Statistics glossary
Browse existing editorial definitions and examples, then follow deeper reading in the subject’s native course. Shared senses can belong to more than one subject. Historical source definitions and tables remain in the 242-entry Physics source glossary.
Statistics printables · Statistics interactive and source resources
Statistics definitions retain their original captured-reading destinations and any editorial qualifications. Current Fall 2026 practice is separately identified in Printables.
182 of 182 editorial senses
68–95–99.7 rule
Statistics · Definition
For a Normal distribution, approximately 68%, 95%, and 99.7% of values lie within one, two, and three standard deviations of the mean.
The rule is an approximation for Normal or sufficiently bell-shaped data, rather than for every distribution.
For example, a Normal model with mean 50 and standard deviation 5 places about 95% of values between 40 and 60.
Alternative hypothesis
Statistics · Definition
An alternative hypothesis specifies the population difference or direction for which a test seeks evidence.
It should be chosen from the research question before inspecting results.
For example, an alternative can state that the population mean travel time is greater than 15 minutes.
Back-to-back stemplot
Statistics · Definition
A back-to-back stemplot compares two data sets using shared stems.
Leaves for one group go on one side and leaves for the other group go on the opposite side.
For example, two classes’ test scores can be compared while retaining their individual values.
Bar graph
Statistics · Definition
A bar graph uses separate bars to compare category counts or proportions.
The bar lengths or heights encode the values, and category order need not be numerical.
For example, separate bars can compare the number of votes for three lunch options.
Bimodal distribution
Statistics · Definition
A bimodal distribution has two main peaks.
Two peaks can suggest different groups, but the graph alone does not establish their cause.
For example, combining commute times from nearby and distant neighborhoods may produce two peaks.
Binomial coefficient
Statistics · Definition
A binomial coefficient counts the ways to select a fixed number of objects from a set when order does not matter.
In a binomial probability calculation, it counts the possible success positions among the trials.
For example, choosing two success positions among four trials can be done in six ways.
Binomial distribution
Statistics · Definition
A binomial distribution models the number of successes in a fixed number of independent trials.
Every trial has two classified outcomes and the same probability of success.
For example, counting red cards in five independent draws with replacement follows a binomial model.
Blinding
Statistics · Definition
Blinding keeps specified participants or study personnel unaware of assigned treatments.
The study should state who is blinded because labels such as single-blind and double-blind can be used differently.
For example, both participants and outcome assessors can be kept unaware of which pill was assigned.
Boxplot
Statistics · Definition
A boxplot displays the median and quartiles with a box and extending whiskers.
A modified boxplot uses whiskers ending at the most extreme observations within the outlier fences and plots flagged observations separately.
For example, parallel boxplots can compare the centers and spreads of two classes’ scores.
Categorical variable
Statistics · Definition
A categorical variable records a group or label for each individual.
Numeric labels still count as categories when arithmetic on them has no meaningful interpretation.
For example, favorite school subject is categorical, even if each subject receives a numbered code.
Causation
Statistics · Definition
Causation means that changing one condition produces a change in an outcome.
Association alone is insufficient because confounding and other explanations may account for the pattern.
For example, a well-conducted randomized fertilizer experiment can support a causal conclusion about growth in its study setting.
Census
Statistics · Definition
A census attempts to collect data from every individual in the defined population.
It avoids sampling variability from selecting only a subset, but can still have coverage, response, and measurement errors.
For example, asking every enrolled student about transportation is a census of that enrollment.
Central limit theorem
Statistics · Definition
For independent identically distributed observations with finite variance, the central limit theorem explains why the standardized sample mean approaches a Normal distribution as sample size increases.
There is no universal sample size that makes every population sufficiently close to Normal; skewness and extreme tails affect the approximation.
For example, averaging many independent waiting times can produce a less skewed distribution of averages than the original waiting-time distribution.
Editorial qualification: The captured n ≥ 30 statement is not a universal guarantee. Under the CLT’s assumptions, the accuracy of a Normal approximation depends on the population and sample size.
Chi-square component
Statistics · Definition
A chi-square component is one cell’s contribution to a categorical test’s total chi-square statistic.
Examining components can show which cells disagree most with the null expectations, without automatically establishing a new significant comparison.
For example, a particularly large component highlights a category worth further contextual investigation.
Chi-square distribution
Statistics · Definition
A chi-square distribution is a nonnegative reference distribution whose shape depends on degrees of freedom.
It is the distribution of a sum of squared independent standard Normal variables.
For example, categorical-count tests compare a suitably constructed chi-square statistic with this reference distribution.
Chi-square goodness-of-fit test
Statistics · Definition
A chi-square goodness-of-fit test compares one categorical variable’s observed counts with a specified distribution.
When the null proportions are fully specified rather than estimated from the data, the degrees of freedom are the number of categories minus 1.
For example, a test can assess whether observed colors fit a manufacturer’s stated color proportions.
Chi-square statistic
Statistics · Definition
For categorical-count tests, the chi-square statistic adds each cell’s squared observed-minus-expected difference divided by its expected count.
Large contributions identify cells with substantial discrepancies relative to their expected counts.
For example, a cell observed 15 times when 10 are expected contributes 2.5 to the statistic.
Chi-square test for homogeneity
Statistics · Definition
A chi-square test for homogeneity compares a categorical variable’s distribution across independently sampled populations or randomized treatment groups.
The null model states that the category proportions are the same across groups.
For example, independently sampled students at several schools can be compared for their transportation-mode distributions.
Chi-square test of association
Statistics · Definition
A chi-square test of association examines the relationship between two categorical variables measured on one sample.
The null model states that the variables are independent in the population.
For example, one random sample of students can be classified by grade level and transportation mode to investigate association.
Chi-square testing
Statistics · Definition
Chi-square categorical tests compare observed counts with expected counts under a null model.
The appropriate test depends on the study design, categorical variables, and hypothesis, with independence and expected-count conditions checked.
For example, one sample with two categorical variables can be analyzed for association using a two-way table.
Cluster sampling
Statistics · Definition
Cluster sampling randomly selects groups of individuals rather than selecting individuals directly across the entire population.
In a one-stage cluster sample, all individuals in the selected clusters are included.
For example, researchers may randomly select classrooms and survey every student in those classrooms.
Coefficient of determination
Statistics · Definition
The coefficient of determination is the fraction of observed response variation accounted for by a fitted regression model.
In simple linear regression with an intercept, it equals the square of the correlation coefficient.
For example, a value of 0.64 means that the fitted linear model accounts for 64% of the response variation in those data.
Combining random variables
Statistics · Definition
Combining random variables forms a new variable from values such as their sum or difference.
Means add or subtract, while the variance of a sum or difference is the sum of the variances when the variables are independent and have finite variances.
For example, two independent daily production counts have a total whose variance is the sum of their individual variances.
Complement of an event
Statistics · Definition
The complement of an event contains every sample-space outcome outside that event.
Its probability is 1 minus the event’s probability.
For example, if a bus is on time with probability 0.8, the complementary event has probability 0.2.
Completely randomized design
Statistics · Definition
A completely randomized design assigns all experimental units to treatments using chance without first separating them into blocks.
The design uses comparison and replication to assess treatment differences.
For example, all eligible plants can be randomly allocated among three fertilizers.
Conditional probability
Statistics · Definition
Conditional probability measures the likelihood of one event when another event is known to have occurred.
It restricts attention to the conditioning event, which must have positive probability.
For example, knowing a drawn card is a face card changes the denominator when finding the probability it is a queen.
Conditional relative frequency
Statistics · Definition
A conditional relative frequency divides a cell count by the total for a specified row or column.
The denominator includes only individuals in the group being conditioned on.
For example, 6 walkers among 10 grade-11 students means that 60% of grade-11 students walk.
Confidence interval
Statistics · Definition
A confidence interval is a data-based range produced by a method with a specified long-run coverage rate for a population parameter.
For a fixed population, the calculated interval either contains the parameter or it does not; the confidence level describes the method over repeated samples.
For example, a 95% method captures the true parameter in about 95% of repeated intervals when its assumptions hold.
Confidence level
Statistics · Definition
A confidence level is the long-run fraction of intervals from a procedure that capture the true parameter under its assumptions.
Holding the data and procedure fixed, increasing the confidence level generally widens the interval.
For example, a 99% interval is usually wider than a 90% interval calculated from the same sample.
Confounding
Statistics · Definition
Confounding occurs when the effects of different variables cannot be distinguished in the observed comparison.
A potential explanation is that another variable is related to both the treatment or explanatory variable and the response.
For example, comparing fertilizer A outdoors with fertilizer B indoors mixes fertilizer and growing-location effects.
Control group
Statistics · Definition
A control group supplies a comparison for evaluating an experimental treatment.
It may receive a placebo, no active treatment, or an established treatment, depending on the study’s design.
For example, a fertilizer experiment may compare a new fertilizer with the usual fertilizer rather than with no fertilizer.
Convenience sample
Statistics · Definition
A convenience sample includes individuals who are easiest to reach.
Easy access does not ensure that the sample represents the target population.
For example, surveying only students sitting nearest the classroom door is convenience sampling.
Correlation coefficient
Statistics · Definition
Pearson’s correlation coefficient measures the direction and strength of a linear relationship between two quantitative variables.
It ranges from −1 to 1, and a value near 0 does not rule out a strong curved relationship.
For example, points closely following a decreasing straight line have correlation near −1.
Critical value
Statistics · Definition
A critical value is a cutoff from a reference distribution chosen for a confidence level or testing rule.
The relevant distribution, degrees of freedom, and one-sided or two-sided procedure determine the cutoff.
For example, a symmetric 95% Normal interval uses a cutoff near 1.96 standard-error units.
Cumulative frequency
Statistics · Definition
Cumulative frequency adds counts through a specified value or interval.
It shows how many observations lie at or below that point in the ordered data.
For example, adding counts for all intervals up to 20 minutes gives the number of recorded waits no longer than 20 minutes.
Grouped cumulative tables follow their stated interval endpoints. The current Topic 1.7 lesson uses the empirical percentage at or below a data value, including equal observations. The preserved reading’s percentile formula instead counts strictly lesser observations. State the convention and keep the comparison consistent, especially with ties.
Cumulative relative frequency
Statistics · Definition
Cumulative relative frequency adds proportions through a specified value or interval.
A cumulative relative frequency graph shows how the accumulated proportion rises from 0 toward 1.
For example, a value of 0.7 at 20 minutes means that 70% of the recorded waits are at or below 20 minutes.
Grouped cumulative tables follow their stated interval endpoints. The current Topic 1.7 lesson uses the empirical percentage at or below a data value, including equal observations. The preserved reading’s percentile formula instead counts strictly lesser observations. State the convention and keep the comparison consistent, especially with ties.
Degrees of freedom
Statistics · Definition
Degrees of freedom describe the independent information remaining after constraints or estimated quantities are accounted for.
The value depends on the procedure, rather than always being the sample size minus 1.
For example, a one-sample t procedure uses n − 1 degrees of freedom, while simple regression inference uses n − 2.
Density curve
Statistics · Definition
A density curve models a continuous distribution with nonnegative height and total area 1.
Probability over an interval is the area under the curve across that interval, rather than its height at a single point.
For example, an interval containing one quarter of the total area has probability 0.25.
Difference between sample means
Statistics · Definition
A difference between sample means compares two groups’ observed averages.
Inference must reflect whether the samples are independent or paired rather than treating every comparison the same way.
For example, two independently sampled classes’ mean travel times can estimate a difference between their population means.
Difference between sample proportions
Statistics · Definition
A difference between sample proportions compares the observed fractions in two samples.
Its order matters, and inference must account for both sample sizes and the independence or dependence of the groups.
For example, subtracting one class’s walking proportion from another’s estimates a difference between their population proportions.
Distribution of data
Statistics · Definition
A distribution describes the values a variable takes and how often those values occur.
For quantitative data, useful features include shape, center, spread, and unusual observations.
For example, a distribution of commuting times shows which times are common and how widely they vary.
Dotplot
Statistics · Definition
A dotplot places a dot for each observation above its numerical value.
Repeated values are usually shown by stacking dots.
For example, three observations equal to 4 produce a stack of three dots above 4.
Equal variance
Statistics · Definition
An equal-variance condition means that the response spread around a regression line is the same across explanatory values.
A residual plot with a funnel-shaped pattern can challenge this assumption.
For example, wider residual scatter for older plants suggests that response variance increases with age.
Event in probability
Statistics · Definition
An event is a set of outcomes from a chance process’s sample space.
The event occurs when the observed outcome belongs to that set.
For example, rolling an even result is the event containing 2, 4, and 6 for a six-sided die.
Expected count
Statistics · Definition
An expected count is the count predicted for a category or cell under the null model.
Goodness-of-fit expectations use the total count times the null proportion, while independence-table expectations use row total times column total divided by the grand total.
For example, a null proportion of 0.2 in 50 trials gives an expected count of 10.
Expected value
Statistics · Definition
Expected value is the probability-weighted mean of a random variable when that mean exists.
It describes a long-run average rather than a value that must occur on one trial.
For example, a fair die has expected value 3.5 even though no face is labeled 3.5.
Experiment
Statistics · Definition
An experiment deliberately imposes treatments and measures responses.
Random assignment, comparison, and replication help distinguish treatment effects from other variation.
For example, assigning plants to fertilizer treatments is an experiment.
Experimental factor
Statistics · Definition
A factor is an explanatory variable deliberately varied in an experiment.
Each assigned setting of that factor is called a level.
For example, fertilizer type is a factor with levels corresponding to the fertilizer choices.
Experimental unit
Statistics · Definition
An experimental unit is the smallest unit independently assigned a treatment.
A human experimental unit is often called a subject or participant.
For example, if fertilizer is assigned to an entire planter, the planter rather than each leaf is the experimental unit.
Explanatory variable
Statistics · Definition
An explanatory variable is used to predict or explain variation in a response variable.
In an experiment, researchers deliberately assign values or treatments related to an explanatory variable.
For example, assigned study time could be an explanatory variable when test score is the response.
Exponential model
Statistics · Definition
An exponential model multiplies the response by the same factor for each equal increase in the explanatory variable.
For positive responses, plotting log y against x can produce a straight line.
For example, a population that doubles every year follows an exponential rather than a power model of time.
Extrapolation
Statistics · Definition
Extrapolation uses a model to predict beyond the range of explanatory values used to fit it.
A pattern observed within the data range may not continue outside that range.
For example, extending a short-term plant-growth line far into the future may give unrealistic heights.
Factor level
Statistics · Definition
A factor level is a particular setting of an experimental factor.
Combining the levels of different factors determines the treatment conditions.
For example, four hours and eight hours can be two levels of a light-duration factor.
Five-number summary
Statistics · Definition
A five-number summary contains the minimum, first quartile, median, third quartile, and maximum.
These values summarize the location and spread of ordered numerical data.
For example, the summary 2, 4, 7, 10, and 15 identifies a median of 7 and an IQR of 6.
Frequency table
Statistics · Definition
A frequency table lists the number of observations in each category or numerical interval.
Its counts add to the total number of observations when the groups do not overlap and cover all observations.
For example, a table can record how many students chose each lunch option.
General addition rule
Statistics · Definition
The general addition rule finds the probability that at least one of two events occurs.
It adds their probabilities and subtracts the probability that both occur, avoiding double counting.
For example, the probability of a card being red or a king must subtract the red kings counted twice.
General multiplication rule
Statistics · Definition
The general multiplication rule finds the probability that two events both occur.
It multiplies the first event’s probability by the second event’s probability conditional on the first.
For example, two red draws without replacement use the updated red proportion for the second draw.
Geometric distribution
Statistics · Definition
A geometric distribution models the trial number of the first success in independent trials with a constant success probability.
In this convention its possible values start at 1, and earlier trials must all be failures.
For example, the number of draws with replacement until the first red card has a geometric distribution.
Histogram
Statistics · Definition
A histogram groups quantitative observations into numerical intervals and displays their frequencies or proportions.
Neighboring bars represent adjacent intervals, so the graph preserves numerical order.
For example, a histogram can group travel times into ten-minute intervals.
Histogram class interval
Statistics · Definition
A histogram class is a numerical interval used to group observations.
The class width determines the span of values within that interval and affects the displayed shape.
For example, intervals from 0 to 10 and from 10 to 20 minutes have width 10 minutes.
Inference for a regression slope
Statistics · Definition
Inference for a regression slope assesses a population linear relationship using the slope fitted to sample data.
It relies on an appropriate design and assumptions about linearity, independent observations, and the response variation around the line.
For example, a t interval can estimate how expected plant height changes per week within the observed age range.
Influential observation
Statistics · Definition
An influential observation substantially changes a fitted model when it is removed.
An unusual point is not necessarily influential, and influence depends on its position and the other data.
For example, a point far along the explanatory axis may strongly change the fitted slope.
Interquartile range
Statistics · Definition
The interquartile range is the third quartile minus the first quartile.
It measures the spread of the middle half of the ordered data.
For example, quartiles of 10 and 18 give an IQR of 8.
Joint distribution
Statistics · Definition
A joint distribution describes the combinations of values of two variables and how often they occur.
For categorical data it can be represented by cell counts or proportions in a two-way table.
For example, the distribution of students across grade-level and transportation-mode combinations is a joint distribution.
Joint frequency
Statistics · Definition
A joint frequency is the count in one cell of a two-way table.
It counts individuals who meet both the row category and the column category.
For example, a cell might count students who are in grade 11 and walk to school.
Joint relative frequency
Statistics · Definition
A joint relative frequency divides one cell’s count by the overall table total.
It describes the proportion of all individuals who have both listed characteristics.
For example, 6 grade-11 walkers among 30 students give a joint relative frequency of 20%.
Law of large numbers
Statistics · Definition
The law of large numbers describes how an average or relative frequency approaches its long-run target as repetitions increase under appropriate assumptions.
It does not require short sequences to look balanced or make earlier outcomes force later outcomes.
For example, the proportion of red cards drawn with replacement tends toward one half over many independent draws.
Least-squares regression
Statistics · Definition
Least-squares regression fits a line predicting a response from an explanatory variable.
The fitted line minimizes the sum of squared vertical residuals between observed and predicted responses.
For example, a fitted line can predict a plant’s height from its age, within the range of observed ages.
Left-skewed distribution
Statistics · Definition
A left-skewed distribution has a longer tail toward smaller values.
Most observations may lie toward the higher end while a few small values extend the left tail.
For example, scores on an easy test can be left-skewed when most scores are high.
Linear transformation of a random variable
Statistics · Definition
A linear transformation multiplies a random variable by a constant and adds another constant.
Adding a constant shifts its mean, while multiplying by a constant scales its standard deviation by the constant’s absolute value.
For example, converting a measured length from meters to centimeters multiplies both its mean and standard deviation by 100.
Logarithmic transformation
Statistics · Definition
A logarithmic transformation replaces positive values by their logarithms.
It can turn an exponential or power relationship into a linear relationship when the appropriate variables are transformed.
For example, plotting the logarithm of bacterial count against time can reveal exponential growth as a line.
Lurking variable
Statistics · Definition
A lurking variable is an unmeasured or unaccounted-for variable that helps explain an observed relationship.
It may create confounding when it is related to both variables being studied.
For example, prior preparation might influence both time spent studying and test score.
Margin of error
Statistics · Definition
A margin of error is the amount added to and subtracted from a point estimate in a symmetric confidence interval.
In the procedures here it combines a critical value with an estimated standard error and addresses modeled sampling variability.
For example, an estimate of 40% with a margin of 3 percentage points gives an interval from 37% to 43%.
Marginal distribution
Statistics · Definition
A marginal distribution describes one variable on its own after the other variable is summed or otherwise accounted for.
In a two-way table, it uses row or column totals, expressed as counts or proportions.
For example, adding all grade-level counts for each transportation mode gives the marginal distribution of transportation mode.
Marginal frequency
Statistics · Definition
A marginal frequency is a row or column total in a two-way table.
It counts one category without separating individuals by the other variable.
For example, the total number of walkers includes walkers from every grade level.
Marginal relative frequency
Statistics · Definition
A marginal relative frequency divides a row or column total by the overall table total.
The collection of these proportions describes one categorical variable on its own.
For example, 12 walkers among 30 students give a marginal relative frequency of 40%.
Matched pairs design
Statistics · Definition
A matched pairs design compares two treatments within matched units or within the same unit.
Assignment within pairs or treatment order is randomized when the design permits, and analysis uses the pairwise differences.
For example, two shoe designs can be compared on the same person with their left-right assignment randomized.
Mean
Statistics · Definition
The mean is a measure of center often called an average.
For a finite data set, the arithmetic mean is the sum of the values divided by their number; for a random variable, the mean is its probability-weighted expected value when that value exists.
Means can be pulled toward extreme values and are not generally resistant.
For example, the arithmetic mean of 2, 4, and 9 is 5.
Median
Statistics · Definition
The median is the middle of an ordered numerical data set.
For an even number of observations, it is the average of the two middle values under the usual convention.
For example, the median of 2, 4, and 20 is 4.
Mode
Statistics · Definition
A mode is a value occurring with the greatest frequency in a data set.
For a density or grouped graph, a mode refers to a location or interval at a peak.
For example, the mode of 2, 2, 3, and 5 is 2.
Multimodal distribution
Statistics · Definition
A multimodal distribution has several main peaks.
Apparent peaks can depend on how a histogram groups observations into intervals.
For example, data collected from several distinct age groups may have several peaks.
Mutually exclusive events
Statistics · Definition
Mutually exclusive events cannot occur together in the same trial.
Their intersection is empty, so the probability that either occurs is the sum of their probabilities.
For example, rolling exactly 1 and rolling exactly 4 are mutually exclusive outcomes of one die roll.
Negative association
Statistics · Definition
Two variables have a negative association when larger values of one tend to accompany smaller values of the other.
The tendency can be linear or curved and does not by itself show cause and effect.
For example, higher fuel efficiency may accompany lower fuel use per mile.
Nonresponse
Statistics · Definition
Nonresponse occurs when selected individuals do not supply usable responses.
It can cause bias if nonrespondents differ systematically from respondents on the study’s variables.
For example, selected students with long commutes may be less likely to answer a transportation survey.
Nonsampling error
Statistics · Definition
Nonsampling errors arise from problems other than selecting a finite sample.
They include inaccurate answers, missed individuals, recording mistakes, and faulty measurement.
For example, entering a recorded age incorrectly can affect a census as well as a sample survey.
Normal distribution
Statistics · Definition
A Normal distribution is a continuous, symmetric bell-shaped distribution determined by its mean and standard deviation.
Many symmetric distributions are not Normal, so symmetry by itself is insufficient.
For example, a Normal model centered at 50 with standard deviation 5 places most values relatively close to 50.
Normal probability plot
Statistics · Definition
A Normal probability plot compares ordered observations with the pattern expected from a Normal distribution.
A roughly straight pattern supports the approximation, while strong curvature or isolated points can reveal departures.
For example, a residual Normal probability plot can help assess a regression model’s Normal-error assumption.
Null hypothesis
Statistics · Definition
A null hypothesis specifies the reference claim or model used to calculate a test’s probabilities.
It commonly states no population difference or a particular parameter value.
For example, a null hypothesis can state that the population mean travel time equals 15 minutes.
Observation
Statistics · Definition
An observation is a recorded value or set of values for an individual or occasion.
The individual may be a person, object, or event rather than a human subject.
For example, one plant’s measured height is an observation in a plant-growth study.
Observational study
Statistics · Definition
An observational study measures individuals without assigning the treatments or conditions being compared.
It can reveal associations, but confounding can prevent a causal conclusion.
For example, recording students’ existing sleep habits and scores is observational when researchers do not assign sleep schedules.
Observed count
Statistics · Definition
An observed count is the actual number of sampled individuals in a category or table cell.
It is compared with the expected count to measure disagreement with a null model.
For example, if 13 sampled students chose walking, the observed count for walking is 13.
One-sample t procedure
Statistics · Definition
A one-sample t procedure makes inference about a population mean using the sample mean and sample standard deviation.
It requires an appropriate study design and a sufficiently suitable distribution of the mean, with particular care for small samples, strong skewness, and outliers.
For example, a random sample of travel times can estimate a population mean when the assumptions are defensible.
One-sample z procedure for a proportion
Statistics · Definition
A one-sample z procedure uses a Normal approximation for inference about one population proportion.
A test uses the hypothesized proportion in its null standard error, while an interval usually estimates the standard error from the sample proportion.
For example, a random sample can be used to test a stated population preference proportion after checking the procedure’s conditions.
One-sided test
Statistics · Definition
A one-sided test seeks evidence for a specified direction of departure from the null claim.
Its relevant extreme outcomes lie in one direction of the test statistic’s distribution.
For example, a greater-than alternative asks whether a population mean exceeds a stated value.
One-way table
Statistics · Definition
A one-way table lists counts or proportions for the categories of one variable.
The categories should cover the intended outcomes without overlap.
For example, a table listing counts for walking, cycling, and riding describes one transportation variable.
Outlier
Statistics · Definition
An outlier is an observation unusually far from the overall pattern.
The 1.5-IQR rule flags possible outliers below the first quartile minus 1.5 IQR or above the third quartile plus 1.5 IQR.
For example, a very long commute may be flagged for investigation without automatically being a recording error.
P-value
Statistics · Definition
A p-value is the probability, under the null model and its assumptions, of results at least as extreme as those observed in the direction or directions specified by the test.
It does not give the probability that the null hypothesis is true or measure the practical size of an effect.
For example, a p-value of 0.03 describes a 3% tail probability under the null model, rather than a 3% probability that the null is correct.
Paired t procedure
Statistics · Definition
A paired t procedure analyzes the mean of differences computed within matched pairs.
It uses one-sample t inference on those differences and requires independence across pairs rather than within each pair.
For example, subtracting each student’s before score from the same student’s after score preserves the pairing.
Percentile
Statistics · Definition
A percentile identifies a location in an ordered distribution by the proportion below or at that location.
Ties and finite samples require a stated convention, so methods can produce slightly different percentile values.
For example, the 80th percentile marks a value with roughly 80% of observations at or below it.
Pie chart
Statistics · Definition
A pie chart displays category proportions as slices of a circle.
A slice’s angle is proportional to its category’s share of the whole.
For example, a category containing one quarter of the observations occupies one quarter of the circle.
Placebo
Statistics · Definition
A placebo resembles an active treatment but lacks the active intervention under study.
It helps provide a comparison for effects associated with receiving treatment and expectations.
For example, an inactive pill can be made to look like the experimental pill.
Placebo effect
Statistics · Definition
A placebo effect is a response associated with expectations or the experience of receiving a treatment.
Observed change in a placebo group can also include natural recovery or other influences, so it is not all automatically a placebo effect.
For example, reported discomfort may change after a participant receives an inactive pill.
Point estimate
Statistics · Definition
A point estimate is the specific numerical value obtained from an estimator for the observed sample.
It gives one estimate without expressing uncertainty by itself.
For example, a sample mean of 14 minutes is a point estimate of the population’s mean travel time.
Point estimator
Statistics · Definition
A point estimator is a rule that uses sample data to produce one estimate of a population parameter.
The rule has a sampling distribution before the sample is observed.
For example, computing the sample mean is a point estimator of the population mean.
Pooled sample proportion
Statistics · Definition
A pooled sample proportion combines successes and sample sizes from two groups.
It estimates a common proportion when a two-proportion null hypothesis states that the population proportions are equal.
For example, 12 successes in 30 trials and 18 in 40 trials give a pooled proportion of 30 divided by 70.
Population
Statistics · Definition
A population is the full set of individuals or measurements a study aims to describe.
It must be defined clearly before deciding whether a sample supports generalization.
For example, all students enrolled in a particular school this year form a population.
Population parameter
Statistics · Definition
A population parameter is a numerical feature of the population or probability model being studied.
It is fixed for that defined population, although its value may be unknown.
For example, the true proportion of all enrolled students who walk to school is a parameter.
Population proportion
Statistics · Definition
A population proportion is the fraction of the entire population with a specified characteristic.
It is a parameter, commonly written p, that a sample proportion can estimate.
For example, the fraction of all enrolled students who walk to school is the population proportion of walkers.
Positive association
Statistics · Definition
Two variables have a positive association when larger values of one tend to accompany larger values of the other.
Individual observations need not follow the tendency perfectly.
For example, taller trees may tend to have wider trunks.
Power model
Statistics · Definition
A power model has the form y = a times x raised to a constant power b.
For positive values, taking logarithms of both variables gives a linear relationship between log y and log x.
For example, the area of a circle is proportional to the square of its radius.
Probability
Statistics · Definition
Probability quantifies how likely an event is under a specified chance model.
It lies between 0 and 1, with larger values representing greater likelihood.
For example, drawing a red card from a well-shuffled standard deck has probability one half.
Probability distribution
Statistics · Definition
A probability distribution describes a random variable’s possible values and their probabilities.
A discrete distribution lists probability masses, while a continuous distribution describes probabilities through areas under a density.
For example, a distribution can list the probabilities of 0, 1, 2, or 3 red cards in three draws.
Probability model
Statistics · Definition
A probability model specifies possible outcomes and probabilities for events formed from those outcomes.
Probabilities are nonnegative, and the probability of the whole sample space is 1.
In a discrete model, individual outcome probabilities sum to 1; in a continuous density model, total area is 1 even though each single value has probability 0.
For example, a fair four-sided die can be modeled with four outcomes each assigned probability one quarter.
Quantitative variable
Statistics · Definition
A quantitative variable records a numerical count or measurement for each individual.
Differences between its values have a meaningful numerical interpretation.
For example, the number of books borrowed and the mass of a backpack are quantitative variables.
Quartile
Statistics · Definition
Quartiles divide ordered quantitative data into four portions.
The first and third quartiles mark approximately the 25th and 75th percentiles, and sample calculation conventions can differ.
For example, a first quartile of 12 minutes means that roughly one quarter of the recorded times are at or below 12 minutes.
Random assignment
Statistics · Definition
Random assignment uses chance to allocate experimental units to treatment conditions.
It tends to balance other influences across groups and supports causal inference in a well-conducted experiment, without guaranteeing identical groups.
For example, shuffled labels can assign plants to two fertilizer treatments.
Random sampling
Statistics · Definition
Random sampling uses a specified chance process to select individuals from a population.
It supports population inference when the sampling frame and data collection are appropriate, but does not guarantee perfect balance or eliminate every source of bias.
For example, a random-number generator can select student identifiers from an enrollment list.
Random variable
Statistics · Definition
A random variable assigns a numerical value to each outcome of a chance process.
Its value is uncertain before the outcome is observed, and its distribution gives the probabilities of possible values.
For example, the number of red cards in three draws is a random variable.
Randomized block design
Statistics · Definition
A randomized block design first groups similar experimental units using a relevant characteristic.
Treatments are then randomly assigned within each block, and comparisons account for the blocking.
For example, fertilizer treatments can be randomized separately within sunny and shaded locations.
Range of data
Statistics · Definition
The range of a data set is its maximum value minus its minimum value.
It measures spread and is sensitive to extreme observations.
For example, values from 3 to 11 have a range of 8.
Regression intercept
Statistics · Definition
A regression intercept is the line’s predicted response when the explanatory variable is zero.
It may lack a useful contextual interpretation when zero lies outside the relevant data range.
For example, an intercept predicts height at age zero even if measurements were collected only for mature plants.
Regression slope
Statistics · Definition
A regression slope is the predicted change in the response for a one-unit increase in the explanatory variable.
Its units are response units per explanatory-variable unit.
For example, a slope of 2 centimeters per week predicts 2 additional centimeters of height for each additional week.
Relative frequency
Statistics · Definition
Relative frequency is a group’s count divided by the total count.
It can be written as a proportion or multiplied by 100 to give a percentage.
For example, 8 choices out of 20 have relative frequency 0.4, or 40%.
Replication
Statistics · Definition
Replication uses multiple independent experimental units for each treatment.
It helps estimate chance variation and increases the information available for comparing treatment responses.
For example, assigning many independently treated plants to each fertilizer is replication, while repeatedly measuring one plant adds a different kind of information.
Residual
Statistics · Definition
A residual is an observed response minus its predicted response.
A positive residual places the observed point above the fitted line, while a negative residual places it below.
For example, an observed height of 18 centimeters and a predicted height of 16 centimeters give a residual of 2 centimeters.
Residual plot
Statistics · Definition
A residual plot displays prediction residuals against explanatory values or fitted responses.
Curvature can suggest that a straight-line model is inappropriate, while changing spread can challenge constant-variance assumptions.
For example, a U-shaped residual pattern suggests that the line misses a curved relationship.
Resistant measure
Statistics · Definition
A resistant measure changes relatively little when a small number of observations become extreme.
Resistance means reduced sensitivity, rather than complete immunity to every change in the data.
For example, changing a largest value from 30 to 300 may leave the median unchanged while substantially increasing the mean.
Response bias
Statistics · Definition
Response bias is a systematic pattern of inaccurate or influenced answers.
Wording, interviewer behavior, memory, and social pressure can affect what people report.
For example, a leading question can encourage respondents to favor a proposed school policy.
Response variable
Statistics · Definition
A response variable records the outcome being studied.
Its relationship to an explanatory variable can be investigated without assuming that the explanatory variable causes it.
For example, plant height is a response when researchers compare fertilizer treatments.
Right-skewed distribution
Statistics · Definition
A right-skewed distribution has a longer tail toward larger values.
Most observations may lie toward the lower end while a few large values extend the right tail.
For example, waiting times can be right-skewed when most waits are short and a few are very long.
Sample
Statistics · Definition
A sample is the subset of a population actually studied.
How it is selected matters for the conclusions that can be drawn about the population.
For example, 60 selected students may be a sample from a school’s full enrollment.
Sample proportion
Statistics · Definition
A sample proportion is the number of sampled individuals with a specified characteristic divided by the sample size.
It is commonly written p-hat and estimates the corresponding population proportion.
For example, 18 walkers among 60 sampled students give a sample proportion of 0.30.
Sample space
Statistics · Definition
The sample space is the set of all possible outcomes of a chance process.
Outcomes must be described at the level relevant to the question being modeled.
For example, the sample space for one draw from a red-blue bag can be red and blue.
Sample statistic
Statistics · Definition
A statistic is a numerical summary calculated from sample data.
Its value can change when a different sample is selected from the same population.
For example, the proportion of walkers among 60 sampled students is a statistic.
Sampling distribution
Statistics · Definition
A sampling distribution describes how a statistic varies across all possible samples drawn under a specified sampling design and sample size.
It is a distribution of statistics rather than of individual observations.
For example, the proportions from repeated random samples of 60 students form a sampling distribution.
Sampling distribution of a sample mean
Statistics · Definition
The sampling distribution of a sample mean describes averages from repeated samples of a fixed size.
For independent identically distributed observations with finite variance, its mean is the population mean and its standard deviation is the population standard deviation divided by the square root of the sample size.
For example, averages from samples of 100 measurements vary less than averages from samples of 25.
Sampling distribution of a sample proportion
Statistics · Definition
The sampling distribution of a sample proportion describes proportions obtained from repeated samples of a fixed size.
A Normal approximation requires an appropriate design and sufficiently large expected success and failure counts.
For example, larger independent samples usually give sample proportions that vary less around the population proportion.
Sampling error
Statistics · Definition
Sampling error is the difference arising because a sample rather than the entire population is observed.
Random sampling variability is distinct from systematic coverage or response bias.
For example, two properly selected random samples can produce different mean commuting times purely by chance.
Sampling frame
Statistics · Definition
A sampling frame is the list or other operational source from which a sample is selected.
It may differ from the target population if some members are omitted or listed incorrectly.
For example, a current enrollment list can serve as a frame for selecting students.
Sampling variability
Statistics · Definition
Sampling variability describes how an estimate changes across repeated samples drawn with the same design.
Larger suitable random samples generally reduce this variability, without automatically correcting systematic bias.
For example, means from samples of 100 students usually vary less than means from samples of 10 students.
Sampling with replacement
Statistics · Definition
Sampling with replacement returns a selected item to the pool before the next selection.
With independent random selections, the available composition and selection probabilities remain unchanged.
For example, replacing a drawn card before reshuffling allows that same card to be drawn again.
Sampling without replacement
Statistics · Definition
Sampling without replacement does not return selected items before later selections.
Selections from a finite population are dependent, although treating them as approximately independent can be reasonable when the sample is a small fraction of the population.
For example, selecting students without replacement ensures that no student is chosen twice.
Scatterplot
Statistics · Definition
A scatterplot displays paired values of two quantitative variables as points.
The horizontal coordinate represents one variable and the vertical coordinate represents the other.
For example, plotting hours studied against test scores helps reveal a pattern without establishing causation.
Significance level
Statistics · Definition
A significance level is a threshold selected before testing to limit Type I errors under the null model.
A p-value at or below that threshold leads to rejection under the stated rule, while a larger p-value leads to failure to reject.
For example, an alpha level of 0.01 demands stronger incompatibility with the null model than a level of 0.05.
Significance test
Statistics · Definition
A significance test assesses how compatible observed data are with a specified null model.
It compares a test statistic or p-value with a decision rule chosen for the study.
For example, a test can assess whether a sample provides evidence that a population walking proportion differs from a stated value.
Simple random sample
Statistics · Definition
A simple random sample of a fixed size gives every possible subset of that size the same chance of selection.
Giving individuals equal selection chances alone does not establish that the design is an SRS.
For example, randomly choosing three identifiers without replacement from ten gives an SRS when every three-identifier set is equally likely.
Simulation
Statistics · Definition
A simulation imitates a chance process using a specified model.
Repeated simulated outcomes can approximate probabilities when the model faithfully represents the situation.
For example, random digits can model repeated draws from a bag with known color proportions.
Skewness
Statistics · Definition
Skewness describes asymmetry in a quantitative distribution’s tails.
A longer right tail indicates right skew, while a longer left tail indicates left skew.
For example, many short waits with a few very long waits produce right skew.
Standard deviation
Statistics · Definition
Standard deviation measures spread around the mean using squared deviations and a square root.
A sample standard deviation uses the sum of squared deviations divided by one less than the sample size before taking the square root.
For example, a larger standard deviation of travel times indicates greater variability, in the original time units.
Standard deviation of residuals
Statistics · Definition
The standard deviation of residuals estimates the typical prediction error around a fitted regression line.
For simple linear regression it is the square root of the sum of squared residuals divided by the sample size minus 2.
For example, a residual standard deviation of 3 points indicates scatter around predictions on the scale of a few score points.
Standard error
Statistics · Definition
A standard error describes the standard deviation of a statistic’s sampling distribution, usually estimated from data.
It measures uncertainty in an estimate rather than the spread of individual observations.
For example, the estimated standard error of a sample mean is the sample standard deviation divided by the square root of the sample size.
Standard Normal distribution
Statistics · Definition
The standard Normal distribution has mean 0 and standard deviation 1.
Standardizing a Normal random variable produces this common probability scale.
For example, a standardized value of 2 lies two standard deviations above the standard Normal mean.
Statistical bias
Statistics · Definition
Statistical bias is a systematic tendency to favor some outcomes or miss a target value.
For an estimator, bias is its long-run average estimate minus the parameter it estimates.
For example, a survey that consistently misses evening workers may systematically misrepresent commuting patterns.
Statistical independence
Statistics · Definition
Statistical independence means that learning one outcome does not change the probability distribution of another.
For two events, it implies that the probability of both occurring equals the product of their probabilities.
For example, independent draws with replacement keep the second draw’s color probabilities unchanged by the first.
Statistical inference
Statistics · Definition
Statistical inference uses data and a study design to draw conclusions beyond the observed values.
Random sampling supports population generalization, while well-conducted random assignment supports treatment-effect conclusions.
For example, a random sample can estimate a school-wide preference without surveying every student.
Statistical power
Statistics · Definition
Statistical power is the probability of rejecting the null hypothesis for a specified true alternative.
It equals 1 minus the Type II error probability for that alternative and can increase with a larger sample or a larger true effect.
For example, a study with power 0.8 detects its specified alternative in about 80% of repeated studies under the model.
Editorial qualification: The captured text describes power as detecting a Type II error. Power is the probability of correctly rejecting the null hypothesis for a specified true alternative, equal to 1 − β.
Statistical significance
Statistics · Definition
A result is statistically significant at a stated level when it meets the test’s rejection rule.
Statistical significance alone does not establish a large effect, practical importance, or causation.
For example, a very large sample can detect a small population difference that has little practical importance.
Stemplot
Statistics · Definition
A stemplot splits each numerical observation into a stem and a leaf.
A key specifies the place values, so the original observations can be recovered.
For example, a key of 3 | 7 = 37 means that stem 3 with leaf 7 represents 37.
Stratified random sampling
Statistics · Definition
Stratified random sampling divides the population into nonoverlapping groups and randomly samples within every group.
The strata can improve representation and precision when their members share a relevant characteristic.
For example, a school may randomly sample students separately within each grade level.
Stratum
Statistics · Definition
A stratum is a nonoverlapping subgroup used in stratified sampling.
A stratified design samples within every stratum rather than selecting only some groups.
For example, each grade level can be a stratum in a school survey.
Student’s t-distribution
Statistics · Definition
A Student’s t-distribution is a symmetric reference distribution with heavier tails than the standard Normal distribution.
Its shape depends on degrees of freedom and approaches the standard Normal shape as those degrees of freedom grow.
For example, inference about a Normal population mean with an unknown standard deviation uses a t reference with one fewer degree of freedom than the sample size.
Survey
Statistics · Definition
A survey collects information by asking individuals questions.
Its quality depends on the sample design, question wording, and responses received.
For example, a school can survey selected families about their transportation needs.
Symmetric distribution
Statistics · Definition
A symmetric distribution has approximately matching shapes on either side of its center.
Symmetry alone does not establish that a distribution is Normal.
For example, a uniform distribution over a centered interval is symmetric but has a flat rather than bell-shaped density.
Systematic random sampling
Statistics · Definition
Systematic random sampling chooses a random starting point and then selects at a fixed interval in an ordered frame.
A repeating pattern in the frame can make the interval an unsuitable choice.
For example, after a random start among the first ten entries, a researcher can select every tenth entry.
T-table
Statistics · Definition
A t-table lists critical values for t-distributions at specified degrees of freedom and tail probabilities.
Its column convention must be checked before using it for a one-sided test or a two-sided interval.
For example, a two-sided 95% interval leaves 2.5% in each tail when selecting its critical value.
Test statistic
Statistics · Definition
A test statistic summarizes how the sample departs from the null model.
Its reference distribution under the null supplies the probabilities used in testing.
For example, a z-statistic expresses a sample proportion’s difference from a hypothesized proportion in null-model standard-error units.
Treatment
Statistics · Definition
A treatment is a specific condition imposed on an experimental unit.
In a multifactor experiment, a treatment is a combination of factor levels.
For example, low fertilizer with eight hours of light is one treatment combination.
Tree diagram
Statistics · Definition
A probability tree represents successive stages of a chance process using branches.
Multiplying the relevant conditional probabilities along a path gives that path’s probability.
For example, a two-draw tree can show how a bag’s second-draw probabilities change when the first item is not replaced.
Two-sample t procedure
Statistics · Definition
A two-sample t procedure compares population means using two independent samples or groups.
The usual unpooled procedure estimates each group’s variance separately, and paired data require a different analysis of pairwise differences.
For example, independently sampled travel times from two schools can estimate the difference between their population means.
Two-sample z procedure for proportions
Statistics · Definition
A two-sample z procedure uses a Normal approximation to compare two population proportions from independent groups.
The equality test pools proportions for its null standard error, while the usual confidence interval uses separate sample proportions.
For example, it can compare preference proportions from independently sampled students in two schools when its conditions hold.
Two-sided test
Statistics · Definition
A two-sided test seeks evidence for departures in either direction from the null claim.
For symmetric z or t tests, extreme evidence appears in both tails of the reference distribution.
For example, a not-equal alternative asks whether a population mean is either lower or higher than a stated value.
Two-way table
Statistics · Definition
A two-way table organizes counts for combinations of two categorical variables.
The cells show combinations, while row and column totals summarize each variable separately.
For example, a table can classify students by grade level and whether they walk to school.
Type I error
Statistics · Definition
A Type I error occurs when a test rejects a true null hypothesis.
The test’s significance level controls its probability under the null model.
For example, a study might report a treatment difference even though the population treatment effects are equal.
Type II error
Statistics · Definition
A Type II error occurs when a test fails to reject a false null hypothesis.
Its probability depends on a specified true alternative and the test’s design.
For example, a study might fail to detect a real treatment difference when the sample is too small.
Unbiased estimator
Statistics · Definition
An unbiased estimator has a sampling-distribution mean equal to the parameter being estimated.
Individual estimates can still differ substantially from the parameter, so unbiased does not mean perfectly accurate.
For example, a properly sampled mean can be unbiased even when small samples produce highly variable estimates.
Undercoverage
Statistics · Definition
Undercoverage occurs when some groups in the target population have little or no opportunity to enter the sample.
A large sample does not remove bias caused by an incomplete sampling frame.
For example, using only a landline directory misses people who have no listed landline.
Unimodal distribution
Statistics · Definition
A unimodal distribution has one main peak.
The peak marks a region where observations or probability are especially concentrated.
For example, a histogram of heights might have one prominent peak around the most common height interval.
Variance
Statistics · Definition
Variance is the square of the standard deviation.
It measures spread in squared units, with the precise averaging rule depending on whether it describes a sample or a probability distribution.
For example, a standard deviation of 3 centimeters corresponds to a variance of 9 square centimeters.
Venn diagram
Statistics · Definition
A Venn diagram represents sets or events with regions inside a sample-space boundary.
Overlapping regions show outcomes shared by two events.
For example, overlapping circles for red cards and kings identify the red kings.
Voluntary response sample
Statistics · Definition
A voluntary response sample consists of people who choose to respond to an open invitation.
People with strong opinions may participate more often than others, creating selection bias.
For example, an unrestricted online poll can overrepresent those especially interested in the issue.
Z-score
Statistics · Definition
A z-score expresses a value’s distance from its distribution’s mean in standard-deviation units.
It is found by subtracting the mean from the value and dividing by the standard deviation.
For example, a z-score of −1.5 places the value 1.5 standard deviations below the mean.
Z-table
Statistics · Definition
A z-table lists probabilities or areas for the standard Normal distribution.
Tables use different conventions, so the headings must be checked to identify which area is listed.
For example, a cumulative-left table gives the area to the left of each z-value.