NUM8ERS · AP Statistics revision
AP Statistics
Formula Sheet
Find the formula. Check the conditions. Explain what your answer means.
A student-friendly reference organised around the five units in our current AP Statistics notes. Each formula comes with its purpose, a reminder about when it applies, and examples you can work through.
Use this guide alongside the topic lessons. Knowing why a formula fits is just as useful as knowing how to substitute the numbers.
Start here
Jump to the skill you need
This guide follows the course framework effective Fall 2026. Older released questions may use nine-unit numbering; match the statistical skill rather than the old unit label.
- Unit 1Data & study design
- Unit 2Probability & distributions
- Unit 3Proportions & chi-square
- Unit 4Means & paired data
- Unit 5Regression
Official exam reference: College Board provides formulas and tables for both exam sections. This NUM8ERS guide adds explanations and examples; it is a revision resource. Find the official reference in the Appendix of the current Course and Exam Description.
Choose the method before the formula
| Question you are answering | Method to consider | Where to look |
|---|---|---|
| How common is a yes/no outcome in one population? | One-proportion -interval or -test | Unit 3 |
| Do two independent groups have different proportions? | Two-proportion -interval or -test | Unit 3 |
| Do category counts fit a distribution, or are two categorical variables associated? | Chi-square test; identify the study design first | Unit 3 |
| What is one population's mean? | One-sample -interval or -test | Unit 4 |
| What is the mean change for matched observations? | Paired -procedure on the differences | Unit 4 |
| Do two independent groups have different means? | Two-sample -interval or -test | Unit 4 |
| How does one quantitative variable help predict another? | Least-squares regression; inspect scatterplot and residuals | Unit 5 |
Interval or test? Use an interval to estimate a parameter with uncertainty. Use a test to assess evidence about a stated population claim.
Read the notation with confidence
| Symbol | Meaning | What to remember |
|---|---|---|
| Sample size; population size | A sample is the observed part of the population. | |
| Sample mean; population mean | The sample statistic estimates the population parameter. | |
| Sample standard deviation; population standard deviation | Both describe variability in individual observations. | |
| Sample proportion; population proportion; null-hypothesised proportion | Choose the right proportion for an interval or a test. | |
| Standard error; margin of error | Standard error estimates a statistic's sampling variability. | |
| Critical values for confidence intervals | The confidence level determines the critical value; a value also depends on degrees of freedom. | |
| Null hypothesis; alternative hypothesis; significance level | State hypotheses about population parameters. | |
| Predicted response; correlation; coefficient of determination | A prediction is different from an observed response. |
Unit 1
Exploring One-Variable Data and Collecting Data
Start by identifying the variable and the group being studied. Numerical summaries describe a distribution; the study design determines what conclusions are supported.
Relative frequency and sample proportion
Here is the count in the category of interest. Divide by the total number of observations to get its share.
Remember: A percentage is . Use the correct total for the group being described.
Mean and median
The mean is the balance point of the data. For the median, order the observations and locate the middle; average the two middle values when the sample size is even.
Remember: The mean is sensitive to extreme values. The median is resistant; choose summaries that fit the shape.
Sample variance and standard deviation
Standard deviation describes a typical scale of distance from the sample mean. It has the same units as the observations.
Remember: This requires . The denominator makes sample variance unbiased under random sampling; sample standard deviation is not exactly unbiased.
Interquartile range and outlier fences
The IQR measures the spread of the middle half of the ordered data. Flag a possible outlier below or above .
Remember: A point exactly on a fence is not outside it. Investigate a flagged value before deciding what it means.
Standardising an observed value
This sample-based score gives the number of sample standard deviations an observation lies above or below the sample mean.
Remember: This requires . A positive score is above the mean; a negative score is below. Standardising alone does not make data normal.
Population mean and standard deviation
Use these definitions when the values for the entire population are available.
Remember: The population formula divides by ; the sample variance formula divides by .
Worked example · Summarise four travel times
Four students have travel times of minutes.
The sample mean is minutes, and the sample standard deviation is about minutes. The median is also minutes. These summaries describe the students observed; whether they represent the school depends on how the sample was selected.
Design reminder: Random sampling supports generalising to the sampled population. Random assignment supports a cause-and-effect comparison of treatments in a well-designed experiment. A large convenience sample does not remove selection bias.
Unit 2
Probability, Random Variables, and Probability Distributions
Translate words into events before you calculate. Distinguish individual outcomes, counts of successes and statistics that vary across repeated samples.
Complement and addition rules
For “at least one”, it can be easier to subtract the probability of none. For “or”, use .
Remember: The “or” includes outcomes in both events. Only omit the overlap term when the events are mutually exclusive.
Conditional probability and multiplication
The condition narrows the denominator to outcomes in , with . The general multiplication rule is .
Remember: For independent events, . Mutually exclusive events with positive probabilities are not independent.
Mean and variability of a discrete random variable
The mean is a probability-weighted long-run average. Variance is , and .
Remember: A valid distribution has and . The expected value need not be a possible individual outcome.
Binomial probability
Use for the number of successes in a fixed number of independent trials with two outcomes and a constant success probability. Here .
Remember: “Exactly” uses a single probability. “At most” sums through the cutoff; “at least” sums from the cutoff, or uses a complement.
Binomial mean and standard deviation
The mean predicts the long-run average count of successes. The standard deviation describes the variability of that count.
Remember: Check the binomial conditions before using either formula. A count and a proportion have different scales.
Normal-model standardisation
For a normal model, convert a value to a standard normal score to find an area, or convert a percentile score back to an original value.
Remember: For here, the second parameter is the standard deviation. A continuous model gives at any single point.
The empirical rule
For an approximately normal distribution, the intervals and contain about and of observations.
Remember: The rule describes an approximately normal distribution. It does not apply to every data set.
Sampling variability: a mean
For independent observations, these describe the centre and spread of the sampling distribution of the sample mean.
Remember: The central limit theorem concerns the distribution of sample means. Larger samples improve the normal approximation; severe skewness or outliers can require a larger sample.
Sampling variability: a proportion
For independent Bernoulli observations, these describe repeated-sample proportions, rather than individual binary outcomes.
Remember: A normal approximation requires and . For sampling without replacement, also check .
Worked example · Exactly three successes
Suppose counts successes in independent trials, each with success probability . The binomial conditions hold.
The probability of exactly three successes is . This is different from the probability of at least three successes, which also includes and .
Standard deviation or standard error? A sampling-distribution standard deviation uses population parameters. A standard error estimates it with sample statistics. For a mean, ; for a proportion, .
Unit 3
Inference for Categorical Data: Proportions
Use proportions for a yes/no outcome and chi-square procedures for category counts. Confidence intervals estimate; significance tests compare the observed result with a null model.
One-proportion confidence interval
Estimate a population proportion. The margin of error is .
Remember: Check random sampling or an appropriate randomised design, independence, and observed counts , .
One-proportion significance test
Test . Choose the tail from the alternative hypothesis before finding the p-value.
Remember: The test uses null counts and , and null-based variability. Do not substitute the sample proportion into this denominator.
Difference of two proportions: interval
Estimate for two independent groups. Keep the order of subtraction consistent with your parameter definition.
Remember: Check observed successes and failures in both groups are each at least . Use an unpooled standard error for the interval.
Difference of two proportions: test
For , estimate the common proportion with , where the numerators are success counts.
Remember: Check pooled expected successes and failures: and for each group. Pool for this equality test, not for the interval.
Sampling distribution of a difference
For independent samples, . Variances add even though the statistics are subtracted.
Remember: These are population-based quantities; estimated standard errors are used when the population proportions are unknown.
Chi-square statistic and goodness of fit
For goodness of fit, expected counts are , where is the null proportion for category . With specified categories, .
Remember: Use counts, not percentages. Check a random sample, independence and every expected count greater than , following the current course framework. The p-value is a right-tail area.
Chi-square: two-way table expected counts
For a table with rows and columns, . Sum chi-square contributions over all cells.
Remember: Independence studies association in one population. Homogeneity compares a categorical distribution across populations or treatments. Check design, independence and every expected count greater than .
Confidence critical values and sample-size planning
For a target margin of error , plan a one-proportion sample with . Here is a planning estimate.
Remember: Use if no estimate is available and round the required sample size up. Express the margin of error as a proportion, not a whole-number percentage.
Worked example · Estimate the proportion who prefer a study app
In a simple random sample of students from a school with at least students, prefer a study app. The independence approximation is reasonable, and the observed success and failure counts, and , meet the large-count condition.
We are confident that between about and of students at this school prefer the app. The interval estimates a population proportion, not the range of individual responses.
Design and independence: State the random sample or random assignment. For samples taken without replacement, check for each relevant population. This sampling fraction check is unnecessary for a randomised experiment. Two-group formulas also require independent groups.
Unit 4
Inference for Quantitative Data: Means
Use a mean procedure for a quantitative response. First decide whether the data form one sample, matched pairs, or two independent samples.
One-sample mean interval
Estimate a population mean when the population standard deviation is unknown. The standard error is .
Remember: Use a random sample or appropriate randomised design. Check independence and the distribution or sample-size condition; use , not a fixed normal critical value.
One-sample mean test
Test against the specified alternative. The statistic measures how far the sample mean is from the null value in standard-error units.
Remember: Use a -distribution with the appropriate degrees of freedom to obtain the p-value. Choose one or two tails to match the alternative.
Paired differences: interval and test
Define each difference consistently, then treat the differences as one sample. The test statistic is , with .
Remember: Here counts pairs. Check the distribution of differences, not the two original distributions separately.
Two independent means: interval
Estimate . The square root term is the standard error of the difference between independent sample means.
Remember: Check conditions for both groups. Use unpooled two-sample inference; do not assume equal population variances.
Two independent means: test
Here is the hypothesised difference , often . State that difference and the subtraction order before calculating.
Remember: Use technology for the Welch degrees of freedom. A conservative table-based choice is ; do not use a paired procedure for unrelated groups.
Sampling distribution of a difference in means
For independent samples, . Estimating the population standard deviations gives the two-sample standard error.
Remember: The population formula describes repeated samples. It is different from the spread of individual observations in either group.
Worked example · Estimate a population mean travel time
A simple random sample of students from a school with at least students has minutes and minutes. Suppose the travel-time distribution is reasonably normal with no concerning outliers.
For a interval, .
We are confident that the school population mean travel time is between about and minutes. The sample standard deviation describes individual travel times; the standard error describes uncertainty in estimating the mean.
Conditions for mean procedures: Explain the random selection or assignment and independence. For sampling without replacement, check . For a small sample, inspect the relevant distribution for skewness and outliers; an approximately normal population supports the procedure. A sample of at least observations is a common course guideline, but severe skewness or outliers still deserve attention. For independent groups, check each; for paired data, check differences.
Unit 5
Regression Analysis
Describe the scatterplot before using a line. A useful linear prediction needs a suitable relationship, sensible input values and residuals that do not show a clear curved pattern.
Predicted response and residual
Use the explanatory variable to predict the response. The residual compares the observed response with its prediction.
Remember: A positive residual means the observation is above the line; a negative residual means it is below. Avoid extrapolating beyond the observed range without justification.
Slope and intercept relationships
These identities link the least-squares line to the sample summaries. Technology supplies the fitted coefficients. The line passes through .
Remember: Interpret slope as the predicted change in the response for a one-unit increase in the explanatory variable. The intercept describes the prediction at , which may not be meaningful.
Correlation and explained variation
Correlation measures direction and strength of a linear association. In simple linear regression, is the proportion of variability in the response explained by its linear relationship with the explanatory variable.
Remember: A value near does not rule out a curved association. Neither correlation nor a large establishes causation.
What least squares minimises
The least-squares line minimises the sum of squared vertical residuals. Use a residual plot to check whether a straight-line description is reasonable.
Remember: Check unusual points, curvature and changing spread. A small residual for one observation does not validate the whole model.
Worked example · Predict and interpret a residual
A fitted model predicts a quiz score from weekly study hours:
For a student who studies hours and scores :
The model predicts points. The student scored points below the prediction. The slope describes an increase of predicted score points per additional weekly study hour. An observational association alone does not show that extra study caused that increase.
Interpret explained variation carefully: If , say that of the variability in quiz scores is explained by the linear relationship with study hours in this model. It does not mean of students are predicted correctly.
Finish with a statistical explanation
Confidence intervals
Name the population parameter, give the interval with units, and state the confidence level. The confidence level describes the long-run success rate of the interval-building method.
Avoid saying that the confidence interval contains a stated percentage of individual observations.
Significance tests
A p-value is the probability, assuming is true, of obtaining a test statistic as extreme or more extreme in the direction of .
If , reject . Otherwise, fail to reject it. State the evidence for the alternative in the context of the population claim.
Errors and power
A Type I error rejects a true . A Type II error fails to reject a false . Power is the chance of correctly rejecting for a particular true alternative.
Here is the Type II error probability for that alternative. Which error matters more depends on the consequences in context.
Precision and sample size
With other factors held fixed, more data usually reduce standard error. A higher confidence level uses a larger critical value and makes an interval wider.
For a fixed population variability, multiplying the sample size by halves the sampling standard deviation. More data do not fix systematic bias.
Common mistakes to catch before you submit
- Wrong denominator: for conditional probability, divide by the total in the given group.
- Wrong proportion: a one-proportion test uses in its standard deviation; an interval estimates variability with .
- Pooling everywhere: use the pooled proportion for a test of equal population proportions, and separate sample proportions for an interval.
- Ignoring pairs: matched observations require a procedure on the differences.
- Adding standard deviations: independent variances add; take the square root afterwards.
- Using percentages in chi-square: use observed and expected counts.
- Equating evidence with certainty: failing to reject does not prove it is true.
- Reporting numbers without meaning: identify the population, response, subtraction order and units.
A quick check of your understanding
Try these before opening the answers.
Why does a one-proportion test use the null proportion?
The test asks how unusual the sample is under the null model. Its sampling variability is therefore calculated assuming , rather than assuming the observed proportion is the population value.
Two classes measure improvement before and after teaching. Which mean procedure fits?
If each student's before and after scores are matched, define a difference for each student and use paired inference on those differences. If the question compares two unrelated groups' improvement scores, use an independent two-sample procedure on those scores.
A residual is positive. Is the observed response above or below the prediction?
It is above the prediction because . The model underpredicted that observation.
Questions students ask
Do I need to memorise every formula?
College Board supplies formulas and tables for the exam. Practise locating the reference you need, identifying each symbol, checking conditions and explaining the result. Our guide adds teaching reminders and examples; consult the official Appendix to see the provided exam reference.
Why are there five units here instead of nine?
The course framework effective Fall 2026 groups content into five units. Our AP Statistics Study Hub follows that sequence. Older released questions and textbooks may have different numbering and additional content, so match resources by skill and course year.
What is the difference between standard deviation and standard error?
Standard deviation describes spread in a distribution. A standard error estimates the standard deviation of a statistic across repeated samples. For example, describes individual observations, while estimates the sampling variability of their mean.
Can I use a normal critical value for a population mean?
In the mean-inference procedures in our current course notes, the population standard deviation is unknown. Estimate it with the sample standard deviation and use a -procedure with the appropriate degrees of freedom. A large sample makes the -distribution closer to the standard normal distribution; it does not change which procedure you should name.
Is a p-value the probability that the null hypothesis is true?
No. A p-value is calculated assuming is true. It describes how unusual the observed result or a more extreme result would be under that assumption, in the direction of the alternative.
Can I print this guide?
Yes. Use your browser's Print option. Expand any worked examples or answers you want included before printing; the print layout simplifies the formula cards.
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