📊AP Stats, Psych & Biostats

Statistics Formula Sheet: Z-Score, t-Test & Confidence Intervals

Every high-yield statistics and probability formula for AP Statistics, Quantitative Methods (QM), Psychology Research, and Biostatistics.

High-Yield Formulas

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Sample Mean & Sample Variance

x̄ = (Σxᵢ)/n, s² = Σ(xᵢ - x̄)² / (n - 1)

When to use: Computing center and spread of a sample dataset.

⚠️ Exam Trap: Divide sample variance by (n - 1) for Bessel’s correction, not N.

Standard Normal Z-Score

Z = (x - μ) / σ, Z_x̄ = (x̄ - μ) / (σ / √n)

When to use: Standardizing normal scores or testing a sample mean when population σ is known.

⚠️ Exam Trap: For a sample mean x̄ of size n, divide σ by √n (the standard error).

One-Sample Student’s t-Statistic

t = (x̄ - μ₀) / (s / √n), df = n - 1

When to use: Hypothesis testing a mean when population standard deviation σ is unknown.

⚠️ Exam Trap: Use t (with df = n - 1) whenever you use sample standard deviation s instead of σ.

Confidence Interval for a Mean or Proportion

x̄ ± t* · (s / √n), p̂ ± z* · √[p̂(1 - p̂)/n]

When to use: Estimating a population mean or proportion at 90% (z*=1.645), 95% (z*=1.96), or 99% (z*=2.576).

⚠️ Exam Trap: Quadrupling sample size n cuts the margin of error in half (because of √n).

Conditional Probability & Bayes’ Theorem

P(A|B) = P(A ∩ B) / P(B) = [P(B|A)·P(A)] / P(B)

When to use: Medical screening sensitivity/specificity and conditional probability trees.

⚠️ Exam Trap: Independent events mean P(A ∩ B) = P(A)P(B); mutually exclusive means P(A ∩ B) = 0.

Chi-Square Statistic (Goodness of Fit)

χ² = Σ [ (Observed - Expected)² / Expected ]

When to use: Testing categorical frequency counts (e.g., genetics Punnett ratios or survey tables).

⚠️ Exam Trap: Always use raw counts (never percentages) inside the Chi-Square formula.

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Frequently Asked Questions

When do I use a Z-test vs. a t-test on a statistics exam?

Use a Z-test for proportions (p̂) or when the population standard deviation (σ) is explicitly known. Use a t-test for means whenever σ is unknown and estimated by the sample standard deviation (s).

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