This article is a practical map of 29 short statistics explainers compiled by Vincent Granville in 2018. It is an index rather than a complete statistics course: use each entry to identify the idea, its usual purpose, and the assumption or distinction that matters before you study the linked explainer.
The topics range from averages and probability to hypothesis tests, regression conditions, time-series models, and information criteria.
Describing data and measurement
Arithmetic mean
Add all observations and divide by the number of observations. The mean uses every value, so unusually large or small values can pull it away from a typical observation.
Average
“Average” is a broad everyday term. It often means the arithmetic mean, but it can also refer to a median or another summary, so state which calculation you used.
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Average deviation
Average deviation usually means the average absolute distance of observations from a chosen center, commonly the mean or median. Because definitions vary, identify the center and whether distances are absolute.
Absolute error and mean absolute error (MAE)
Absolute error is the size of a prediction’s miss: |actual − predicted|. MAE is the mean of those absolute errors across observations, expressed in the same units as the outcome and giving each miss equal linear weight.
Accuracy and precision
Accuracy is closeness to the true or accepted value; precision is repeatability. Measurements can be precise but systematically wrong, accurate on average but scattered, both, or neither.
Attribute variable / passive variable
An attribute (or passive) variable records a characteristic that already exists, such as age or blood type, rather than something assigned by a researcher. It can describe groups, but it does not by itself establish a causal effect.
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Average inter-item correlation
This is the mean correlation among items intended to measure the same construct, such as questions on a questionnaire. Higher values can indicate consistency, although extremely high correlations may mean items are redundant.
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Probability, distributions, and graphical meaning
68–95–99.7 rule
For an approximately normal distribution, about 68% of values fall within one standard deviation of the mean, 95% within two, and 99.7% within three. It is an approximation that depends on a roughly bell-shaped distribution.
Bell curve (normal curve)
The normal curve is a symmetric, mound-shaped probability distribution defined by its mean and standard deviation. Real data need not be normal; the model is useful only when its fit and the analysis assumptions are reasonable.
Bernoulli distribution
A Bernoulli variable has one trial and two outcomes, commonly coded success/failure or 1/0, with success probability p. Repeated independent Bernoulli trials lead to the binomial model.
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Bayes’ theorem updates a prior probability with evidence to obtain a posterior probability: P(A|B) = P(B|A)P(A)/P(B). The prior and the evidence model matter; a test’s accuracy alone does not equal the probability that a positive result is true.
Area between two z values on opposite sides of the mean
Convert each observation to a z score, locate both scores on the standard normal curve, and subtract the appropriate cumulative areas. When the scores lie on opposite sides of zero, the central area is the sum of the two tail-to-mean areas.
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Area to the right of a z score
For a standard normal variable, the area to the right of z is the probability of observing a value greater than that z score. It equals one minus the cumulative area to the left.
Area Principle
In a statistical graphic, the visual area used for a quantity should be proportional to the quantity represented. Violating this principle—such as stretching a bar’s area far more than its value—can exaggerate differences.
Sampling assumptions and tests
10% condition in statistics
For sampling without replacement, a common independence guideline is that the sample should be no more than 10% of the population. Sampling more than 10% can make observations dependent enough to invalidate standard error calculations.
Assumption of independence
Independence means one observation or error does not provide information about another, given the study design and model. Repeated measures, clusters, families, and time-ordered data often violate it and require methods that account for dependence.
Assumption of normality / normality test
Normality may be required for a model’s errors or a small-sample test—not necessarily for every raw variable. Use plots and subject-matter knowledge alongside formal tests, because normality tests can be overly sensitive in large samples and underpowered in small ones.
Assumptions and conditions for regression
Regression analyses commonly check linearity, independent errors, constant error variance, and appropriately distributed residuals for the intended inference. Influential observations, omitted structure, and poorly measured variables can matter even when a residual plot looks acceptable.
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Bartlett’s test
Bartlett’s test evaluates whether several groups have equal variances. It is sensitive to departures from normality, so a robust alternative or graphical assessment may be preferable when group distributions are non-normal.
Balanced and unbalanced designs
A balanced design has equal numbers of observations in each group or treatment combination; an unbalanced design does not. Balance simplifies comparisons and can improve efficiency, while valid analysis of unbalanced data depends on the model and missing-data pattern.
Models, regression, and time series
Adjusted R-squared
Adjusted R-squared modifies ordinary R-squared for the number of predictors and sample size. It can decrease when a new predictor adds little explanatory value, but it is not a test of causality or a guarantee that the model will predict well on new data.
ANCOVA
Analysis of covariance (ANCOVA) compares group means while statistically controlling for one or more continuous covariates. The interpretation depends on a suitable linear relationship, comparable slopes when required, independent errors, and correctly specified groups and covariates.
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Autoregressive model
An autoregressive model predicts a time-series value from its own earlier values. The number of lags, stationarity, seasonality, and changing variance determine whether the model is appropriate.
Augmented Dickey–Fuller (ADF) test
The ADF test examines whether a time series has a unit root, a common indication of non-stationarity. Its result depends on choices such as including an intercept or trend and the lag length; failing to reject the null is not proof that every feature of the series is stable.
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Attributable risk / attributable proportion
Attributable risk is the difference in incidence between an exposed group and a comparison group. The attributable proportion expresses the excess incidence as a fraction of incidence in the exposed group. Causal wording requires a defensible study design and control of confounding.
Akaike’s Information Criterion (AIC)
AIC compares fitted models by rewarding goodness of fit and penalizing added parameters. Lower AIC is preferred among the candidate models fitted to the same data and likelihood framework; it is a relative selection tool, not a universal measure of truth.
Bayesian Information Criterion (BIC)
BIC also balances fit against model complexity, with a penalty that increases with sample size and is typically stronger than AIC’s. Lower BIC is preferred for the compared models under the same data and likelihood assumptions.
Benjamini–Hochberg procedure
The Benjamini–Hochberg procedure ranks p-values and sets a cutoff to control the expected false-discovery rate across a family of tests. The chosen target level and dependence structure affect its behavior; it does not make every individual significant result certain to be true.
Bessel’s correction
When estimating a population variance from a sample, dividing the squared deviations by n − 1 rather than n corrects the typical downward bias caused by estimating the sample mean. This correction applies to the usual unbiased sample-variance estimator, not to every variance calculation.
Quick Recap
How to use this index
- Start with the concept that matches your immediate task: summarize data, model a relationship, analyze a time series, compare risks, or adjust for multiple tests.
- Check the assumptions named in the relevant entry before interpreting a p-value, confidence interval, coefficient, or model-comparison score.
- Keep neighboring ideas separate: a mean is one kind of average; accuracy differs from precision; AIC and BIC are relative criteria; and a statistical test cannot replace the design assumptions that justify it.
- Use the original 29-topic index by Vincent Granville (published October 24, 2018) as a signpost to individual explainers, then consult a full introductory statistics text for worked calculations.
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