Statistics Glossary

For a preplanned two-sided test at α = 0.05, a p-value of 0.04 crosses the threshold. It does not imply a 5% probability that this particular decision is wrong.

To ask whether mean package fill differs from 500 g, write H₁: μ ≠ 500 g. Choose a one-sided direction before inspecting the sample if that direction is justified.

For three groups and nine observations, a classical one-way ANOVA uses numerator df 2 and denominator df 6. A significant overall test does not identify which pairs differ.

A bar chart can compare counts of tea, coffee, and juice orders. Separate bars represent categories, rather than contiguous numerical intervals.

A survey of only weekday customers may systematically miss weekend preferences. A larger weekday sample does not automatically remove that selection problem.

With ten independent trials and success probability 0.2 on each, the expected success count is np = 2. Sampling without replacement from a small population changes the model.

A box can show Q1 = 10, median = 15, and Q3 = 20. Whisker and outlier conventions vary, so identify the convention rather than assuming whiskers are always the minimum and maximum.

Payment method is nominal; a satisfaction response from unhappy to happy is ordinal. Averaging numeric category codes is not automatically meaningful.

Counting every enrolled student is a census of that school’s enrollment. Missing or incorrectly recorded students can still create error.

Averages of many independent die rolls become approximately normal even though individual rolls remain discrete. There is no universal sample-size cutoff that works for every population.

If Z₁ and Z₂ are independent standard normal variables, Z₁² + Z₂² has a chi-square distribution with 2 df. Categorical tests use one application of this distribution.

A 2×3 independence table has df = (2−1)(3−1) = 2. Check expected counts and independent units before using the approximation.

In predicted score = 60 + 3 × hours, 3 is a slope coefficient with units of score points per hour. It describes a fitted association unless a causal interpretation is supported.

A 95% t interval for a mean uses an estimate plus or minus a margin. Across repetitions, its method has about 95% coverage under the assumptions; it does not contain 95% of individuals.

A modeled waiting time can take any nonnegative real value, although a timer may record it rounded to seconds. The recording precision is not the same as the model’s support.

For paired X and Y with r = −0.8, the observed linear pattern decreases. A U-shaped relationship can have r = 0 despite strong dependence.

Positive covariance indicates that above-mean X tends to accompany above-mean Y. Its magnitude depends on units, unlike standardized Pearson correlation.

A one-sample t procedure with 16 observations uses 15 df. A paired test counts pairs, and Welch’s independent-group df may be fractional.

When predicting waiting time from order size, waiting time is the response. Naming it dependent does not prove order size causes the observed differences.

The mean of 2, 3, 3, 4, 8 is 4 and the median is 3. These describe the observed values without automatically generalizing to a population.

The number of orders in an hour is a count taking nonnegative integers. A numerical customer ID is a label, not a count variable.

Two datasets can share a mean while one has a long right tail. Plotting the distribution reveals what a single summary misses.

An observed 4-point mean difference is an effect estimate in original units. Dividing by a relevant SD gives a standardized measure whose definition should be labeled.

A fair six-sided die has expected value 3.5, although no single roll equals 3.5. Expectation is a probability-weighted mean, not the most likely outcome.

A classical ANOVA with three groups and nine observations uses F with df (2, 6). Swapping the numerator and denominator df changes the distribution.

If coffee occurs in 30 of 50 orders, its frequency is 30 and relative frequency is 0.60. These are observed summaries, not guaranteed population probabilities.

A histogram groups waits into intervals, such as 0–2 and 2–4 minutes. If bin widths differ, density heights rather than raw counts preserve area interpretation.

A claim that population mean wait equals 10 minutes is a statistical hypothesis. Define the population and measurement before testing it.

In a model predicting score from study hours, hours is a predictor. It need not be independent of other predictors or randomly assigned.

A sample mean and a confidence interval can estimate a population mean under a suitable design and model. Inference does not repair an unrepresentative sample.

If Q1 = 10 and Q3 = 18, the IQR is 8. Quartile algorithms can differ in small samples, so state the convention when reproducing results.

A normal distribution has kurtosis 3, or excess kurtosis 0. Different software can report either convention, and sample estimates are sensitive to extremes.

A fitted line ŷ = 10 + 2x predicts a two-unit higher mean response for a one-unit higher predictor. An individual response still varies around the fitted mean.

A mean interval 20 ± 2.13 has margin 2.13. It reflects the specified uncertainty procedure, not a maximum possible error across all sources.

For 2, 3, 3, 4, 8, the mean is 20/5 = 4. Combine group means by weighting with group sizes when the target is the individual-level mean.

For 2, 3, 4, 8, the median is (3+4)/2 = 3.5. Ties mean it is not always literally true that half the values are strictly below it.

For 2, 3, 3, 4, 8, the mode is 3. If every value is unique, courses often report no mode; a strict maximum-frequency definition instead gives a complete tie.

A central 95% normal interval uses approximately μ ± 1.96σ. The familiar μ ± 2σ interval contains about 95.45%, and only under a normal model.

H₀: μ = 500 g is a target-value claim, not necessarily a claim of zero. A p-value is conditional on this model; it is not the model’s probability of being true.

A wait of 60 minutes among mostly 2–5 minute waits merits investigation. Do not discard it merely because it changes the mean or p-value.

If p = 0.03, results at least as extreme as observed have probability 3% under the null model and assumptions. This does not mean a 3% probability that the null is true.

Population mean wait μ is a parameter; sample mean wait x̄ is a statistic used to estimate it. The target population must be defined.

A 90th-percentile wait is a cutoff at or below which about 90% of the relevant distribution lies. Ties and sample quantile conventions affect exact empirical calculations.

For a café study, the population might be all eligible orders during one month. Changing the branch or period changes the population being described.

A fair die assigns probability 1/6 to rolling a six. In a continuous model, an exact point can have probability zero without being logically impossible.

Q2 is the median; Q1 and Q3 mark the lower and upper quarters. Small-sample interpolation or median-of-halves algorithms can give different results.

In a simple random sample, eligible units have equal selection probabilities. Random selection does not guarantee every realized sample is perfectly representative.

For 2, 3, 3, 4, 8, the range is 8−2 = 6. It uses only the extremes and can change greatly with one unusual observation.

A regression model links a response to predictors. Check the functional form, dependence, variation, and influential observations before interpreting coefficients.

Regression can estimate mean score as a function of hours and prior attainment. Adding predictors does not automatically remove confounding or improve out-of-sample prediction.

If an observed score is 88 and the fitted score is 86.7, its residual is 1.3. A residual is not the same as the unobserved population-model error.

Eighty recorded orders can form a sample of a month’s eligible orders. A convenience sample may not represent the full month.

Forty people each measured twice give 40 pairs for a paired analysis, not 80 independent people. Precision depends on information and design, not just row count.

Repeated independent samples of 25 observations produce a distribution of sample means. Its spread differs from the spread of individual observations.

A long right tail often pulls the mean above the median. Mean–median ordering alone does not prove a distribution’s shape.

For sample values 6, 8, 10, 12, 14, sample SD is √10 ≈ 3.1623. SD has the original units; variance has squared units.

With sample SD 10 and n = 25 independent observations, estimated SE of the mean is 2. SE measures sampling variability and does not include systematic bias.

Z ~ N(0,1) has P(Z ≤ 1.96) ≈ 0.9750. Standardizing a skewed variable does not make it standard normal.

A design can target 80% power for a specified effect at a specified α. It does not have one universal power for every possible nonzero effect.

A preplanned test with p = 0.04 crosses α = 0.05 but not α = 0.01. Neither comparison says how large or practically valuable the effect is.

For 15 df, a central 95% t interval uses cutoff 2.1314, larger than the normal cutoff 1.9600. Unknown population SD does not become known when n exceeds 30.

A paired t-test analyzes within-unit differences; Welch’s t-test compares independent means. Same-length columns alone do not justify pairing.

Rejecting a true claim that μ = 500 g is a Type I error. α controls a procedure’s rate under its null assumptions, not the probability a specific rejection is wrong.

Failing to reject μ = 500 g when the true mean is 495 g is a Type II error for that alternative. A nonsignificant result does not prove no difference.

The sample squared-deviation sum for 6, 8, 10, 12, 14 is 40; sample variance is 40/4 = 10. Describing the complete five-value population instead gives 40/5 = 8.

A value 62 relative to mean 50 and SD 8 has z = 1.5. Its normal upper-tail probability is 0.0668 only if the normal model is justified.