Appendix A: Advanced Placement (AP) Statistics Formulas

This web page shows the formulas that are provided to students at the Advanced Placement (AP) Statistics Exam.

Students do not need to memorize these formulas for the exam, but they should know how to use the formulas. Each formula links to a web page that explains how to use the formula.

Descriptive Statistics

• Sample mean = x = ( Σ x_i ) / n
• Sample standard deviation = s = sqrt [ Σ ( x_i - x )² / ( n - 1 ) ]
• Pooled sample standard error = SE_pooled = sqrt [ (n₁ - 1) * s₁² + (n₂ - 1) * s₂² ] / (n₁ + n₂ - 2) ]
• Simple linear regression line: ŷ = b₀ + b₁x
• Regression coefficient = b₁ = Σ [ (x_i - x) (y_i - y) ] / Σ [ (x_i - x)²]
• Regression slope intercept = b₀ = y - b₁ * x
• Sample correlation coefficient = r = [ 1 / (n - 1) ] * Σ { [ (x_i - x) / s_x ] * [ (y_i - y) / s_y ] }
• Regression coefficient = b₁ = r * (s_y / s_x)
• Standard error of regression slope = s_b1 = sqrt [ Σ(y_i - ŷ_i)² / (n - 2) ] / sqrt [ Σ(x_i - x)² ]

Probability

• Rule of addition: P(A ∪ B) = P(A) + P(B) - P(A ∩ B)
• Rule of multiplication: P(A ∩ B) = P(A) P(B|A)
• Expected value of X = E(X) = μ_x = Σ [ x_i * P(x_i) ]
• Variance of X = Var(X) = σ² = Σ [ x_i - E(x) ]² * P(x_i) = Σ [ x_i - μ_x ]² * P(x_i)

The following formulas assume that X has a binomial distribution, the probability of success is P, the number of trials is n, and the number of successes are x. And _nC_x refers to combinations.

• Binomial formula: P(X = x) = b(x; n, P) = _nC_x * P^x * (1 - P)^{n - x} = _nC_x * P^x * Q^{n - x}
• Mean of binomial distribution = μ_x = n * P
• Standard deviation of binomial distribution = σ_x = sqrt[ n * P * ( 1 - P ) ]
• Mean of sampling distribution of the proportion = μ_p = P
• Standard deviation of sampling distribution of the proportion = σ_p = sqrt[ P * ( 1 - P ) / n ]

The following formulas assume that x is the mean of a simple random sample of size n from an infinitely-large population, having a mean of μ and a standard deviation of σ.

• Mean of sampling distribution of the mean = μ_x = μ
• Standard deviation of the sampling distribution of the mean = σ_x = σ/sqrt(n)

Inferential Statistics

• Standardized test statistic = (Statistic - Parameter) / (Standard deviation of statistic)
• Confidence interval: Sample statistic + Critical value * Standard error of statistic
• Standardized test statistic = (Statistic - Parameter) / (Standard deviation of statistic)
• Standard deviation of sample mean = σ_x = σ/sqrt(n)
• Standard deviation of proportion = σ_p = sqrt[ P * (1 - P)/n ] = sqrt( PQ / n )
• Standard deviation of difference of sample means = σ_d = sqrt[ (σ₁² / n₁) + (σ₂² / n₂) ]
• Standard deviation of difference of sample proportions = σ_d = sqrt{ [P₁(1 - P₁) / n₁] + [P₂(1 - P₂) / n₂] }
• Standard deviation of difference of sample proportions:
  σ_d = sqrt[P * (1 - P)] * sqrt[ (1 / n₂) + (1 / n₂)]       when P₁ = P₂
• Chi-square test statistic = Χ² = Σ[ (Observed - Expected)² / Expected ]