AP stats formulas and templates

Standard Error of a Proportion

SE = sqrt[ p̂ (1 – p̂) / n ]

Standard Error of a Mean

SE = s / sqrt(n)

Standard Error of a Difference of Two Means

SE = sqrt[ (s1^2 / n1) + (s2^2 / n2) ]

Standard Error of a Difference of Two Proportions

SE = sqrt[ (p̂1 (1–p̂1)/n1) + (p̂2 (1–p̂2)/n2) ]

Standard Error of a Slope

SE_b = s / sqrt[ Σ(x–x̄)^2 ]

Test Statistic for a One-Sample t-Test

t = (x̄ – μ₀) / (s / sqrt(n))

Test Statistic for a Two-Sample t-Test

t = ( (x̄1 – x̄2) – (μ1 – μ2) ) / SE

Margin of Error for Proportion

ME = z* × SE

Margin of Error for Mean

ME = t* × SE

Chi-Square Statistic

χ² = Σ (Observed – Expected)² / Expected

Expected Counts in Chi-Square

Expected = (Row Total × Column Total) / Grand Total

Residual in Regression

residual = actual y – predicted y

Describing a Distribution (SOCS)

“The distribution of [variable] is roughly [shape], with a center around [center]. It has a spread of [spread]. There [are/are not] any clear outliers.”

Example:

“The distribution of test scores is roughly symmetric, with a center around 75 points. It has a spread of about 15 points. There do not appear to be any outliers.”

Interpreting a Confidence Interval

“We are [confidence level]% confident that the true [parameter] lies between [lower bound] and [upper bound].”

Example:

“We are 95% confident that the true proportion of students who prefer online learning is between 0.42 and 0.58.”

Interpreting a p-value

“If the null hypothesis is true, there is a [p-value]% chance of obtaining a result as extreme or more extreme than the observed result.”

Example:

“If the true mean is 500, there is a 0.03 probability of getting a sample mean of 510 or higher.”

Concluding a Hypothesis Test

“Because the p-value is [less/greater] than α = [significance level], we [reject/fail to reject] the null hypothesis. There is [strong/not enough] evidence that [alternative hypothesis in context].”

Example:

“Because the p-value is less than 0.05, we reject the null hypothesis. There is strong evidence that the new medication increases recovery rate.”

Interpreting r² (Coefficient of Determination)

“[r²]% of the variation in [response variable] can be explained by the linear relationship with [explanatory variable].”

Example:

“82% of the variation in exam scores can be explained by the number of hours studied.”

Interpreting Slope of Regression Line

“For each one-unit increase in [x], the predicted [y] changes by approximately [slope].”

Example:

“For each additional hour studied, the predicted exam score increases by about 4 points.”

Interpreting Standard Deviation

“The typical distance between the observed [variable] and the mean is about [standard deviation].”

Example:

“The typical distance between individual weights and the mean weight is about 3.2 pounds.”