Anscombe's Quartet

The cover image about Anscombe’s Quartet was generated by ChatGPT, and it used the following prompt: “The digital design highlights the title ‘Anscombe’s Quartet’ in bold, white sans-serif letters, centered against a dynamic abstract backdrop. The image is split into four colorful quadrants, each showcasing unique textures and patterns—ranging from painterly hues and curved lines to scattered circles and dots.”

Introduction

While listening to a presentation today, I happened to hear a term——Anscombe’s Quartet, a term I had never encountered before, yet it has significant implications in both statistics and data visualization.

History

Anscombe’s quartet consists of four datasets constructed by British statistician Francis Anscombe in 1973. These four datasets share nearly identical statistical properties but differ dramatically when visualized graphically.

Anscombe’s Quartet

Here are the four sets of data, each containing 11 pairs of $x$ and $y$ values:

We’ll now analyze this dataset using the R programming language.

Data Input

We enter the data as matrices in R for ease of manipulation:

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# dataset 1
x1 <- c(10.0, 8.0, 13.0, 9.0, 11.0, 14.0, 6.0, 4.0, 12.0, 7.0, 5.0)
y1 <- c(8.04, 6.95, 7.58, 8.81, 8.33, 9.96, 7.24, 4.26, 10.84, 4.82, 5.68)

# dataset 2
x2 <- c(10.0, 8.0, 13.0, 9.0, 11.0, 14.0, 6.0, 4.0, 12.0, 7.0, 5.0)
y2 <- c(9.14, 8.14, 8.74, 8.77, 9.26, 8.10, 6.13, 3.10, 9.13, 7.26, 4.74)

# dataset 3
x3 <- c(10.0, 8.0, 13.0, 9.0, 11.0, 14.0, 6.0, 4.0, 12.0, 7.0, 5.0)
y3 <- c(7.46, 6.77, 12.74, 7.11, 7.81, 8.84, 6.08, 5.39, 8.15, 6.42, 5.73)

# dataset 4
x4 <- c(8.0, 8.0, 8.0, 8.0, 8.0, 8.0, 8.0, 19.0, 8.0, 8.0, 8.0)
y4 <- c(6.58, 5.76, 7.71, 8.84, 8.47, 7.04, 5.25, 12.50, 5.56, 7.91, 6.89)

# collactions
x = cbind(x1, x2, x3, x4)
y = cbind(y1, y2, y3, y4)

# print
cat('x:\n')
print(x)

cat('y:\n')
print(y)

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x:
      x1 x2 x3 x4
 [1,] 10 10 10  8
 [2,]  8  8  8  8
 [3,] 13 13 13  8
 [4,]  9  9  9  8
 [5,] 11 11 11  8
 [6,] 14 14 14  8
 [7,]  6  6  6  8
 [8,]  4  4  4 19
 [9,] 12 12 12  8
[10,]  7  7  7  8
[11,]  5  5  5  8
y:
         y1   y2    y3    y4
 [1,]  8.04 9.14  7.46  6.58
 [2,]  6.95 8.14  6.77  5.76
 [3,]  7.58 8.74 12.74  7.71
 [4,]  8.81 8.77  7.11  8.84
 [5,]  8.33 9.26  7.81  8.47
 [6,]  9.96 8.10  8.84  7.04
 [7,]  7.24 6.13  6.08  5.25
 [8,]  4.26 3.10  5.39 12.50
 [9,] 10.84 9.13  8.15  5.56
[10,]  4.82 7.26  6.42  7.91
[11,]  5.68 4.74  5.73  6.89

Mean, Variance, and Correlation Coefficient

We examine the mean, variance, and correlation coefficients of the data.

Interestingly, the means and variances are nearly identical up to two decimal places. The correlation coefficients for each dataset pair, such as $(x_1, y_1)$, are also nearly identical.

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# means
cat('means:\n')
colMeans(x)
colMeans(y)

# variances
cat('variances:\n')
apply(x, 2, var)
apply(y, 2, var)

# correlation coefficients
cat('correlation coefficients:\n')
cor(x, y)

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means:
x1 x2 x3 x4 
 9  9  9  9 
      y1       y2       y3       y4 
7.500909 7.500909 7.500000 7.500909 
variances:
x1 x2 x3 x4 
11 11 11 11 
      y1       y2       y3       y4 
4.127269 4.127629 4.122620 4.123249 
correlation coefficients:
           y1         y2         y3         y4
x1  0.8164205  0.8162365  0.8162867 -0.3140467
x2  0.8164205  0.8162365  0.8162867 -0.3140467
x3  0.8164205  0.8162365  0.8162867 -0.3140467
x4 -0.5290927 -0.7184365 -0.3446610  0.8165214

Linear Regression

We apply linear regression to each dataset. All four datasets yield similar regression equations:

$$ y = 3 + 0.5 x. $$

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lm1 <- lm(y1 ~ x1)
lm2 <- lm(y2 ~ x2)
lm3 <- lm(y3 ~ x3)
lm4 <- lm(y4 ~ x4)

coefficients_matrix <- rbind(
  coef(lm1),
  coef(lm2),
  coef(lm3),
  coef(lm4)
)

rownames(coefficients_matrix) <- c("lm1", "lm2", "lm3", "lm4")
colnames(coefficients_matrix) <- c("Intercept", "Slope")

coefficients_matrix

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    Intercept     Slope
lm1  3.000091 0.5000909
lm2  3.000909 0.5000000
lm3  3.002455 0.4997273
lm4  3.001727 0.4999091

This is quite astonishing! Without further checks, one might assume the datasets are essentially the same, but they are not.

Scatter Plots

Here are the scatter plots for the four datasets. They tell a different story.

Dataset 1 resembles a linear relationship; Dataset 2 shows a clear non-linear trend; Dataset 3 includes an outlier that affects the regression; Dataset 4 consists mostly of constant $x$ values with a single outlier.

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par(mfrow = c(2, 2))

plot(x1, y1, main = "Dataset 1", pch = 19)
abline(lm1, col = "red")

plot(x2, y2, main = "Dataset 2", pch = 19)
abline(lm2, col = "red")

plot(x3, y3, main = "Dataset 3", pch = 19)
abline(lm3, col = "red")

plot(x4, y4, main = "Dataset 4", pch = 19)
abline(lm4, col = "red")

Boxplots

Boxplots show that the quartile distributions of each dataset vary as well.

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boxplot(cbind(x, y),
        main = "Boxplots of Anscombe's Quartet",
        horizontal = TRUE
)

Conclusion

Anscombe’s Quartet teaches us a critical lesson: we should not rely solely on statistical summaries. Instead, we must leverage tools like data visualization and diverse analytical perspectives to fully understand our data. Statistical figures are essential, but without proper context and visualization, they can be misleading.

Further Learning

• The R Notebook HTML file used in this article.

References

• Anscombe’s Quartet. (September 23, 2021). Wikipedia, The Free Encyclopedia. Retrieved June 3, 2025 from https://zh.wikipedia.org/zh-tw/安斯库姆四重奏