Ultimate AP Statistics Formula and Exam Cheat Sheet

Posted on Oct 6, 2026 in Statistics

Random:

everyone has an equal chance.

Systematic:

choose every kth person.

Stratified:

divide into groups, randomly sample from each group.

Cluster:

divide into groups, randomly select entire groups.

Convenience:

choose whoever is easiest to reach. Qualitative = Categories = Eye color, Quantitative =  Numbers = Height, Discrete = Counted = Number of siblings, Continuous = Measured = Weight. Common bias:

Selection bias:

sample doesn’t represent population.

Voluntary response:

people choose whether to participate.

Non-response:

selected people don’t respond.

Response bias:

wording/instructions influence answers. GRAPHS & DESCRIPTIVE STATISTICS:
Bar/Pie → qualitative data Histogram → quantitative data, Stem-and-leaf → quantitative, Dotplot → quantitative, Time-series → data over time.

Center

Mean:

x̄ = Σx / n, Mean is not resistant to outliers. Median is resistant.
Mode = most frequent value.

Range:

Range=Maximum−Minimum.

Sample variance: s^2 = Σ (xi – x̄)² / (n – 1). Sample standard deviation:

s = sqrt( sum(x_i – x_bar)^2 / (n – 1) ). IQR: IQR = Q3- Q1. Outliers: Lower fence: Q1−1.5 (IQR); Upper fence: Q3+1.5(IQR). Values outside the fences = potential outliers.

Z-SCORES:

Formula: z = (x – μ) / σ. x = individual value, μ\mu = population mean, σ\sigma = population standard deviation:

Remember: z > 0:

above the mean, z < 0:
below the mean, z = 0:
exactly at the mean.

REGRESSION Equation: y = a + bx or y = mx + b:

a = y-intercept, b = slope, y = predicted y-value.

Correlation r:

Between −1 and +1.
Positive → variables increase together. Negative → one increases as the other decreases. Closer to ±1 → stronger relationship. Closer to 0 → weaker relationship. R² = proportion of variation.

IMPORTANT:

Correlation does not prove causation.

PROBABILITY-

Complement Rule: P(Ac)=1-P(A). Think:

“Not A”. Addition Rule:

P(A or B)=P(A)+P(B)−P(A and B). If A and B are mutually exclusive:
P(A or B)=P(A)+P(B).

Probability Rules:

0 ≤ P(A) ≤ 1. 0 = impossible. 1 = certain. 

RANDOM VARIABLES & BINOMIAL- Discrete Random Variable:

A random variable that takes specific/countable values. For a probability distribution: 0≤P(x)≤1, and: ∑P(x)=1.

Expected Value:

μ=∑[xP(x)].

Variance:

σ2=∑[x2P(x)]−μ2.

Standard Deviation:

σ=√σ2.

BINOMIAL DISTRIBUTION:

Four Requirements- Remember BINS:
B = Binary: only 2 outcomes, I = Independent trials, N = Number of trials is fixed, S = Same probability of success.

Mean

Μ=np.

Standard Deviation:

σ=√np(1−p)

NORMAL DISTRIBUTION – Standard Normal:

μ=0,σ=1. Z Formula: z=x−μ/σ

SAMPLING DISTRIBUTIONS

Sample Mean

Mean:

μxˉ=μ\mu_{\bar{x}}=\mu

Standard error:

SE=σnSE=\frac{\sigma}{\sqrt n}

Important

As n increases, the standard error gets smaller.

Sample Proportion

Mean:

μp^=p\mu_{\hat p}=p

Standard error:

SE=p(1−p)nSE=\sqrt{\frac{p(1-p)}{n}}

Conditions:

np≥10np\geq10

and

n(1−p)≥10n(1-p)\geq10

9. CONFIDENCE INTERVALS

General Structure

Estimate±Margin of Error\boxed{\text{Estimate}\pm\text{Margin of Error}}

Think:

Confidence Interval = Estimate ± ME

Mean — σ Known

Use z:

ME=z∗σnME=z^*\frac{\sigma}{\sqrt n}

Mean — σ Unknown

Use t:

ME=t∗snME=t^*\frac{s}{\sqrt n}

Degrees of freedom:

df=n−1df=n-1

MEMORY TRICK

Σ known → Z

Σ unknown → T

Proportion

ME=z∗p^(1−p^)nME=z^*\sqrt{\frac{\hat p(1-\hat p)}{n}}

Confidence Level

Higher confidence → wider interval

Larger sample size → narrower interval

Interpretation

Correct:

“We are 95% confident that the true population parameter is between ___ and ___.”

Don’t say:

“There is a 95% chance the parameter is in the interval.”

10. HYPOTHESIS TESTING

Null Hypothesis

H0H_0

Usually contains:

==

Alternative Hypothesis

HaH_a

Can contain:

<,>,≠<,\quad>,\quad\neq

Decision Rule

Compare:

P-valuevs.αp\text{-value}\quad\text{vs.}\quad\alpha

Usually:

α=0.05\alpha=0.05

If:

p≤αp\leq\alpha

➡️ Reject H0H_0

If:

p>αp>\alpha

➡️ Fail to reject H0H_0

Never say you “proved H0H_0 true.”

Errors

Type I Error

Reject H0H_0 when H0H_0 is actually true.

Think:

False positive

Type II Error

Fail to reject H0H_0 when H0H_0 is actually false.

Think:

False negative

11. WHICH TEST DO I USE?

This is probably one of the most important parts to memorize.

One Mean

Σ known:

➡️ One-sample z-test

Σ unknown:

➡️ One-sample t-test

One Proportion

➡️ One-proportion z-test

Two Independent Means

Two completely separate groups.

Example:

Compare average test scores of Class A vs. Class B.

Standard error:

SE=s12n1+s22n2SE=\sqrt{\frac{s_1^2}{n_1}+\frac{s_2^2}{n_2}}

Paired Means

Same people measured twice or naturally matched pairs.

Examples:

• Before vs. After

• Weight before vs. After

• Same students’ scores before and after tutoring

First calculate the differences, then perform a one-sample t-test on those differences.

Two Proportions

Compare proportions from two independent groups.

Pooled proportion:

p^pool=x1+x2n1+n2\hat p_{pool}= \frac{x_1+x_2}{n_1+n_2}

⭐ TEST SELECTION FLOWCHART

What type of data?

Numerical/Mean?

One group?

→ σ known → z

→ σ unknown → t

Two groups?

→ Independent → two-mean procedure

→ Paired/before-after → paired t

Proportion?

One group?

→ one-proportion z

Two independent groups?

→ two-proportion z

🧮 CALCULATOR CHEAT SHEET

🚨 FORMULAS TO MEMORIZE

Z-score

z=x−μσ\boxed{z=\frac{x-\mu}{\sigma}}

Mean Standard Error

SE=σn\boxed{SE=\frac{\sigma}{\sqrt n}}

Proportion Standard Error

SE=p(1−p)n\boxed{SE=\sqrt{\frac{p(1-p)}n}}

Binomial Mean

μ=np\boxed{\mu=np}

Binomial SD

σ=np(1−p)\boxed{\sigma=\sqrt{np(1-p)}}

IQR

IQR=Q3−Q1\boxed{IQR=Q_3-Q_1}

Regression

y^=a+bx\boxed{\hat y=a+bx}

Confidence Interval

estimate±ME\boxed{\text{estimate}\pm\text{ME}}

Mean CI, σ known

xˉ±z∗σn\boxed{\bar{x}\pm z^*\frac{\sigma}{\sqrt n}}

Mean CI, σ unknown

xˉ±t∗sn\boxed{\bar{x}\pm t^*\frac{s}{\sqrt n}}

Proportion CI

p^±z∗p^(1−p^)n\boxed{\hat p\pm z^*\sqrt{\frac{\hat p(1-\hat p)}n}}

Two-mean SE

SE=s12n1+s22n2\boxed{SE=\sqrt{\frac{s_1^2}{n_1}+\frac{s_2^2}{n_2}}}

🧠 LAST-MINUTE MEMORY TRICKS

PDF = exact

CDF = cumulative

Normalcdf = probability

InvNorm = value

Σ known = Z

Σ unknown = T

P ≤ α = Reject

P > α = Fail to Reject

Type I = Reject true H0H_0

Type II = Don’t reject false H0H_0

Bigger n = smaller SE

Bigger confidence = wider CI

Correlation ≠ causation

Paired = differences first

Your study guide specifically emphasizes that the final is cumulative and that you should be able to identify the correct procedure, know the calculator function, and state the answer in context.

If your professor allows a one-page notes sheet, I can also turn this into a super-condensed 1-page cheat sheet with only the formulas, calculator commands, and “which test do I use?” rules so you can print it and use it while studying.

Descriptive Statistics: x̄ = ∑x / n | x̄ = sample mean, ∑x = sum of all values, n = sample size, Range=Max−Min. IQR=Q3−Q1, Q1 = first quartile, Q3 = third quartile.

Outlier fences:

Q1−1.5(IQR) Q3+1.5(IQR). Sample Variance & Standard Deviation, s2=∑(x−x̄)2/n−1 s = √s2, s2 = sample variance, s = sample standard deviation, x = individual value, x̄ = sample mean, n = sample size.

Z-Score:

z=x−μ/σ, z = z-score, x = individual value, μ\mu = population mean, σ\sigma = population standard deviation.

Regression:

y=a+bx or y=mx+b, y = predicted y-value, a = y-intercept, b = slope, x = explanatory variable, r² = proportion of variation explained:
 r = correlation, r² = coefficient of determination.

Probability:

P(Ac)=1−P(A)\boxed{P(A^c)=1-P(A)}, P(A)P(A) = probability of A, P(Ac)P(A^c) = probability of NOT A. P(A or B)=P(A)+P(B)−P(A and B), P(A and B) = probability both occur.

Random Variables:

μ=∑[xP(x)], μ\mu = expected value/mean, x = possible outcome, P(x) = probability of outcome, σ2=∑[x2P(x)]−μ2, σ2 = variance, σ\sigma = standard deviation, σ=√σ2.

Binomial:

μ=np σ=√np(1−p), n = number of trials, p = probability of success, 1−p = probability of failure, μ\mu = mean, σ\sigma = standard deviation.

Normal Distribution:

z=x−μ/σ, μ\mu = mean, σ\sigma = standard deviation.
Sampling Distribution — Mean: μx̄=μ SE=σ/√n, μx̄ = mean of sample means, SE = standard error, σ\sigma = population SD, n = sample size.

Sampling Distribution — Proportion:

μp̂=p SE=√p(1−p)/n, p = population proportion, n = sample size.

Confidence Intervals:

CI = estimate ± ME, CI = confidence interval, ME = margin of error.

Mean, σ known:

ME=z* σ/√n.

Mean, σ unknown:

ME=t* s/√n, z* = critical z-value, t* = critical t-value, s = sample SD, df=n−1.

Proportion: ME=z*√p̂(1−p̂)/n. Hypothesis Testing:

p-value≤α⇒Reject H0, p-value>α⇒Fail to Reject H0, H0 = null hypothesis, Ha = alternative hypothesis, α\alpha = significance level, p-value = probability used to make the decision.

Two Independent Means:

SE=s12n1+s22n2, s1,s2 = sample SDs, n1,n2 = sample sizes.

Two Proportions:

p̂ pool=x1+x2 / n1+n2, p̂ pool = pooled proportion, x1,x2x = number of successes, n1,n2 = sample sizes.

Paired Data:

d=x1−x2, d = difference within each pair, x1,x2 = paired observations. Then use the one-sample t procedure on the differences.