-95-99.7)

In a normal distribution, about 68%, 95%, and 99.7% of values fall within 1, 2, and 3 standard deviations of the mean.

Categorical variable

Records which group or category an individual falls into (e.g., color, brand). Summarized with counts and proportions.

Quantitative variable

Takes numerical values for which arithmetic makes sense (e.g., height, time). Summarized with center and spread.

SOCS

The checklist for describing a distribution: Shape, Outliers (unusual features), Center, Spread — always stated in context.

Mean

The arithmetic average and balance point of a distribution. Pulled toward the tail of a skew and toward outliers.

Median

The middle value of an ordered data set. Resists outliers, so it is preferred for skewed distributions.

Mode

The most frequent value(s) in a data set; a distribution can be unimodal, bimodal, or multimodal.

Range

The difference between the maximum and minimum values: a simple, non-resistant measure of spread.

Standard deviation

The typical distance of values from the mean: $s = \sqrt{\dfrac{\sum (x - \bar{x})^2}{n - 1}}$. Larger means more spread.

Variance

The square of the standard deviation; the average squared deviation from the mean.

Interquartile range (IQR)

The spread of the middle 50% of the data: $\text{IQR} = Q_3 - Q_1$. Resistant to outliers.

1.5 x IQR rule

Flags outliers: a value below $Q_1 - 1.5\,\text{IQR}$ or above $Q_3 + 1.5\,\text{IQR}$ is an outlier.

Five-number summary

Minimum, Q1, median, Q3, maximum — the basis of a boxplot.

Percentile

The percentage of values in a distribution at or below a given value.

Skewed right

A long tail on the right (high) side; the mean is typically greater than the median.

Skewed left

A long tail on the left (low) side; the mean is typically less than the median.

Resistant measure

A statistic not strongly affected by outliers (median, IQR), unlike the mean and standard deviation.

Histogram

A graph of a quantitative variable's distribution using bars over equal-width intervals (bins) of values.

Boxplot

A display of the five-number summary; a modified boxplot also marks outliers separately.

Normal distribution

A symmetric, bell-shaped distribution described by its mean $\mu$ and standard deviation $\sigma$.

Standardizing

Converting values to z-scores so distributions with different units can be compared on a common scale.

Cumulative relative frequency

The running total of relative frequencies; an ogive plots it to read percentiles.

Effect of changing units (linear transformation)

Adding a constant shifts center but not spread; multiplying by a constant scales both center and spread.

Scatterplot

A graph of two quantitative variables, with the explanatory variable on the x-axis and the response on the y-axis.

Explanatory vs. response variable

The explanatory variable (x) may explain or predict changes in the response variable (y).

Correlation coefficient (r)

Measures the strength and direction of a LINEAR relationship; $-1 \le r \le 1$. The sign matches the slope.

Direction, form, strength

How to describe a scatterplot association: positive/negative direction, linear/curved form, and weak/strong strength.

Correlation is not causation

A strong r shows association, not cause and effect; only a randomized experiment supports a causal claim.

Least-squares regression line (LSRL)

The line $\hat{y} = a + bx$ that minimizes the sum of squared residuals.

Slope of the LSRL

The predicted change in y for each one-unit increase in x; interpret it in context with units.

y-intercept of the LSRL

The predicted value of y when x = 0; sometimes not meaningful in context.

Residual

Observed minus predicted: $y - \hat{y}$. A positive residual means the model under-predicted.

Residual plot

A plot of residuals vs x; random scatter supports a linear model, while a curved pattern means a line is the wrong fit.

Coefficient of determination (r-squared)

$r^2$ — the fraction of variation in y that is explained by the linear regression model.

Slope formula from r

$b = r \cdot \dfrac{s_y}{s_x}$ — slope equals the correlation times the ratio of the standard deviations.

Extrapolation

Predicting beyond the range of the observed x-values; unreliable because the pattern may not hold there.

Influential point

A point (often with extreme x) that, if removed, markedly changes the regression line or correlation.

High-leverage point

An observation with an x-value far from the mean of x; it can strongly pull the regression line.

Standard deviation of residuals (s)

The typical size of a prediction error from the regression line.

Effect of an outlier on r

A single outlier can substantially raise or lower the correlation, since r is not resistant.

Two-way table

A table of counts for two categorical variables; used to find joint, marginal, and conditional distributions.

Conditional distribution (two-way table)

The distribution of one categorical variable for a fixed category of the other ('given that').

Marginal distribution

The totals in the margins of a two-way table — the distribution of one variable ignoring the other.

Population

The entire group of individuals about which we want information.

Sample

The subset of the population from which we actually collect data.

Census

An attempt to collect data from every individual in the population.

Simple random sample (SRS)

A sample chosen so that every possible group of n individuals has an equal chance of being selected.

Stratified random sample

Divide the population into homogeneous strata, then take an SRS within each stratum.

Cluster sample

Divide the population into clusters, randomly select whole clusters, and sample everyone within them.

Systematic sample

Choose a random starting point in an ordered list, then take every kth individual after it.

Convenience sample

Selecting individuals who are easiest to reach; prone to bias and not suitable for inference.

Voluntary response sample

People choose themselves to respond; typically biased toward those with strong opinions.

Bias

A systematic tendency of a method to favor certain outcomes; a larger sample does not remove it.

Undercoverage

When some groups in the population are left out of the sampling process.

Nonresponse bias

When selected individuals cannot be contacted or refuse to participate, differing from those who do.

Response bias

Inaccurate answers caused by question wording, interviewer effects, or sensitive topics.

Observational study

Measures variables without imposing a treatment; can show association but not causation.

Experiment

Deliberately imposes treatments on units; with random assignment it can establish cause and effect.

Confounding variable

A lurking variable associated with both the explanatory variable and the response, blocking causal claims.

Random assignment

Using chance to assign units to treatments; balances out lurking variables and enables causal inference.

Control group

A baseline group (often given a placebo or standard treatment) for comparison.

Placebo effect

A response to a fake treatment driven by belief; controlled with a placebo and blinding.

Blinding

Keeping subjects (single-blind) and/or those measuring the response (double-blind) unaware of treatment assignments.

Replication

Applying each treatment to many units so chance differences average out.

Blocking

Grouping units by a known variable before random assignment to reduce its effect on the comparison.

Sampling vs. assignment

Random sampling lets you generalize to a population; random assignment lets you infer causation.

Probability

The long-run relative frequency of an outcome over many repetitions of a random process; between 0 and 1.

Law of large numbers

As the number of trials grows, the proportion of successes approaches the true probability.

Complement rule

$P(A^c) = 1 - P(A)$ — the probability an event does not happen.

Addition rule (general)

$P(A \cup B) = P(A) + P(B) - P(A \cap B)$.

Mutually exclusive (disjoint) events

Cannot occur at the same time, so $P(A \cap B) = 0$.

Independent events

One occurring does not change the other's probability, so $P(A \cap B) = P(A)P(B)$.

Mutually exclusive vs. independent

Disjoint events are usually NOT independent — if one happens, the other can't, so probabilities change.

Conditional probability

$P(A \mid B) = \dfrac{P(A \cap B)}{P(B)}$ — the probability of A given that B occurred.

Multiplication rule (general)

$P(A \cap B) = P(A) \cdot P(B \mid A)$.

Test for independence

Events A and B are independent if $P(A \mid B) = P(A)$ (equivalently $P(A \cap B) = P(A)P(B)$).

Random variable

A variable whose numerical value is determined by the outcome of a random process.

Discrete vs. continuous random variable

Discrete takes countable values (whole numbers); continuous takes any value in an interval.

Expected value (mean of a random variable)

$\mu_X = \sum x \cdot P(x)$ — the long-run average outcome.

Variance of a random variable

$\sigma_X^2 = \sum (x - \mu_X)^2 P(x)$; the standard deviation is its square root.

Mean of a sum of random variables

$\mu_{X+Y} = \mu_X + \mu_Y$ — means always add.

Variance of independent X plus or minus Y

For independent variables, variances ADD for both sums and differences: $\sigma_{X \pm Y}^2 = \sigma_X^2 + \sigma_Y^2$.

Binomial setting (BINS)

Binary outcomes, Independent trials, fixed Number of trials n, Same probability p each trial.

Binomial mean and SD

For $X \sim B(n, p)$: mean $np$, standard deviation $\sqrt{np(1-p)}$.

Binomial probability formula

$P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}$.

Geometric setting

Independent trials with the same success probability p, counting the number of trials until the first success.

Geometric mean

The mean number of trials until the first success is $\dfrac{1}{p}$.

Normal approximation conditions

A binomial may be approximated by a normal model when $np \ge 10$ and $n(1-p) \ge 10$.

Probability distribution

A list of all possible values of a random variable with their probabilities, which must sum to 1.

Tree diagram

A branching display of sequential outcomes with their probabilities; multiply along branches.

Mutually exclusive events and the addition rule

If A and B are disjoint, $P(A \cup B) = P(A) + P(B)$.

Continuous random variable probability

For a continuous variable, probability equals area under its density curve; $P(X = x) = 0$ for a single point.

Discrete probability sum

For any discrete random variable, the probabilities of all outcomes sum to exactly 1.

Combining means and SDs (scaling)

If $Y = a + bX$: $\mu_Y = a + b\mu_X$ and $\sigma_Y = |b|\,\sigma_X$.

Parameter

A number describing a population (e.g., $\mu, p, \sigma$) — usually unknown.

Statistic

A number computed from a sample (e.g., $\bar{x}, \hat{p}, s$) used to estimate a parameter.

Sampling distribution

The distribution of a statistic over all possible samples of a fixed size from the population.

Standard error

The standard deviation of a sampling distribution; for the sample mean it is $\dfrac{\sigma}{\sqrt{n}}$.

Unbiased estimator

A statistic whose sampling-distribution mean equals the true parameter it estimates.

Sampling distribution of x-bar (mean)

Mean $\mu$, standard deviation $\dfrac{\sigma}{\sqrt{n}}$; approximately normal by the CLT for large n.

Sampling distribution of p-hat (proportion)

Mean p, standard deviation $\sqrt{\dfrac{p(1-p)}{n}}$; approximately normal when Large Counts holds.

Large Counts condition

For p-hat to be approximately normal: $np \ge 10$ and $n(1-p) \ge 10$.

10% condition

When sampling without replacement, the sample must be no more than 10% of the population for independence.

Effect of sample size on spread

The standard error shrinks as n grows — proportional to $\dfrac{1}{\sqrt{n}}$, not $\dfrac{1}{n}$.

Why the CLT matters

It lets us use normal-based inference for means even when the population is skewed, given a large enough sample.

Sampling variability

The natural variation in a statistic from one random sample to another.

Bias vs. variability of an estimator

Bias = center of the sampling distribution off target; variability = how spread out the estimates are.

Sampling distribution of a difference

The distribution of $\bar{x}_1 - \bar{x}_2$ or $\hat{p}_1 - \hat{p}_2$ across all pairs of samples.

Standard error of x-bar1 minus x-bar2

$\sqrt{\dfrac{\sigma_1^2}{n_1} + \dfrac{\sigma_2^2}{n_2}}$ (variances add for independent samples).

Standard error of p-hat1 minus p-hat2

$\sqrt{\dfrac{p_1(1-p_1)}{n_1} + \dfrac{p_2(1-p_2)}{n_2}}$.

Normal condition for means

A sampling distribution of x-bar is approximately normal if the population is normal OR n is large (CLT).

Simulation of a sampling distribution

Repeatedly drawing random samples and recording the statistic builds an approximate sampling distribution.

Confidence interval

An interval estimate, statistic plus or minus margin of error, for an unknown population parameter.

Margin of error

Critical value times standard error; the half-width of a confidence interval.

Interpreting a confidence interval

'We are C% confident the true parameter lies between the bounds' — C% describes the method, not one interval.

Interpreting the confidence level

In repeated sampling, about C% of the intervals built this way would capture the true parameter.

One-proportion z-interval

$\hat{p} \pm z^* \sqrt{\dfrac{\hat{p}(1-\hat{p})}{n}}$.

Effect of confidence level on width

A higher confidence level uses a larger critical value, producing a wider interval.

Effect of sample size on width

A larger sample lowers the standard error, producing a narrower interval.

Null hypothesis (H0)

The default claim of no effect, no difference, or no relationship that a test seeks evidence against.

Alternative hypothesis (Ha)

The claim being investigated — that an effect, difference, or relationship exists (one- or two-sided).

Significance level (alpha)

The p-value cutoff for rejecting H0; it also equals the probability of a Type I error.

Decision rule

If $p \le \alpha$, reject H0; if $p > \alpha$, fail to reject H0 — you never 'accept' H0.

One-proportion z-test statistic

$z = \dfrac{\hat{p} - p_0}{\sqrt{p_0(1-p_0)/n}}$, using the hypothesized $p_0$ for the standard error.

Power of a test

The probability a test correctly rejects a false null hypothesis: $1 - \beta$.

Ways to increase power

Increase the sample size, the true effect size, or the significance level $\alpha$.

Type I and Type II trade-off

Lowering $\alpha$ reduces Type I errors but raises the chance of Type II errors, all else equal.

p-value is NOT

It is not the probability that H0 is true; it is the probability of the data assuming H0 is true.

Statistical vs. practical significance

A tiny effect can be statistically significant with a huge sample yet have little real-world importance.

Two-proportion z-interval

$(\hat{p}_1 - \hat{p}_2) \pm z^* \sqrt{\dfrac{\hat{p}_1(1-\hat{p}_1)}{n_1} + \dfrac{\hat{p}_2(1-\hat{p}_2)}{n_2}}$.

Pooled proportion

In a two-proportion z-test, the combined sample proportion $\hat{p}_c$ used to estimate the common standard error under H0.

Conditions for proportion inference

Random sample/assignment, the 10% condition, and Large Counts (expected successes and failures at least 10).

Critical value (z*)

The number of standard errors for a given confidence level (e.g., $z^* = 1.96$ for 95%).

One-sided vs. two-sided test

A one-sided Ha tests a single direction (>, <); a two-sided Ha tests for any difference and doubles the tail area.

Confidence interval and significance test link

A value outside a C% two-sided interval would be rejected by the matching test at $\alpha = 1 - C$.

Standard normal distribution

A normal distribution with mean 0 and standard deviation 1; z-scores follow it.

Stating a conclusion in context

Reject or fail to reject H0, then translate it into a statement about the real-world question and variable.

Why use t instead of z for means

The population standard deviation $\sigma$ is unknown, so we estimate it with s, requiring the t-distribution.

t-distribution

Bell-shaped like the normal but with heavier tails, set by its degrees of freedom; approaches normal as df grows.

Degrees of freedom (one-sample t)

$df = n - 1$ for a one-sample t procedure.

One-sample t-interval

$\bar{x} \pm t^* \dfrac{s}{\sqrt{n}}$.

One-sample t-test statistic

$t = \dfrac{\bar{x} - \mu_0}{s/\sqrt{n}}$.

Conditions for one-sample t

Random sample, 10% condition for independence, and an approximately normal population or large n (CLT).

Matched-pairs t-test

Analyzes the differences within naturally linked pairs (before/after, twins) with a one-sample t on $\mu_d$.

Recognizing paired data

Two measurements tied to the same individual or matched unit signal a paired (not two-sample) design.

Two-sample t-test

Compares the means of two INDEPENDENT groups; the null is that the two population means are equal.

Two-sample t-interval

$(\bar{x}_1 - \bar{x}_2) \pm t^* \sqrt{\dfrac{s_1^2}{n_1} + \dfrac{s_2^2}{n_2}}$.

Conservative degrees of freedom

For a two-sample t by hand, use $df = \min(n_1 - 1, n_2 - 1)$ — the smaller sample size minus 1.

Margin of error for a mean

$t^* \dfrac{s}{\sqrt{n}}$; shrink it by increasing n or lowering the confidence level.

Interpreting a t-interval for a mean

'We are C% confident the true mean lies between the bounds,' stated with the variable and units.

Paired t hypotheses

$H_0: \mu_d = 0$ versus an alternative $\mu_d \ne 0$ (or one-sided), where $\mu_d$ is the mean difference.

Two-sample t null hypothesis

$H_0: \mu_1 = \mu_2$ — the two population means are equal (no difference).

Effect of larger sample on a t-interval

A larger sample lowers the standard error and narrows the interval, all else equal.

Interpreting a t statistic

How many estimated standard errors the sample mean is from the hypothesized mean.

Robustness of t procedures

t procedures are fairly robust to non-normality, especially for larger samples, but sensitive to strong outliers.

Checking normality for t

With small n, examine a dotplot/boxplot/normal probability plot for strong skew or outliers before using t.

Standard error of the mean (estimated)

$SE_{\bar{x}} = \dfrac{s}{\sqrt{n}}$ — the sample standard deviation s divided by $\sqrt{n}$.

z for proportions, t for means

Use z procedures for categorical proportions and t procedures for quantitative means.

Paired vs. two-sample

Paired: one column of differences (linked data). Two-sample: two separate, independent groups.

Why pairing helps

Pairing controls for individual variation, often producing a more powerful test than two independent samples.

Chi-square (chi-squared) statistic

$\chi^2 = \sum \dfrac{(O - E)^2}{E}$, summing over all cells — how far observed counts fall from expected.

Chi-square goodness-of-fit test

Tests whether ONE categorical variable's distribution matches a claimed distribution.

Goodness-of-fit degrees of freedom

$df = (\text{number of categories}) - 1$.

Expected count (goodness-of-fit)

$E = n \times p_i$ — sample size times the hypothesized proportion for each category.

Chi-square test of independence

Uses ONE sample classified by TWO categorical variables to test whether they are associated.

Chi-square test of homogeneity

Compares the distribution of one categorical variable across SEVERAL populations or groups.

Two-way table degrees of freedom

$df = (r - 1)(c - 1)$, where r and c are the numbers of rows and columns.

Expected count (two-way table)

$E = \dfrac{(\text{row total})(\text{column total})}{\text{grand total}}$.

Chi-square conditions

Random sample(s), the 10% condition, and ALL expected counts at least 5.

Chi-square is right-tailed

Larger $\chi^2$ means observed counts are farther from expected, giving a smaller (one-tailed) p-value.

Chi-square null hypothesis (independence)

H0: the two categorical variables are independent (no association) in the population.

Chi-square null hypothesis (homogeneity)

H0: the distribution of the categorical variable is the same across all groups.

Components of chi-square

Each cell's contribution $\dfrac{(O-E)^2}{E}$; the largest components show where observed and expected differ most.

Independence vs. homogeneity setup

Independence: one sample, two variables. Homogeneity: multiple samples/groups, one variable.

Conclusion for a chi-square test

Reject or fail to reject H0, then state the evidence for/against association or a difference in context.

When chi-square instead of z

Use chi-square for COUNTS across categories; use z for a single proportion or comparing two proportions.

Effect of sample size on chi-square

Larger samples increase expected counts and the chi-square statistic, making real associations easier to detect.

Inference for the slope

Inference about the population slope $\beta$ of the true regression line relating x and y.

Slope null hypothesis

$H_0: \beta = 0$ — there is no linear relationship between x and y in the population.

Slope t-test statistic

$t = \dfrac{b}{SE_b}$ — the sample slope divided by its standard error.

Slope degrees of freedom

$df = n - 2$, because two parameters (slope and intercept) are estimated.

Slope confidence interval

$b \pm t^* \cdot SE_b$ — an interval estimate for the true slope $\beta$.

Meaning of rejecting H0: beta = 0

Evidence that a real linear association exists between the explanatory and response variables.

LINER conditions

Linear relationship, Independent observations, Normal residuals, Equal variance, Random data collection.

Reading regression output (slope)

The slope is the 'Coef'/'Estimate' on the predictor's row; 'SE Coef' is its standard error.

Standard deviation of residuals (s) in output

Reported as 's'; the typical size of a prediction error from the regression line.

p-value in regression output

The two-sided probability for the slope's t statistic, testing $H_0: \beta = 0$.

Standard error of the slope

$SE_b$ measures how much the estimated slope would vary across repeated samples.

Checking the linear condition

A residual plot with random scatter (no pattern) supports the linear condition for slope inference.

Checking equal variance

Residuals should show roughly constant spread across x (no fanning) for slope inference.

Confidence interval for slope interpretation

'We are C% confident the true slope (predicted change in y per unit x) is between the bounds,' in context.

True regression line

The population line $\mu_y = \alpha + \beta x$ that the sample LSRL estimates.

Why slopes need inference

A nonzero sample slope could arise from sampling variability; a test checks whether the relationship is real.

Marginal vs. conditional in chi-square

Compare conditional distributions across groups to see the association a chi-square test formalizes.

Four-step inference framework

State (parameter and hypotheses), Plan (procedure and conditions), Do (statistic and p-value), Conclude (in context).