Anscombe's Quartet

In 1973 the statistician Francis Anscombe published four tiny datasets — eleven (x, y) points each — built to make a single, stubborn point: summary statistics can be identical while the data they summarize is nothing alike. Run the standard battery of descriptive numbers over the four sets and they come out the same to two decimals — same mean of x, same mean of y, same variances, same correlation, same least-squares line. Plot them and the illusion shatters: one is a clean linear trend, one a smooth curve, one a perfect line dragged off course by a single outlier, and one a vertical stack of points fixed by a lone far-flung observation.

The quartet is the canonical argument for looking at data before trusting a model fit to it. A Probability Distribution is more than its first two moments, and a correlation coefficient is a single number squeezed out of a whole cloud of points — squeezing discards exactly the structure that tells you whether the number means anything.

The four datasets

The datasets live in data/anscombe-1.csv through data/anscombe-4.csv, each a column of x and a column of y. Sets I, II, and III share the same eleven x values; set IV holds x constant at 8 for ten points and places the eleventh far out at 19. By construction the differences are invisible to the usual statistics:

# evidence/anscombe-summary/{inputs: data/anscombe-{1,2,3,4}.csv (4th -> set4.csv), run}
# Per-dataset mean x, mean y, sample variance x & y, Pearson correlation,
# and the least-squares slope & intercept. The columns barely move across rows.
files = ["anscombe-1.csv", "anscombe-2.csv", "anscombe-3.csv", "set4.csv"]
for i, path in enumerate(files, start=1):
    mx, my, vx, vy, r, slope, intercept = stats(*load(path))
    print("%3d  %6.2f  %6.2f  %5.2f  %5.2f  %5.3f  %5.2f  %9.2f"
          % (i, mx, my, vx, vy, r, slope, intercept))

The faint wobble in the last column — the correlation reads 0.816 for the first three sets and 0.817 for the fourth — is real but immaterial: rounded to the two decimals a reader actually reports, every dataset carries the identical correlation.

The one number everyone quotes

The headline statistic of the quartet is its shared correlation. It is the same — to the precision anyone reports it — across all four datasets, which is exactly why it is such a treacherous summary on its own:

# evidence/anscombe-correlation/{inputs: data/anscombe-{1,2,3,4}.csv, run}
# Assert the four correlations agree to two decimals, then print the shared value.
vals = [corr(*load("anscombe-%d.csv" % i)) for i in (1, 2, 3, 4)]
rounded = {"%.2f" % v for v in vals}
assert len(rounded) == 1, "datasets disagree on correlation: %s" % rounded
print(rounded.pop())

A correlation near 0.82 sounds like a strong, trustworthy linear relationship — and for Set I it is. For Set II it hides a curve; for Set III it has been inflated by a single outlier; for Set IV it is essentially meaningless, manufactured by one point. The number is the same in every case.

Look at the data

Here is the whole argument in one picture. Each panel plots one dataset against the same least-squares line y = 3.00 + 0.50\,x that every one of them produces. The summary statistics are identical; the shapes could hardly be more different.

Why it matters

Anscombe's quartet is a compact rebuttal to "the data is summarized by its statistics." The same caution runs through the rest of this section. The Central Limit Theorem promises that averages converge to a bell curve, and the Law of Large Numbers promises that sample means approach the truth — but both speak about aggregates, and an aggregate can be blind to shape, skew, and outliers in exactly the way the quartet exploits. A least-squares line is itself a projection — the Linear Algebra of minimizing squared residuals — and like any projection it throws information away; the quartet is four different pre-images of the same low-dimensional shadow.

The modern descendant of Anscombe's idea is the "Datasaurus" dozen, where a cartoon dinosaur and a starburst and a bullseye all share the same means, variances, and correlation. The lesson is unchanged from 1973: compute the statistics, but plot the data.

See also

• Probability Distribution
• Central Limit Theorem
• Law of Large Numbers
• Linear Algebra
• Benford's Law