AP Statistics Formula Sheet — Complete Test Reference A complete AP Statistics reference covering descriptive statistics, probability and counting, common distributions, sampling distributions, confidence intervals, and hypothesis testing. The College Board provides a formula sheet on the exam; this sheet covers everything you should know.
Descriptive Statistics Sample Mean
x ˉ = 1 n ∑ i = 1 n x i \bar{x} = \frac{1}{n}\sum_{i=1}^{n} x_i x ˉ = n 1 i = 1 ∑ n x i Sample Variance
s 2 = 1 n − 1 ∑ ( x i − x ˉ ) 2 s^2 = \frac{1}{n - 1}\sum (x_i - \bar{x})^2 s 2 = n − 1 1 ∑ ( x i − x ˉ ) 2 Sample Standard Deviation
Z-Score
z = x − μ σ z = \frac{x - \mu}{\sigma} z = σ x − μ IQR
IQR = Q 3 − Q 1 \text{IQR} = Q_3 - Q_1 IQR = Q 3 − Q 1 Outlier Rule
x < Q 1 − 1.5 ⋅ IQR or x > Q 3 + 1.5 ⋅ IQR x < Q_1 - 1.5 \cdot \text{IQR} \text{ or } x > Q_3 + 1.5 \cdot \text{IQR} x < Q 1 − 1.5 ⋅ IQR or x > Q 3 + 1.5 ⋅ IQR Probability Rules Addition Rule (General)
P ( A ∪ B ) = P ( A ) + P ( B ) − P ( A ∩ B ) P(A \cup B) = P(A) + P(B) - P(A \cap B) P ( A ∪ B ) = P ( A ) + P ( B ) − P ( A ∩ B ) Multiplication Rule (Independent)
P ( A ∩ B ) = P ( A ) P ( B ) P(A \cap B) = P(A) P(B) P ( A ∩ B ) = P ( A ) P ( B ) Conditional Probability
P ( A ∣ B ) = P ( A ∩ B ) P ( B ) P(A \mid B) = \frac{P(A \cap B)}{P(B)} P ( A ∣ B ) = P ( B ) P ( A ∩ B ) Bayes' Theorem
P ( A ∣ B ) = P ( B ∣ A ) P ( A ) P ( B ) P(A \mid B) = \frac{P(B \mid A) P(A)}{P(B)} P ( A ∣ B ) = P ( B ) P ( B ∣ A ) P ( A ) Complement
P ( A c ) = 1 − P ( A ) P(A^c) = 1 - P(A) P ( A c ) = 1 − P ( A ) Random Variables & Expected Value Expected Value (Discrete)
E [ X ] = μ X = ∑ x i p ( x i ) E[X] = \mu_X = \sum x_i p(x_i) E [ X ] = μ X = ∑ x i p ( x i ) Variance
Var ( X ) = σ X 2 = ∑ ( x i − μ X ) 2 p ( x i ) \operatorname{Var}(X) = \sigma_X^2 = \sum (x_i - \mu_X)^2 p(x_i) Var ( X ) = σ X 2 = ∑ ( x i − μ X ) 2 p ( x i ) Standard Deviation
σ X = Var ( X ) \sigma_X = \sqrt{\operatorname{Var}(X)} σ X = Var ( X ) Linear Transform
E [ a X + b ] = a μ X + b , Var ( a X + b ) = a 2 σ X 2 E[aX + b] = a\mu_X + b,\ \operatorname{Var}(aX + b) = a^2 \sigma_X^2 E [ a X + b ] = a μ X + b , Var ( a X + b ) = a 2 σ X 2 Sum of Independent RVs
Var ( X + Y ) = σ X 2 + σ Y 2 \operatorname{Var}(X + Y) = \sigma_X^2 + \sigma_Y^2 Var ( X + Y ) = σ X 2 + σ Y 2 Common Distributions Binomial PMF
P ( X = k ) = ( n k ) p k ( 1 − p ) n − k P(X = k) = \binom{n}{k} p^k (1 - p)^{n - k} P ( X = k ) = ( k n ) p k ( 1 − p ) n − k Binomial Mean / SD
μ = n p , σ = n p ( 1 − p ) \mu = np,\ \sigma = \sqrt{np(1 - p)} μ = n p , σ = n p ( 1 − p ) Geometric PMF
P ( X = k ) = ( 1 − p ) k − 1 p P(X = k) = (1 - p)^{k - 1} p P ( X = k ) = ( 1 − p ) k − 1 p Geometric Mean
μ = 1 p \mu = \frac{1}{p} μ = p 1 Normal PDF
f ( x ) = 1 σ 2 π e − ( x − μ ) 2 / ( 2 σ 2 ) f(x) = \frac{1}{\sigma \sqrt{2 \pi}} e^{-(x - \mu)^2 / (2 \sigma^2)} f ( x ) = σ 2 π 1 e − ( x − μ ) 2 / ( 2 σ 2 ) Standardization
Z = X − μ σ ∼ N ( 0 , 1 ) Z = \frac{X - \mu}{\sigma} \sim N(0, 1) Z = σ X − μ ∼ N ( 0 , 1 ) Sampling Distributions Sample Mean Distribution (CLT)
X ˉ ∼ N ( μ , σ n ) for large n \bar{X} \sim N\!\left(\mu,\ \tfrac{\sigma}{\sqrt{n}}\right) \text{ for large } n X ˉ ∼ N ( μ , n σ ) for large n Standard Error of the Mean
S E x ˉ = s n SE_{\bar{x}} = \frac{s}{\sqrt{n}} S E x ˉ = n s Standard Error of a Proportion
S E p = p ( 1 − p ) n SE_p = \sqrt{\frac{p(1 - p)}{n}} S E p = n p ( 1 − p ) Standard Error (Difference of Means)
S E = s 1 2 n 1 + s 2 2 n 2 SE = \sqrt{\tfrac{s_1^2}{n_1} + \tfrac{s_2^2}{n_2}} S E = n 1 s 1 2 + n 2 s 2 2 Confidence Intervals Mean (Known σ)
x ˉ ± z ∗ ⋅ σ n \bar{x} \pm z^* \cdot \tfrac{\sigma}{\sqrt{n}} x ˉ ± z ∗ ⋅ n σ Mean (Unknown σ, t-Interval)
x ˉ ± t ∗ ⋅ s n \bar{x} \pm t^* \cdot \tfrac{s}{\sqrt{n}} x ˉ ± t ∗ ⋅ n s Proportion
p ^ ± z ∗ p ^ ( 1 − p ^ ) n \hat{p} \pm z^* \sqrt{\tfrac{\hat{p}(1 - \hat{p})}{n}} p ^ ± z ∗ n p ^ ( 1 − p ^ ) Difference of Means
( x ˉ 1 − x ˉ 2 ) ± t ∗ s 1 2 n 1 + s 2 2 n 2 (\bar{x}_1 - \bar{x}_2) \pm t^* \sqrt{\tfrac{s_1^2}{n_1} + \tfrac{s_2^2}{n_2}} ( x ˉ 1 − x ˉ 2 ) ± t ∗ n 1 s 1 2 + n 2 s 2 2 Difference of Proportions
( p ^ 1 − p ^ 2 ) ± z ∗ p ^ 1 ( 1 − p ^ 1 ) n 1 + p ^ 2 ( 1 − p ^ 2 ) n 2 (\hat{p}_1 - \hat{p}_2) \pm z^* \sqrt{\tfrac{\hat{p}_1(1-\hat{p}_1)}{n_1} + \tfrac{\hat{p}_2(1-\hat{p}_2)}{n_2}} ( p ^ 1 − p ^ 2 ) ± z ∗ n 1 p ^ 1 ( 1 − p ^ 1 ) + n 2 p ^ 2 ( 1 − p ^ 2 ) Hypothesis Tests Test Statistic (Mean, z-test)
z = x ˉ − μ 0 σ / n z = \frac{\bar{x} - \mu_0}{\sigma / \sqrt{n}} z = σ / n x ˉ − μ 0 Test Statistic (Mean, t-test)
t = x ˉ − μ 0 s / n t = \frac{\bar{x} - \mu_0}{s / \sqrt{n}} t = s / n x ˉ − μ 0 Test Statistic (Proportion)
z = p ^ − p 0 p 0 ( 1 − p 0 ) / n z = \frac{\hat{p} - p_0}{\sqrt{p_0(1 - p_0)/n}} z = p 0 ( 1 − p 0 ) / n p ^ − p 0 Chi-Square Test Statistic
χ 2 = ∑ ( O − E ) 2 E \chi^2 = \sum \frac{(O - E)^2}{E} χ 2 = ∑ E ( O − E ) 2 Two-Sample t Test
t = x ˉ 1 − x ˉ 2 s 1 2 n 1 + s 2 2 n 2 t = \frac{\bar{x}_1 - \bar{x}_2}{\sqrt{\tfrac{s_1^2}{n_1} + \tfrac{s_2^2}{n_2}}} t = n 1 s 1 2 + n 2 s 2 2 x ˉ 1 − x ˉ 2 Linear Regression Regression Line
y ^ = b 0 + b 1 x \hat{y} = b_0 + b_1 x y ^ = b 0 + b 1 x Slope
b 1 = r ⋅ s y s x b_1 = r \cdot \frac{s_y}{s_x} b 1 = r ⋅ s x s y Intercept
b 0 = y ˉ − b 1 x ˉ b_0 = \bar{y} - b_1 \bar{x} b 0 = y ˉ − b 1 x ˉ Correlation Coefficient
r = 1 n − 1 ∑ ( x i − x ˉ s x ) ( y i − y ˉ s y ) r = \frac{1}{n - 1}\sum\!\left(\tfrac{x_i - \bar{x}}{s_x}\right)\!\left(\tfrac{y_i - \bar{y}}{s_y}\right) r = n − 1 1 ∑ ( s x x i − x ˉ ) ( s y y i − y ˉ ) Coefficient of Determination
