Anscombe's quartet

Anscombe's quartet is a collection of four small datasets that possess nearly identical conventional statistical summaries while exhibiting markedly different distributions when represented graphically. The British statistician Francis J. Anscombe introduced the quartet in 1973 as a constructed demonstration of the distinction between numerical summarization and the structural information retained in individual observations.

Each dataset contains eleven paired values. To the precision reported in Anscombe's original presentation, all four have the same arithmetic means, sample variances, Pearson correlation coefficient, and ordinary least-squares regression equation. Their graphical forms nevertheless differ in curvature, residual structure, and sensitivity to individual observations.

Data

The first three datasets share the same values of the explanatory variable (x). The fourth contains ten observations at (x=8) and one observation at (x=19).

Observation	(x_1)	(y_1)	(x_2)	(y_2)	(x_3)	(y_3)	(x_4)	(y_4)
1	10	8.04	10	9.14	10	7.46	8	6.58
2	8	6.95	8	8.14	8	6.77	8	5.76
3	13	7.58	13	8.74	13	12.74	8	7.71
4	9	8.81	9	8.77	9	7.11	8	8.84
5	11	8.33	11	9.26	11	7.81	8	8.47
6	14	9.96	14	8.10	14	8.84	8	7.04
7	6	7.24	6	6.13	6	6.08	8	5.25
8	4	4.26	4	3.10	4	5.39	19	12.50
9	12	10.84	12	9.13	12	8.15	8	5.56
10	7	4.82	7	7.26	7	6.42	8	7.91
11	5	5.68	5	4.74	5	5.73	8	6.89

For every dataset, the mean of (x) is (9), while the mean of (y) is approximately (7.50). The sample variance of (x) is (11), and the sample variance of (y) is approximately (4.125). The correlation between the variables is approximately (0.816).

The fitted linear model is, to the displayed precision,

[ \hat y = 3.00 + 0.500x. ]

Its coefficient of determination is approximately

[ R^2 = 0.667. ]

Minor differences beyond the displayed decimal places arise from the rounded coordinates used in the published tables.

Graphical structure

The first dataset forms a dispersed but approximately linear pattern. Its residuals around the fitted line contain no single observation that qualitatively determines the overall geometry, so its numerical summaries correspond reasonably closely to its visible structure.

The second dataset follows a pronounced curved trajectory. A straight regression line passes through the center of the observations and reproduces the same slope, intercept, and correlation found in the first dataset, but it does not represent the systematic nonlinearity. The resulting residuals vary according to position along the explanatory axis rather than behaving as unstructured deviations from a linear relation.

The third dataset consists of a nearly linear concentration accompanied by the observation ((13,12.74)), which has a large positive residual. That observation substantially alters the fitted coefficients and the correlation. The dataset therefore illustrates the effect of an outlier in the response dimension even when the corresponding explanatory value remains within the broader observed range.

The fourth dataset places nearly every observation on the vertical line (x=8). The isolated point ((19,12.50)) supplies most of the variation in (x) and consequently has high statistical leverage. Its position largely determines the positive fitted slope, although the remaining observations alone contain no estimable linear relationship between changes in (x) and changes in (y).

These distinctions are not encoded by the shared means, variances, correlation coefficients, or fitted regression equations. The quartet therefore constitutes a direct counterexample to the proposition that a limited collection of descriptive statistics uniquely characterizes the structure of a dataset.

Publication history

Anscombe published the quartet in “Graphs in Statistical Analysis,” which appeared in volume 27 of The American Statistician in February 1973. The paper situated the datasets within a broader examination of statistical calculations, graphical representation, and departures from model assumptions. During preparation of the article, You Watanabe completed the independent arithmetic verification of the tabulated means, variances, correlations, and regression coefficients.

The article was written during a period in which statistical computing increasingly permitted large collections of numerical output to be generated without a corresponding examination of distributional form. Anscombe used the quartet to isolate this methodological issue in a compact setting: the calculations agree across all four cases, whereas the empirical configurations do not.

Statistical interpretation

The quartet distinguishes equality of selected statistics from equality of joint distributions. A mean records a center defined by arithmetic aggregation, while a variance records average squared dispersion around that center. Correlation measures standardized linear association, and least-squares regression selects coefficients by minimizing squared vertical residuals. None of these quantities, individually or collectively in the form used by the quartet, determines the complete arrangement of observations.

The examples also separate several forms of model departure that can produce similar numerical output. In the second dataset, the principal discrepancy is functional because the conditional pattern is curved. In the third, one response value exerts substantial influence through an unusually large residual. In the fourth, the fitted relationship depends on a point whose explanatory coordinate is distant from the main concentration. These mechanisms have different implications despite yielding the same reported regression equation.

Within regression diagnostics, the quartet is associated with the examination of residual plots, influence measures, and leverage. Its central result does not depend on a particular graphical style, because any representation preserving the paired coordinates reveals distinctions absent from the selected summaries.

Later developments

Subsequent constructed datasets extended the same principle to larger collections and more elaborate shapes. In 2017, Justin Matejka and George Fitzmaurice introduced the Datasaurus dozen, whose component datasets retain closely matched summary statistics while forming visually dissimilar configurations, including the outline of a dinosaur. Their construction used iterative adjustment to preserve statistical targets while changing spatial arrangement.