AP Statistics Cumulative Review

Exam Study Guide

AP Stats Review

Unit Overview

This review covers the core concepts of collecting, analyzing, and drawing conclusions from data.

Descriptive Statistics — Organizing and summarizing data
Sampling & Experimentation — Methods for collecting data
Probability & Distributions — Random variables and the normal curve
Inference — Confidence intervals and hypothesis testing

Descriptive Statistics

The Basics of Data

Variables are characteristics whose values change. Data comes in two main types:

Categorical (Qualitative) — Basic characteristics (e.g., color, gender).

Numerical (Quantitative) — Measurements or observations.
- Discrete: Listable sets or counts.
- Continuous: Any value over an interval (measurements).

Describing Distributions (S.O.C.S.)

When describing numerical graphs, always remember S.O.C.S. and write in context!

Shape — Symmetrical, skewed (left/right), uniform, bimodal.

Outliers — Gaps, clusters, or unusual data points.

Center — The middle of the data (Mean, Median, Mode).

Spread — Variability (Range, IQR, Standard Deviation).

Measures of Center & Spread

Measure	Description	Resistance
Mean (\(\mu\) or \(\bar{x}\))	The average value.	Non-Resistant
Median	The 50th percentile / middle point.	Resistant
Range	Max \(-\) Min.	Non-Resistant
IQR	\(Q3 - Q1\) (Middle 50%).	Resistant
Standard Deviation (\(\sigma\) or \(s\))	Typical deviation from the mean.	Non-Resistant

Resistant means the measure is not heavily affected by outliers.

The Normal Curve & Z-Scores

The Normal Curve is a bell-shaped, symmetrical distribution.

A Z-score tells you how many standard deviations an observation is from the mean. \(z = \frac{x - \mu}{\sigma}\)

The Empirical Rule (68-95-99.7):

68% of data is within \(\pm 1\sigma\).
95% of data is within \(\pm 2\sigma\).
99.7% of data is within \(\pm 3\sigma\).

Interactive: Normal Distribution

norm_data = Array.from({length: 200}, (_, i) => {
  let x = -10 + (i * 0.1);
  let y = (1 / (norm_sigma * Math.sqrt(2 * Math.PI))) * Math.exp(-0.5 * Math.pow((x - norm_mu) / norm_sigma, 2));
  return {x: x, y: y};
})

Plot.plot({
  aspectRatio: 1.5,
  x: {domain: [-10, 10], label: "Value (x)"},
  y: {label: "Density"},
  grid: true,
  marks: [
    Plot.areaY(norm_data, {x: "x", y: "y", fill: "#3498db", fillOpacity: 0.3}),
    Plot.line(norm_data, {x: "x", y: "y", stroke: "#2980b9", strokeWidth: 3}),
    // Mean line
    Plot.ruleX([norm_mu], {stroke: "#e74c3c", strokeWidth: 2, strokeDasharray: "4"}),
    Plot.text([[norm_mu, 0.45]], {text: [`μ = ${norm_mu}`], fill: "#e74c3c", fontSize: 20})
  ]
})

Sampling & Experimentation

Collecting Data

A Census is a complete count of the population. It is often expensive or impossible.

Instead, we use a Sample (a subset of the population).

Simple Random Sample (SRS) — Every unit and every set of units has an equal chance of selection.

Stratified — Divide into homogeneous groups (strata), then SRS each group.

Cluster — Select a random location and sample ALL individuals there.

Systematic — Choose a random starting point, then pick every \(n\)th person.

Bias in Sampling

Bias favors a certain outcome. Avoid these common pitfalls:

Voluntary Response: People choose to participate (often strong opinions).
Convenience Sampling: Asking people who are easy to reach.
Undercoverage: Leaving out groups in the selection process.
Non-response: Selected individuals cannot or will not participate.
Response Bias: False answers or leading questions.

Experimental Design

An Observational Study observes outcomes without applying a treatment, while an Experiment actively imposes a treatment to measure a response.

Principles of Experimental Design: 1. Control — Keep extraneous variables constant. 2. Randomization — Use chance to assign subjects to treatments, reducing bias. 3. Replication — Use many subjects to quantify natural variation.

The only way to show cause and effect is with a well-designed, well-controlled experiment.

Probability & Random Variables

Probability Rules

Probability governs the rules of chance. All probabilities are \(0 < P < 1\).

\(P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B)\)

Mutually Exclusive (Disjoint): Events cannot happen at the same time (\(P(A \text{ and } B) = 0\)).

Independent: Knowing one event doesn’t change the outcome of another. \(P(A \text{ and } B) = P(A) \cdot P(B)\)

Conditional Probability: \(P(A | B) = \frac{P(A \text{ and } B)}{P(B)}\)

Special Discrete Distributions

Binomial Distribution:

Two outcomes (Success/Failure).
Fixed number of trials (\(n\)).
Independent trials with constant probability (\(p\)).
\(\mu_x = np\)

Geometric Distribution:

Same conditions as Binomial, but NO fixed number of trials.
Random variable is when the FIRST success occurs.

Inference: Confidence Intervals & Tests

Sampling Distributions & The CLT

A Sampling Distribution is the distribution of all possible values of a statistic from all possible samples.

\(\mu_{\bar{x}} = \mu_x \quad \text{and} \quad \sigma_{\bar{x}} = \frac{\sigma_x}{\sqrt{n}}\)

\(\mu_{\hat{p}} = p \quad \text{and} \quad \sigma_{\hat{p}} = \sqrt{\frac{pq}{n}}\)

Central Limit Theorem (CLT): When \(n\) is sufficiently large (\(n > 30\)), the sampling distribution of the sample mean is approximately normal, even if the population is not.

Confidence Intervals

Used to estimate an unknown population parameter.

\(\text{C.I.} = \text{Statistic} \pm (\text{Critical Value}) \cdot (\text{Standard Deviation of Statistic})\)

To decrease the Margin of Error: * Decrease confidence level (smaller critical value). * Decrease standard deviation (\(s\)). * Increase sample size (\(n\)).

Interpretation: “We are [x]% confident that the true [parameter] of [context] is between [a] and [b].”

Hypothesis Testing

Testing whether a value occurs by random chance.

Null Hypothesis (\(H_0\)): Statement of “no effect” or “no difference”.
Alternate Hypothesis (\(H_a\)): Statement suspected of being true.
P-Value: Probability of obtaining observed results assuming \(H_0\) is true.

Conclusion: Since the p-value is \(< \alpha\), I reject \(H_0\). There is sufficient evidence to suggest \(H_a\).

Errors & Power

Every decision has the possibility of making an error.

Type I Error (\(\alpha\)) — Rejecting \(H_0\) when \(H_0\) is actually true.

Type II Error (\(\beta\)) — Failing to reject \(H_0\) when \(H_0\) is false.

Power (\(1 - \beta\)) — Probability of correctly rejecting a false \(H_0\).
- Increase Power by: Increasing \(\alpha\), increasing \(n\), or increasing effect size (\(\mu_0 - \mu_a\)).

Chi-Square & Regression

Chi-Square (\(\chi^2\)) Tests

Used to test counts of categorical data. All curves are right-skewed. Expected counts = \((\text{row total} \cdot \text{column total}) / \text{grand total}\).

Goodness of Fit — 1 Sample, 1 Variable. Does observed fit expected?
Independence — 1 Sample, 2 Variables. Are variables independent?
Homogeneity — 2+ Samples, 1 Variable. Are distributions the same?

Least Squares Regression Line (LSRL)

Models bivariate continuous data to minimize residuals.

\(\hat{y} = a + bx\)

Residual: \(y - \hat{y}\) (Actual \(-\) Predicted). A residual plot with no pattern indicates a linear relationship.
Correlation (\(r\)): Strength and direction of a linear relationship \([-1, 1]\).
Coefficient of Determination (\(r^2\)): Proportion of variation in \(y\) explained by the LSRL of \(x\) and \(y\).
Extrapolation: Using LSRL to predict outside the data range (don’t do it!).

Assumption Checks for Inference

Before running tests, verify assumptions:

Test	Assumptions to Check
Proportions (\(z\))	Random sample, \(np \ge 10\), \(n(1-p) \ge 10\), Pop \(\ge 10n\).
Means (\(t\))	Random sample, approximately normal (or \(n > 30\)), no outliers.
Categorical (\(\chi^2\))	Random sample, ALL expected counts \(> 5\).
Regression (slope \(t\))	Linear relation, residual plot has no pattern, constant standard deviation of responses.

Good Luck on the AP Exam!

Remember:

Always write in context.
Define your parameters.
Check your assumptions.
Label your graphs.