Formulas Used on the AP Statistics Exam
This web page shows the formulas that are provided to students at the Advanced Placement 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 [ (n_{1} - 1) * s_{1}^{2} + (n_{2} - 1) * s_{2}^{2} ] / (n_{1} + n_{2} - 2) ]
- Simple linear regression line: ŷ = b_{0} + b_{1}x
- Regression coefficient = b_{1} = Σ [ (x_{i} - x) (y_{i} - y) ] / Σ [ (x_{i} - x)^{2}]
- Regression slope intercept = b_{0} = y - b_{1} * x
- Sample correlation coefficient = r = [ 1 / (n - 1) ] * Σ { [ (x_{i} - x) / s_{x} ] * [ (y_{i} - y) / s_{y} ] }
- Regression coefficient = b_{1} = 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) = σ^{2} = 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 _{n}C_{x} refers to combinations.
- Binomial formula: P(X = x) = b(x; n, P) = _{n}C_{x} * P^{x} * (1 - P)^{n - x} = _{n}C_{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
- 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 error for the difference between two sample means = SE_{d} = sqrt[ (s_{1}^{2} / n_{1}) + (s_{2}^{2} / n_{2}) ]
- Standard error for the difference between two sample proportions = SE_{d} = sqrt{ [P_{1}(1 - P_{1}) / n_{1}] + [P_{2}(1 - P_{2}) / n_{2}] }
- Chi-square test statistic = Χ^{2} = Σ[ (Observed - Expected)^{2} / Expected ]