AP Statistics Vocabulary

10% Condition

A guideline stating that when sampling without replacement, the sample size should be no more than 10% of the population size for the observations to be treated as approximately independent.

Similar definitions: 10% rule

Example: "The          was satisfied because the sample of 200 was less than 10% of the 5,000-student school."

Addition Rule

For any two events, P(A or B) = P(A) + P(B) − P(A and B). For mutually exclusive events, P(A or B) = P(A) + P(B).

Similar definitions: sum rule

Example: "The          was applied to find the probability that a card drawn was a heart or a face card."

Alternative Hypothesis

The hypothesis that contradicts the null hypothesis and represents what the researcher is trying to find evidence for. It can be one-sided or two-sided.

Similar definitions: research hypothesis, $H_a$

Example: "The          stated that students using the new software would score higher than those using traditional methods."

Bar Chart

A graphical display using rectangular bars to show the frequency or relative frequency of each category in a categorical variable.

Similar definitions: bar graph

Example: "The          displayed the number of students in each grade, with each grade represented by a separate bar."

Bias

A systematic error in a statistic or study design that consistently overestimates or underestimates the true population parameter.

Similar definitions: systematic error

Example: "Because the sample excluded lower-income households, the estimate of average income contained         ."

Binomial Distribution

A probability distribution modeling the number of successes in a fixed number of independent trials, each with the same probability of success.

Similar definitions: binomial model

Example: "The number of heads in 10 coin flips follows a          with n = 10 and p = 0.5."

Blinding

An experimental design technique in which subjects, researchers, or both are unaware of which treatment a subject is receiving, in order to reduce bias in measurement.

Similar definitions: masking

Example: "The study used double          so that neither patients nor evaluators knew who received the real drug."

Blocking

A technique in experimental design that groups similar experimental subjects together into blocks before random assignment, in order to control for variation from a known source.

Similar definitions: block design

Example: "The researcher used          by separating male and female participants before randomly assigning treatments."

Boxplot

A graphical display showing the five-number summary (minimum, Q1, median, Q3, maximum) of a dataset, useful for comparing distributions and identifying outliers.

Similar definitions: box-and-whisker plot

Example: "Comparing         s for the two classes revealed that Class A had a higher median but greater spread."

Categorical Data

Data that consists of distinct groups or categories such as gender, race, or political affiliation. Categorical variables can be nominal (no inherent order) or ordinal (with a natural ranking).

Similar definitions: qualitative data

Example: "The researcher recorded favorite music genre as          because the values were labels rather than numbers."

Census

A study that attempts to collect data from every member of the population rather than from a sample.

Similar definitions: complete enumeration

Example: "The national          counted every resident in the country to determine population totals."

Central Limit Theorem

A theorem stating that the sampling distribution of the sample mean approaches a normal distribution as the sample size increases, regardless of the shape of the population distribution, provided n is sufficiently large.

Similar definitions: CLT

Example: "Because of the         , the distribution of sample means from a skewed population became approximately normal when n = 50."

Chi-Square Goodness of Fit Test

A hypothesis test that compares the observed frequencies of a single categorical variable to expected frequencies based on a hypothesized distribution.

Similar definitions: GoF test

Example: "A          was used to determine whether the observed counts of M&M colors matched the company's stated distribution."

Chi-Square Statistic

A test statistic calculated as $\Sigma(O - E)^2 / E$, where $O$ is the observed frequency and $E$ is the expected frequency for each cell or category.

Similar definitions: $\chi^2$ statistic

Example: "The          of 12.4 with 3 degrees of freedom produced a p-value small enough to reject the null hypothesis."

Chi-Square Test of Homogeneity

A hypothesis test that compares the distribution of a categorical variable across two or more distinct populations to determine if the distributions are the same.

Similar definitions: test of equal distributions

Example: "The          was used to compare the political party preferences of voters in three different states."

Chi-Square Test of Independence

A hypothesis test that determines whether two categorical variables measured on the same group of individuals are associated or independent.

Similar definitions: test of association

Example: "The          revealed a statistically significant association between gender and preferred study method."

Coefficient of Determination

A value from 0 to 1 that measures the proportion of variation in the response variable that is explained by the linear regression model with the explanatory variable.

Similar definitions: r-squared, $R^2$

Example: "An          of 0.81 meant that 81% of the variation in test scores was explained by hours of study."

Complement

The set of all outcomes in the sample space that are NOT in the event of interest. The probability of the complement equals 1 minus the probability of the event.

Similar definitions: complementary event

Example: "The          of rolling a 6 is rolling any number from 1 to 5, with probability 5/6."

Conditional Distribution

The distribution of one variable in a two-way table restricted to a particular value or category of the other variable.

Similar definitions: conditional proportion

Example: "The          of subject preference among females showed that 60% preferred science over humanities."

Conditional Probability

The probability that an event A occurs given that another event B has already occurred, written as P(A|B).

Similar definitions: given probability

Example: "The          of drawing a king given the first card drawn was an ace was calculated using the reduced sample space."

Confidence Interval

A range of plausible values for a population parameter, constructed from sample data at a specified confidence level (e.g., 95%), meaning that in repeated sampling the interval would capture the true parameter that percentage of the time.

Similar definitions: interval estimate

Example: "The 95%          for the mean was (42.1, 47.9), suggesting the true population mean was likely between those values."

Confidence Level

The long-run percentage of confidence intervals, constructed using the same method from repeated random samples, that would contain the true population parameter. Common levels are 90%, 95%, and 99%.

Similar definitions: confidence percentage

Example: "A          of 95% means that if the sampling procedure were repeated many times, 95% of the resulting intervals would contain the true parameter."

Confounding Variable

An extraneous variable that is associated with both the explanatory and response variables, making it difficult to determine the true cause-and-effect relationship.

Similar definitions: confounder

Example: "Ice cream sales and drowning rates both increase in summer, making temperature a          that explains the apparent association."

Continuous Random Variable

A random variable that can take on any value within an interval or range, with no gaps between possible values.

Similar definitions: continuous variable

Example: "The exact weight of a randomly selected apple is a          because it can take any value within a range."

Control Group

The group in an experiment that does not receive the treatment, serving as a baseline for comparison with the treatment group.

Similar definitions: baseline group, comparison group

Example: "The          received a sugar pill so that any difference in outcomes could be attributed to the actual drug."

Convenience Sample

A non-random sample consisting of individuals who are easiest to reach or most readily available, which typically produces biased results.

Similar definitions: haphazard sample

Example: "Asking only students in the front row for opinions is an example of a          because it excludes most of the class."

Correlation Coefficient

A numerical value ranging from -1 to 1 that measures the strength and direction of the linear association between two quantitative variables. Values near ±1 indicate a strong relationship; values near 0 indicate a weak or no linear relationship.

Similar definitions: Pearson r, r-value

Example: "With a          of 0.92, there was a strong positive linear relationship between study hours and exam scores."

Critical Value

The value from a reference distribution ($z^*$ or $t^*$) that corresponds to the desired confidence level or significance level, used to construct confidence intervals or define rejection regions.

Similar definitions: $z^*$, $t^*$

Example: "For a 95% confidence interval using the normal distribution, the          is $z^* = 1.96$."

Degrees of Freedom

A parameter controlling the shape of the t-distribution or chi-square distribution. For a one-sample t-test, degrees of freedom equal n − 1.

Similar definitions: df

Example: "With a sample size of 20, the          for the one-sample t-test was 19."

Discrete Random Variable

A random variable that can take on a countable number of distinct values, typically whole numbers.

Similar definitions: discrete variable

Example: "The number of customers entering a store each hour is a          because it can only be a whole number."

Dot Plot

The arithmetic average of a dataset, calculated by summing all values and dividing by the number of values. It is sensitive to extreme values and outliers.

Similar definitions: arithmetic mean, average

Example: "The          salary of the employees was pulled upward by a few very high earners."

Double-Blind Experiment

An experiment in which neither the subjects nor the researchers measuring outcomes know which treatment each subject is receiving, eliminating both subject and researcher bias.

Similar definitions: double-masked study

Example: "The clinical trial was a          because neither the patients nor the doctors evaluating them knew who received the active drug."

Empirical Rule

A rule stating that for a normal distribution, approximately 68% of data falls within 1 standard deviation of the mean, 95% within 2, and 99.7% within 3.

Similar definitions: 68-95-99.7 rule, three-sigma rule

Example: "Using the         , a teacher predicted that about 95% of students would score within 2 standard deviations of the mean."

Event

A subset of the sample space containing one or more outcomes of interest in a probability experiment.

Similar definitions: outcome set

Example: "Getting an even number when rolling a die is an          consisting of the outcomes {2, 4, 6}."

Expected Frequency

The count predicted for a cell in a chi-square test under the null hypothesis, calculated as (row total imes column total) / grand total.

Expected Value

The long-run average value of a random variable over many repetitions of the random process, calculated as the sum of each value multiplied by its probability.

Similar definitions: mean of a random variable, E(X)

Example: "The          of the number of heads in 4 coin flips is 2."

Experiment

A study in which researchers deliberately impose a treatment on subjects and observe the response in order to establish cause-and-effect relationships.

Similar definitions: controlled experiment, randomized experiment

Example: "In the         , the researchers randomly assigned subjects to receive either the new drug or a placebo."

Explanatory Variable

The input variable (plotted on the x-axis) that is used to explain or predict changes in the response variable. Also called the independent variable.

Similar definitions: independent variable, predictor variable

Example: "In the study of fertilizer and plant growth, the amount of fertilizer applied was the         ."

Extrapolation

Using a regression model to make predictions for values of the explanatory variable outside the range of the observed data, which can produce unreliable or misleading results.

Similar definitions: out-of-range prediction

Example: "Predicting a student's college GPA from a kindergarten test score would be an example of dangerous         ."

Five-Number Summary

A set of five descriptive statistics — minimum, first quartile (Q1), median, third quartile (Q3), and maximum — that describe the distribution of a dataset.

Similar definitions: quartile summary

Example: "The          for the test scores was {52, 71, 83, 91, 100}."

Geometric Distribution

A probability distribution modeling the number of trials needed to obtain the first success in a sequence of independent Bernoulli trials.

Similar definitions: geometric model

Example: "The number of rolls of a die before rolling a 6 for the first time follows a         ."

Histogram

A bar graph used for quantitative data in which each bar represents the frequency or relative frequency of data values falling within a specified interval (bin).

Similar definitions: frequency histogram

Example: "The          of exam scores revealed a roughly symmetric, bell-shaped distribution."

Independence Condition

A condition for inference requiring that individual observations are independent of one another. It is satisfied by random sampling or random assignment and verified using the 10% condition when sampling without replacement.

Similar definitions: independence requirement

Example: "The          was met because students were randomly selected and the sample was less than 10% of the population."

Independent Events

Two events are independent if the occurrence of one does not affect the probability of the other.

Similar definitions: statistically independent events

Example: "Flipping a coin and rolling a die are          because the result of one does not change the probability of the other."

Inference for Regression

The use of hypothesis tests and confidence intervals to determine whether a linear relationship exists in the population and to estimate the population slope.

Similar definitions: regression inference

Example: "Using         , the student tested whether there was a significant linear relationship between temperature and ice cream sales."

Influential Point

A data point that has a large impact on the position or slope of the regression line, typically a point with an extreme x-value.

Similar definitions: high-leverage point

Example: "Removing the single          changed the slope of the regression line from 0.8 to 0.3."

Intercept (Regression)

The predicted value of the response variable when the explanatory variable equals zero. It is the point where the regression line crosses the y-axis, denoted $b_0$ in the sample and $\beta_0$ in the population.

Similar definitions: y-intercept, $b_0$

Example: "The          of 15 suggested that even with zero hours of study, the model predicted a score of 15."

Interquartile Range (IQR)

The difference between the third quartile (Q3) and the first quartile (Q1), representing the spread of the middle 50% of data. It is resistant to outliers.

Similar definitions: IQR, midspread

Example: "The          of 18 indicated that the middle 50% of scores spanned 18 points."

Large Counts Condition

A condition for using the normal approximation for proportions, requiring that both np \nge 10 and n(1 - p) \nge 10.

Similar definitions: success-failure condition

Example: "With $n = 100$ and $p = 0.4$, the          was met since both $40$ and $60$ are at least $10$."

Law of Large Numbers

A principle stating that as the number of trials of a random process increases, the relative frequency of an event approaches its true probability.

Similar definitions: LLN

Example: "The          explains why the proportion of heads approaches 0.5 as more coin flips are observed."

Least-Squares Regression Line

The line that minimizes the sum of the squared vertical distances (residuals) between the observed data points and the line, used to model the linear relationship between two quantitative variables.

Similar definitions: line of best fit, LSRL, regression line

Example: "The          ŷ = 3.2x + 10 was used to predict exam scores from hours of study."

Linear Regression Slope

The value representing the change in the predicted response variable for each one-unit increase in the explanatory variable, denoted $b_1$ in the sample and $\beta_1$ in the population.

Similar definitions: regression coefficient, $b_1$

Example: "The estimated          of 4.5 means that each additional hour of study is associated with a 4.5-point increase in predicted exam score."

Lurking Variable

A variable that influences both the explanatory and response variables but is not included in the study, potentially creating a false impression of association.

Similar definitions: hidden variable

Example: "The apparent relationship between shoe size and reading ability in children was explained by age, a         ."

Margin of Error

The maximum likely difference between the sample estimate and the true population parameter, equal to the critical value multiplied by the standard error.

Similar definitions: ME

Example: "The poll reported 54% support with a          of ±3%, meaning the true proportion was likely between 51% and 57%."

Marginal Distribution

The distribution of one variable in a two-way table, found by looking at the row totals or column totals.

Similar definitions: marginal frequency, marginal proportion

Example: "The          of preferred subject showed that 55% of all students preferred math over English."

Matched Pairs Design

An experimental or observational design in which subjects are paired based on similar characteristics and each member receives a different treatment, or in which each subject receives both treatments in random order. Differences within each pair are analyzed.

Similar definitions: paired design, within-subjects design

Example: "In the         , each participant tasted both brands of coffee and the differences in ratings were analyzed."

Mean

The arithmetic average of a dataset, calculated by summing all values and dividing by the number of values. It is sensitive to extreme values and outliers.

Similar definitions: arithmetic mean, average

Example: "The          salary of the employees was pulled upward by a few very high earners."

Median

The middle value in an ordered dataset that divides the distribution into two equal halves. It is resistant to outliers, making it preferable for skewed distributions.

Similar definitions: middle value

Example: "Because the data was right-skewed, the teacher reported the          rather than the mean as a better measure of center."

Multiplication Rule

For independent events A and B, P(A and B) = P(A) imes P(B). For dependent events, P(A and B) = P(A) imes P(B|A).

Mutually Exclusive Events

Two or more events that cannot occur at the same time; if one event occurs, the other cannot.

Similar definitions: disjoint events

Example: "Rolling a 3 and rolling a 5 on a single die roll are          because only one number can appear."

Nonresponse Bias

A type of bias that occurs when individuals selected for a survey systematically differ from those who do not respond, affecting the representativeness of results.

Similar definitions: nonparticipation bias

Example: "The survey suffered from          because the people most likely to skip the survey had very different opinions from those who responded."

Normal Distribution

A continuous, symmetric, bell-shaped probability distribution completely described by its mean and standard deviation. Many natural phenomena and sampling distributions are approximately normal.

Similar definitions: Gaussian distribution, bell curve

Example: "Adult heights follow a roughly          with a mean of 5'9" and a standard deviation of 3 inches."

Normal Probability Plot

A graph used to assess whether a dataset follows a normal distribution. If the points fall approximately along a straight line, the data can be considered approximately normal.

Similar definitions: normal quantile plot, Q-Q plot

Example: "The          showed points falling close to a straight line, supporting the use of a t-procedure."

Null Hypothesis

The default assumption in a hypothesis test, typically stating that there is no effect, no difference, or no association between variables.

Similar definitions: $H_0$, H-naught, default hypothesis

Example: "The          stated that the mean test score of students using the new curriculum was equal to 75."

Observational Study

A study in which researchers observe subjects and measure variables of interest without attempting to influence or manipulate the subjects.

Similar definitions: survey study

Example: "The researchers conducted an          by recording students' sleep habits and grades without assigning any sleep schedules."

One-Proportion z-Interval

A confidence interval used to estimate a population proportion, computed as $\hat{p} \pm z^*\left(\sqrt{\frac{\hat{p}(1-\hat{p})}{n}}\right)$, when the Large Counts and 10% conditions are met.

One-Proportion z-Test

A hypothesis test used to determine whether a population proportion differs from a specified value, using the z-distribution when the Large Counts and 10% conditions are met.

Similar definitions: one-sample z-test for proportion

Example: "The quality control manager used a          to test whether the defect rate exceeded the claimed 2%."

One-Sample t-Interval

A confidence interval used to estimate a population mean when the population standard deviation is unknown, computed as x̄ ± t*(s/√n) with n − 1 degrees of freedom.

Similar definitions: t-interval for mean

Example: "The researcher used a          to estimate the average commute time in the city based on a sample of 40 commuters."

One-Sample t-Test

A hypothesis test used to determine whether the mean of a single population differs from a specified value, using the t-distribution when the population standard deviation is unknown.

Similar definitions: single-sample t-test

Example: "The researcher used a          to determine whether the average daily calorie intake differed from the recommended 2,000."

Outlier

A data point that is significantly different from the rest of the dataset, typically identified using the 1.5 × IQR rule or z-scores.

Similar definitions: extreme value, anomaly

Example: "The value of 98 was flagged as an          because it fell more than 1.5 × IQR above Q3."

P-Value

The probability of obtaining a test statistic as extreme as or more extreme than the one observed, assuming the null hypothesis is true. A small p-value provides evidence against the null hypothesis.

Similar definitions: observed significance level

Example: "With a          of 0.03, the researchers rejected the null hypothesis at the 0.05 significance level."

Paired t-Test

A hypothesis test for comparing two related measurements or treatments by analyzing the differences within each pair, treating them as a single sample.

Similar definitions: dependent samples t-test, matched pairs t-test

Example: "The researcher used a          to determine whether students scored differently on a test before and after a tutoring session."

Parameter

A numerical value that describes a characteristic of an entire population, such as the population mean (μ) or population standard deviation (σ).

Similar definitions: population characteristic

Example: "The average height of all adults in a country is a          because it describes the entire population."

Percentile

The value below which a given percentage of observations fall. For example, the 80th percentile is the value below which 80% of the data lies.

Similar definitions: centile

Example: "A score at the 90th          means that 90% of all scores were below that value."

Placebo

An inactive treatment (such as a sugar pill) given to the control group in an experiment to account for the psychological effect of receiving a treatment.

Similar definitions: inert treatment, dummy treatment

Example: "The          allowed researchers to separate the true drug effect from the psychological effect of receiving any treatment."

Placebo Effect

The phenomenon in which subjects show a measurable response to an inactive treatment simply because they believe they are receiving a real treatment.

Similar definitions: suggestion effect

Example: "The researchers accounted for the          by giving the control group a sugar pill that looked identical to the actual medication."

Pooled Proportion

A combined estimate of the common proportion under the null hypothesis in a two-sample proportion test, calculated by combining the successes and sample sizes from both groups.

Similar definitions: combined proportion

Example: "The          was used as the estimate of p when calculating the standard error in the two-proportion z-test."

Population

The entire group of individuals, objects, or events that a researcher is interested in studying.

Similar definitions: universe of subjects

Example: "The          for the study was all registered voters in the United States."

Power

The probability that a hypothesis test correctly rejects a false null hypothesis. Power equals 1 − β and increases with larger sample sizes and larger effect sizes.

Similar definitions: statistical power

Example: "Increasing the sample size from 30 to 200 greatly increased the          of the test to detect a real difference."

Probability

A number between 0 and 1 that quantifies the likelihood of an event occurring, where 0 means impossible and 1 means certain.

Similar definitions: likelihood, chance

Example: "The          of rolling a six on a fair die is 1/6."

Probability Distribution

A description (table, graph, or formula) of all possible values of a random variable along with the probability associated with each value.

Similar definitions: probability model

Example: "The          for a fair six-sided die assigns probability 1/6 to each of the values 1 through 6."

Quantitative Data

Data that involves numerical measurements or counts such as height, weight, or number of siblings. It can be discrete (whole numbers only) or continuous (any value within a range).

Similar definitions: numerical data

Example: "The students' exam scores were collected as          because they are numerical and can be compared meaningfully."

Quartile

Values that divide an ordered dataset into four equal parts. Q1 is the 25th percentile, Q2 is the median, and Q3 is the 75th percentile.

Similar definitions: quartile values

Example: "The first          (Q1) of 23 indicated that 25% of students scored below 23."

Random Assignment

The process of using chance to allocate subjects to treatment groups in an experiment, ensuring that differences in outcomes are due to the treatment rather than preexisting differences.

Similar definitions: randomization

Example: "The researchers used          to place participants into either the exercise or no-exercise group."

Random Variable

A variable whose value is determined by the outcome of a random process, assigning a numerical value to each outcome in a sample space.

Similar definitions: stochastic variable

Example: "Let X be the number of heads in three coin flips; X is a          because its value depends on chance."

Randomness Condition

A condition for inference requiring that the data were produced by a random sample or random assignment, ensuring results can be generalized and that inference is valid.

Similar definitions: random sample condition

Example: "The          was satisfied because participants were randomly selected from the school's enrollment list."

Range

The difference between the maximum and minimum values in a dataset. It provides a simple measure of total spread but is sensitive to outliers.

Similar definitions: total spread

Example: "The          of the test scores was 45, calculated by subtracting the lowest score from the highest."

Replication

The use of multiple subjects in each treatment group of an experiment to reduce the effect of chance variation and increase the reliability of results.

Similar definitions: repetition

Example: "The experiment used          by assigning 50 subjects to each treatment group rather than just one."

Residual

The difference between an observed value of the response variable and the value predicted by the regression line. Positive residuals indicate underprediction; negative residuals indicate overprediction.

Similar definitions: error, prediction error

Example: "The student's actual score was 5 points above the predicted value, giving a          of +5."

Residual Plot

A graph of residuals versus the explanatory variable or predicted values, used to check whether a linear model is appropriate. A random scatter with no pattern indicates a good fit.

Similar definitions: diagnostic plot

Example: "The          showed a curved pattern, indicating that a linear model was not appropriate for the data."

Response Variable

The output variable (plotted on the y-axis) that is being explained or predicted by the explanatory variable. Also called the dependent variable.

Similar definitions: dependent variable, outcome variable

Example: "The plant's height was the          because it was what the researchers measured as the outcome."

Sample

A subset of the population that is selected for study and used to make inferences about the population.

Similar definitions: subset

Example: "The researchers selected a          of 500 students from the school's enrollment list."

Sample Proportion

The fraction of individuals in a sample having a particular characteristic, calculated as the number of successes divided by the sample size (x/n).

Similar definitions: p-hat, p̂

Example: "In a sample of 200 voters, 110 favored the measure, giving a          of 0.55."

Sample Space

The set of all possible outcomes of a random experiment or process.

Similar definitions: outcome space, universal set

Example: "The          for flipping a coin twice is {HH, HT, TH, TT}."

Sampling Bias

A systematic error that occurs when some members of the population are more likely to be selected for the sample than others, leading to results that do not accurately represent the population.

Similar definitions: selection bias

Example: "Only surveying students in the cafeteria introduced         , since students who bring lunch from home were not represented."

Sampling Distribution

The probability distribution of a statistic (such as the sample mean or sample proportion) calculated from all possible samples of a given size taken from a population.

Similar definitions: distribution of a sample statistic

Example: "The          of the sample mean becomes approximately normal as sample size increases, by the Central Limit Theorem."

Sampling Variability

The natural variation in sample statistics from one random sample to another, even when all samples are drawn from the same population.

Similar definitions: natural variation

Example: "The          in survey results means that different random samples of the same size will produce slightly different estimates."

Scatterplot

A graph that displays the relationship between two quantitative variables, with each point representing a pair of measurements for one individual.

Similar definitions: scatter diagram, scatter graph

Example: "The          revealed a moderately strong positive linear association between shoe size and height."

Significance Level

The threshold probability set before conducting a hypothesis test, below which the p-value leads to rejection of the null hypothesis. Commonly set at 0.05 or 0.01.

Simple Random Sample (SRS)

A sampling method in which every member of the population has an equal chance of being selected, and every possible sample of a given size is equally likely.

Similar definitions: SRS, random sample

Example: "The teacher used a          by numbering all students and using a random number generator to select 30."