Anscombe's quartet

Anscombe's quartet is a collection of four synthetic datasets, each comprising eleven pairs of observations (x, y), devised by British statistician Francis J. Anscombe in 1973 to highlight the critical role of graphical methods in statistical analysis.^[1] Despite sharing virtually identical summary statistics—including a mean of x = 9, a mean of y = 7.50, a variance of x = 11.00, a variance of y = 4.12, a correlation coefficient of r = 0.816, and the same least-squares regression line y = 3 + 0.5x—the datasets yield profoundly different scatter plots that reveal distinct data structures and relationships.^[1] This deliberate construction underscores how numerical summaries alone can mask underlying patterns, anomalies, or non-linearities, advocating for visualization as an essential preliminary step in data exploration.^[1] The quartet's four datasets, labeled I through IV, exemplify varied graphical behaviors:

Dataset I displays a scattered but approximately linear relationship between x and y, aligning well with the common regression line and supporting straightforward linear modeling.
Dataset II reveals a nonlinear, upward-curving pattern resembling a parabola, where the linear regression provides a poor fit despite matching summary statistics.
Dataset III follows a linear trend similar to Dataset I but is dominated by a single high-leverage outlier at (13, 12.74), which disproportionately influences the regression without altering the overall statistics.
Dataset IV consists almost entirely of points clustered vertically at x = 8 (with y values varying around 6–9), except for one distant outlier at (19, 12.50), making the apparent linear relationship illusory and the regression line irrelevant.

These contrasts, drawn directly from Anscombe's original tabulations, demonstrate how outliers, curvature, or clustering can evade detection through algebraic computations alone.^[1] Since its publication, Anscombe's quartet has served as a foundational teaching tool in statistics and data science, reinforcing the principle that "graphs are essential to good statistical analysis" and influencing modern practices in exploratory data analysis.^[1] It remains pertinent in an era of big data and automated modeling, reminding practitioners to prioritize visual inspection to avoid misleading conclusions from aggregated metrics.^[2]

History and Origin

Creation by Francis Anscombe

Francis John Anscombe (1918–2001), a prominent British statistician, created Anscombe's quartet in 1973 as a pedagogical tool to underscore the pitfalls of relying solely on numerical statistical analyses.^[3] Born in England and educated at Trinity College, Cambridge, where he earned his B.A. in mathematics in 1939, Anscombe went on to lecture in statistics at the university, shaping the field through his emphasis on rigorous interpretive methods.^[4] He taught at Cambridge before moving to positions at Princeton and Yale, where he founded the statistics department in 1963.^[5] Anscombe's motivation stemmed from the rapid adoption of high-speed computers in the early 1970s, which facilitated complex numerical computations but often encouraged analysts to overlook visual inspection of data.^[3] In his view, this technological shift exacerbated a pre-existing tendency to prioritize "exact" numerical outputs over the interpretive insights provided by graphs, leading to potentially erroneous conclusions.^[3] He constructed the quartet to demonstrate that identical summary statistics could mask fundamentally different underlying relationships.^[3] The quartet appeared in Anscombe's seminal paper "Graphs in Statistical Analysis," published in The American Statistician.^[3] Through this work, he advocated for computers to generate both calculations and graphs, stating: "make both calculations and graphs. Both sorts of output should be studied; each will contribute to understanding."^[3] This creation reflected Anscombe's broader contributions to statistical computing and quality control, where he pioneered methods to ensure reliable data interpretation amid growing computational power.^[5]

Publication and Initial Reception

Anscombe's quartet was first introduced in the article "Graphs in Statistical Analysis" by Francis J. Anscombe, published in The American Statistician, Volume 27, Issue 1, pages 17–21, in February 1973.^[6] In the paper, Anscombe advocated for the routine use of graphs as a "simple but powerful" diagnostic tool in statistical analysis, emphasizing their ability to reveal structures hidden by numerical summaries alone. He presented the quartet as an illustrative exhibit: four datasets engineered to yield nearly identical simple linear regression outputs—including the same means, variances, correlation coefficients, and regression equations—yet displaying markedly different underlying data configurations when plotted. This demonstration underscored the risks of relying solely on summary statistics, particularly in identifying outliers, nonlinearity, and other anomalies that could invalidate model assumptions.^[3] The paper received positive uptake within the statistical community, with no notable controversies arising from its claims. It was praised for its straightforward presentation and practical focus, making complex ideas accessible to both practitioners and educators. The quartet has since become a standard pedagogical tool, frequently cited in textbooks and incorporated into courses on exploratory data analysis to highlight the indispensability of visualization.^[7]

Description of the Datasets

Overview of the Quartet

Anscombe's quartet refers to a collection of four bivariate datasets, each comprising pairs of variables x and y with 11 observations, constructed to possess nearly identical summary statistical measures while exhibiting fundamentally different relationships between the variables.^[1] These datasets were introduced by statistician Francis J. Anscombe in his 1973 paper to underscore the limitations of relying solely on numerical summaries in statistical analysis.^[1] The core purpose of the quartet is to illustrate how datasets that appear statistically equivalent based on aggregate metrics—such as means, variances, and correlation coefficients—can nonetheless reveal markedly distinct underlying structures when subjected to graphical examination.^[1] This demonstration highlights the potential pitfalls of descriptive statistics alone, emphasizing the necessity of visualization to uncover patterns that might otherwise remain obscured.^[1] The datasets are conventionally labeled as sets I, II, III, and IV, each representing a unique relational archetype: set I approximates a linear association, set II displays a nonlinear curvature, set III incorporates a vertical outlier disrupting the trend, and set IV features a leverage point outlier in a largely scattered distribution.^[1]

Detailed Data for Each Dataset

Anscombe's quartet comprises four distinct datasets, each containing eleven paired observations of variables x and y, designed to share the same basic statistical properties despite markedly different underlying relationships. The exact numerical values for these datasets, as originally presented, are detailed below.

Dataset I

This dataset exhibits a roughly linear positive relationship between x and y.

x	y
10	8.04
8	6.95
13	7.58
9	8.81
11	8.33
14	9.96
6	7.24
4	4.26
12	10.84
7	4.82
5	5.68

Dataset II

This dataset features a nonlinear, parabolic relationship between x and y.

x	y
10	9.14
8	8.14
13	8.74
9	8.77
11	9.26
14	8.10
6	6.13
4	3.10
12	9.13
7	7.26
5	4.74

Dataset III

This dataset shows a strong linear relationship but is influenced by an outlier.

x	y
10	7.46
8	6.77
13	12.74
9	7.11
11	7.81
14	8.84
6	6.08
4	5.39
12	8.15
7	6.42
5	5.73

Dataset IV

This dataset includes a vertical line of points at x = 8 with one outlier at x = 19.

x	y
8	6.58
8	5.76
8	7.71
8	8.84
8	8.47
8	7.04
8	5.25
19	12.50
8	5.56
8	7.91
8	6.89

Anscombe constructed these datasets by manually adjusting the y values to achieve identical summary statistics—such as means, variances, and correlation coefficients—across all four, while deliberately varying the functional forms and distributions to highlight graphical differences.^[3]

Summary Statistics

Shared Statistical Properties

Anscombe's quartet consists of four datasets, each with 11 paired observations of variables x and y, engineered to exhibit nearly identical summary statistics despite profound differences in their underlying structures.^[1] Across all datasets, the x variable has a mean of exactly 9 and a sample variance of exactly 11, with no outliers in three of the datasets and the statistics unaffected by the apparent outlier in the fourth.^[1] The y variable shares a mean of 7.50 and a sample variance of approximately 4.125 across the datasets.^[1] The Pearson correlation coefficient r between x and y is approximately 0.816 in each case, indicating a similar strength of linear association.^[1] Fitting a simple linear regression model to each dataset produces nearly identical parameters: a slope of approximately 0.50, an intercept of approximately 3.00, a standard error of the estimate of approximately 1.24, and a t-statistic of approximately 4.24 for the slope (with p < 0.01 in all instances).^[1] These shared properties, equal within rounding error, demonstrate how standard numerical summaries can obscure critical variations in data distribution and relationships.^[1]

Computation of Key Metrics

The key summary statistics for Anscombe's quartet are computed using standard formulas for a dataset of n=11 pairs (x_i, y_i), demonstrating how the datasets are engineered to produce identical numerical results despite their structural differences.^[3] The sample mean for the x values across all four datasets is calculated as \bar{x} = \frac{1}{n} \sum_{i=1}^n x_i. For each dataset, the x values sum to 99, yielding \bar{x} = 99 / 11 = 9. The sample mean for y is similarly \bar{y} = \frac{1}{n} \sum_{i=1}^n y_i = 7.5, as the y values in each dataset sum to 82.5. These means provide the central tendency but mask the varying distributions of the points.^[3] The sample variance for y is given by s_y^2 = \frac{1}{n-1} \sum_{i=1}^n (y_i - \bar{y})^2, which evaluates to approximately 4.125 in all datasets (with the x variance fixed at 11 due to identical x values in the first three datasets and adjusted in the fourth). This equality arises from the deliberate construction where the sum of squared deviations from the mean, \sum (y_i - \bar{y})^2 \approx 41.25, is the same across datasets. For the x values in datasets I–III, \sum (x_i - \bar{x})^2 = 110, confirming s_x^2 = 110 / 10 = 11. In dataset IV, the x values are mostly 8 with one outlier at 19, but the deviations balance to yield the same sum of squares.^[3] The Pearson correlation coefficient r measures linear association and is computed as r = \frac{\sum_{i=1}^n (x_i - \bar{x})(y_i - \bar{y}) }{ \sqrt{ \sum_{i=1}^n (x_i - \bar{x})^2 \sum_{i=1}^n (y_i - \bar{y})^2 } }, resulting in r \approx 0.816 for all datasets. This uniformity stems from matching the cross-product sum \sum (x_i - \bar{x})(y_i - \bar{y}) \approx 55 in each case, divided by \sqrt{110 \times 41.25} \approx 67.38.^[3] For linear regression, the model is y = \beta_0 + \beta_1 x, where the slope \beta_1 = r \frac{s_y}{s_x} and intercept \beta_0 = \bar{y} - \beta_1 \bar{x}. With r \approx 0.816, s_y \approx 2.03, and s_x \approx 3.32, \beta_1 = 0.816 \times (2.03 / 3.32) \approx 0.5 and \beta_0 = 7.5 - 0.5 \times 9 = 3.0, identical for all datasets. To illustrate, consider dataset I with x = [10, 8, 13, 9, 11, 14, 6, 4, 12, 7, 5] and y = [8.04, 6.95, 7.58, 8.81, 8.33, 9.96, 7.24, 4.26, 10.84, 4.82, 5.68]: the deviations (x_i - 9) and (y_i - 7.5) produce squared sums of 110 and approximately 41.25, respectively, and cross-products summing to approximately 55, balancing the formulas to match the other datasets.^[3]

Graphical Analysis

Scatter Plot Visualizations

To visualize the datasets in Anscombe's quartet, construct scatter plots by plotting the x values along the horizontal axis and the y values along the vertical axis, with each of the 11 data points represented as a marker for the corresponding dataset.^[3] When viewed this way, the points in each dataset form markedly distinct patterns, despite the datasets sharing nearly identical linear regression lines numerically.^[3] For Dataset I, the points align closely to a straight line with a positive slope and moderate scatter around it, suggesting an approximately linear relationship.^[3] In Dataset II, the points trace a clear parabolic curve, rising and then falling, indicative of a nonlinear pattern.^[3] Dataset III features points that follow an approximately linear trend with moderate scatter, except for one point that deviates substantially in the y-direction at (13, 12.74), creating an outlier that influences the overall trend.^[3] For Dataset IV, the points form a tight vertical cluster near x=8, disrupted by a single high-leverage point at (19, 12.50), which pulls the regression line away from the main group.^[3] These scatter plots reveal aspects of non-normality and heteroscedasticity in the relationships that remain hidden in the summary statistics.^[3]

Interpretation of Visual Patterns

The scatter plot for Dataset I reveals a clear linear relationship between the variables, with points distributed in a manner that supports the assumptions of ordinary least squares regression. The residuals appear roughly normally distributed and homoscedastic, indicating that the fitted linear model adequately captures the underlying pattern without significant violations of key statistical assumptions.^[1] In contrast, the visualization of Dataset II exposes a pronounced nonlinearity, as the points form a parabolic curve rather than aligning with a straight line. Despite the high coefficient of determination (r² ≈ 0.67), the linear fit poorly describes the data, underscoring the limitations of summary statistics in detecting curved relationships and suggesting the appropriateness of polynomial regression models instead.^[1] Dataset III's plot highlights the distorting effect of a single outlier at (13, 12.74), which pulls the regression line away from the otherwise tight linear cluster of points. Removing this influential point substantially changes the regression line (slope decreases from 0.5 to ≈0.35), but the correlation coefficient r remains high (≈0.78) for the remaining points, demonstrating how outliers can alter model parameters while association stays strong.^[1] For Dataset IV, the scatter plot shows a leverage point at (19, 12.5) that disproportionately influences the regression line, while the remaining points cluster vertically at x=8 with y values varying around 7.5. This configuration reveals no meaningful linear trend in the bulk of the data, rendering the linear model inappropriate and emphasizing the risks of leverage in skewing fits.^[1] A central insight from these visualizations is that the coefficient of determination r² quantifies the proportion of variance explained by a linear model, but it does not assess the overall quality or nature of the relationship between variables, as evidenced by the quartet's identical r² values across disparate patterns.^[1]

Implications for Statistical Practice

Dangers of Relying on Summary Statistics

Anscombe's quartet demonstrates the profound risks associated with depending exclusively on summary statistics for data analysis, as the four datasets exhibit identical means, variances, linear correlation coefficients, and regression equations despite fundamentally divergent underlying structures revealed through visualization.^[1] This equivalence in numerical summaries can mask critical features of the data, fostering misleading interpretations that compromise the validity of statistical inferences. One primary danger is the concealment of outliers and leverage points, as seen in datasets III and IV, where a single anomalous observation exerts disproportionate influence on the regression line, potentially leading to overconfident predictions that fail to represent the majority of the data points.^[1] In dataset III, the outlier drives the apparent linear trend, while in dataset IV, vertical alignment of most points with an isolated leverage point creates a spurious correlation, both scenarios underscoring how summary statistics obscure these influential anomalies. Another risk involves overlooking nonlinearity, exemplified by dataset II's quadratic relationship, which summary metrics interpret as linear and thus yield biased inferences or necessitate unapplied data transformations for accurate modeling.^[1] Furthermore, the identical summary statistics promote a false sense of equivalence among the datasets, implying they could be interchangeably used in modeling without consequence, yet their scatter plots expose incompatible patterns that render such substitutions invalid.^[1] This illusion of interchangeability heightens the peril in practical applications, where unexamined statistics might propagate errors. In the 1970s computing era, the rise of automated statistical packages often treated analysis as a "black box" process, prioritizing output over inspection and thereby amplifying these pitfalls—a caution Anscombe's work explicitly addresses.^[1] For instance, in fields like medicine, reliance on such summaries without graphical checks could result in erroneous treatment or policy decisions, as demonstrated in biomedical research contexts where overlooked data structures lead to flawed conclusions about relationships between variables. Similar consequences arise in economics, where misinterpreted correlations might inform misguided fiscal policies based on hidden nonlinearities or outliers.^[1]

Role of Visualization in Data Analysis

Visualization serves as a fundamental tool in exploratory data analysis (EDA), facilitating the identification of data anomalies, the evaluation of key statistical assumptions such as linearity and normality, and the selection of appropriate models by revealing underlying patterns that numerical summaries alone cannot capture. In particular, graphical techniques enable analysts to visually inspect relationships between variables, detect outliers or influential points, and assess the suitability of transformations or alternative approaches before proceeding to formal modeling. Anscombe's quartet underscores a critical lesson for statistical workflows: plotting data, especially through scatterplots, is essential prior to fitting models, as it exposes structures—like non-linear trends or leverage points—that conventional tests such as t-tests or ANOVA may fail to detect despite identical summary statistics across the datasets. This example illustrates the quartet's deceptive uniformity in numerical properties, emphasizing visualization's ability to diagnose such discrepancies and prevent misguided inferences. In contemporary practice, visualization is routinely integrated with statistical software like R and Python, where built-in datasets such as Anscombe's quartet serve as teaching tools for implementing graphical checks. Anscombe advocated this approach emphatically, stating, "Before anything else is done, we should scatterplot the y values against the x values and see what sort of relation there is—if any," highlighting graphs as an indispensable preliminary step to any formal analysis. His work contributed to the evolution of EDA frameworks, as advanced by John W. Tukey, who formalized graphical exploration as a core statistical methodology. Today, these principles are embedded in data science curricula, where visualization is taught as a standard protocol to enhance interpretive reliability. A specific recommendation emerging from this perspective is the routine use of residual plots following regression analysis to validate model fits, identify violations of assumptions, and uncover patterns like heteroscedasticity or non-linearity that could compromise conclusions. By prioritizing such diagnostics, analysts can refine models iteratively, ensuring that statistical procedures align with the data's true characteristics.