Eleven pairs. r = 0.816, p = 0.0022, a 95% interval from 0.424 to 0.951. A large correlation by any benchmark, significant with room to spare. On paper the report is one sentence long. Then the second data set gives r = 0.816. So does the third. The fourth gives 0.817.
These four sets are Anscombe's quartet, published by the statistician Francis Anscombe to make one point: the same summary numbers can sit on top of completely different data. Each set has the same mean of X (9), almost the same mean of Y (7.50), the same Pearson correlation and the same fitted line, about y = 3.00 + 0.500x. Plotted, they look nothing alike. One is a noisy line, one a smooth curve, one a perfect line with a single stray point, and one a vertical column of ten points plus one far away.
This article treats the quartet as a small project. The brief: four columns of paired data arrive, and before anyone reports a correlation, find out which of the four r actually describes. The tools are the statistics calculators added this week: the Pearson correlation calculator, the skewness calculator and the kurtosis calculator. No plotting software, on purpose. The question is how far the numbers alone get you.
The brief and the data
All four sets have eleven pairs. Sets I, II and III share the same X column, and set IV has its own.
X for sets I to III: 10, 8, 13, 9, 11, 14, 6, 4, 12, 7, 5. X for set IV: 8, 8, 8, 8, 8, 8, 8, 19, 8, 8, 8.
Y for set I: 8.04, 6.95, 7.58, 8.81, 8.33, 9.96, 7.24, 4.26, 10.84, 4.82, 5.68. Set II: 9.14, 8.14, 8.74, 8.77, 9.26, 8.10, 6.13, 3.10, 9.13, 7.26, 4.74. Set III: 7.46, 6.77, 12.74, 7.11, 7.81, 8.84, 6.08, 5.39, 8.15, 6.42, 5.73. Set IV: 6.58, 5.76, 7.71, 8.84, 8.47, 7.04, 5.25, 12.50, 5.56, 7.91, 6.89.
The budget for the check was one pass through each calculator per column, about a minute of pasting per set. The target was simple: a yes or no for each set on the question "can r = 0.816 be reported as a description of this relationship?"
Step by step
Step 1: correlation for all four sets
Pasting each pair of columns into the Pearson calculator, with a two-sided test at alpha 0.05, gives the table below. The p-value, the t statistic and the interval are practically identical. The only column that moves is Spearman's rank correlation, printed next to Pearson's r.
| Set | Pearson r | p, two-sided | 95% interval | Spearman | Warning in the result |
|---|---|---|---|---|---|
| I | 0.816 | 0.0022 | 0.424 to 0.951 | 0.818 | none |
| II | 0.816 | 0.0022 | 0.424 to 0.951 | 0.691 | none |
| III | 0.816 | 0.0022 | 0.424 to 0.951 | 0.991 | none |
| IV | 0.817 | 0.0022 | 0.425 to 0.951 | 0.500 | Pearson and Spearman disagree by 0.32 |
One set out of four raised a flag. For set IV, Spearman is 0.500 against Pearson's 0.817, a gap of 0.32, and the calculator says so. For set III the result stays quiet. That deserves a closer look, because set III is the one with the obvious stray point.
Why no warning? The calculator recomputes r without each pair in turn and speaks up when a single pair moves r by 0.2 or more, or when dropping it flips the significance verdict. In set III the stray pair is (13, 12.74). Without it, the other ten points lie on a perfect line and r becomes 1.000. That is a change of 0.184, just under the threshold, and the verdict stays "significant" either way. Spearman, at 0.991, is 0.175 above Pearson, again under the 0.2 gap that triggers a note. The rule did what it was set up to do. It was simply not set up for an outlier that makes a strong relationship look weaker, rather than one that manufactures a relationship out of nothing.
Step 2: skewness of each Y column
Correlation looks at the pairs together. Skewness looks at one column at a time and asks whether one side has a longer tail. Each Y column went into the skewness calculator as a sample, alpha 0.05.
Set III lit up at once: G1 = 1.855, highly skewed to the right, D'Agostino's test p = 0.0078. The result named the value responsible, 12.74, and said that without it the skewness would be 0.202. Set IV looked similar: G1 = 1.507, p = 0.0262, with 12.5 named and 0.011 without it. Set II came out skewed the other way, G1 = -1.316, with p = 0.0489, barely under the threshold and with no single value to blame. Set I sat at -0.065, p = 0.9182, symmetric by any standard.
Step 3: kurtosis of the same columns
Kurtosis measures tail weight, how much of the variance comes from a few values far from the mean. With eleven values the calculator reports it without a test, since the tests need at least twenty, but the number and the note about the farthest value are still there.
| Set | Skewness G1 | D'Agostino p | Excess kurtosis G2 | Single value named |
|---|---|---|---|---|
| I | -0.065 | 0.9182 | -0.535 | no single value flagged |
| II | -1.316 | 0.0489 | 0.846 | no single value flagged |
| III | 1.855 | 0.0078 | 4.384 | 12.74: skewness 0.202, kurtosis -0.849 without it |
| IV | 1.507 | 0.0262 | 3.151 | 12.5: skewness 0.011, kurtosis -1.237 without it |
Sets III and IV look alike here. Both Y columns carry heavy tails, 4.384 and 3.151, the neighborhood of a Laplace distribution, and both collapse to light tails once one value leaves: -0.849 and -1.237. Set I is light-tailed at -0.535. Set II, at 0.846, is unremarkable on its own.
Step 4: the X columns
Checking Y alone would have missed half of set IV's story. Its X column is ten copies of 8 and a single 19. The skewness calculator gives 3.317 with p below 0.0001, and the kurtosis calculator gives 11.000, heavier than an exponential distribution. The shared X column of sets I to III is an evenly spaced run from 4 to 14: skewness 0.000 and excess kurtosis -1.200, the exact value of a flat, uniform spread.
Step 5: removing the named points
The last step was to take out the point each check had named and run the correlation again. Set III without the pair (13, 12.74) gives r = 1.000 and a t statistic of 1077.352: the remaining ten points are a line, and the stray pair was hiding it. Set IV without the pair (19, 12.50) gives no correlation at all. The Pearson calculator refuses, because every remaining X is 8, the standard deviation of X is zero, and r is undefined. The entire 0.817 came from one point.
The ledger
Here is the plan against what the checks found, set by set.
| Set | What r promised | What the checks found | Difference |
|---|---|---|---|
| I | r = 0.816, a clean linear trend | both columns close to symmetric, no flag anywhere | none: r can be reported |
| II | r = 0.816, a clean linear trend | Y skewed left (p = 0.0489); the plot shows a curve | r describes the wrong shape |
| III | r = 0.816, a clean linear trend | one Y value carries the skewness and kurtosis; without its pair r = 1.000 | r understates a perfect line |
| IV | r = 0.817, a clean linear trend | X skewness 3.317, kurtosis 11.000; without one pair r is undefined | r exists only because of one point |
| Total | four reportable correlations | one reportable correlation | three of four need a different summary |
One set in four. That is the honest score for r = 0.816 across the quartet, and the one set where it holds is also the one where every check came back clean.
Four lessons
Lesson 1: significance says nothing about shape
The p-value of 0.0022 was identical in all four sets, including the one where the correlation rests on a single point. The second mode of the Pearson calculator, which interprets an r you already have, makes the reason plain: typed in as r = 0.816 with 11 pairs, it returns the same p and reports that 6 pairs would already be enough for significance. Significance is a function of r and n alone. The data behind them never enter the calculation.
Lesson 2: automatic warnings have thresholds, and thresholds have blind spots
The Pearson calculator's own checks caught set IV and missed set III. The skewness and kurtosis of the columns caught both. Neither method alone is complete, and set II, the curve, slipped past almost everything: its skewness test landed at 0.0489, one step from "not significant", and no single value was named. A curve is a pattern in the pairs, not in either column, so column checks see it only faintly.
Lesson 3: check X as well as Y
Set IV's Y column looked like set III's. Only its X column, with a skewness of 3.317 and a kurtosis of 11.000, showed that the design itself was broken: ten measurements at one setting and one at another. No correlation can describe a relationship that was sampled at two points.
Lesson 4: the named value is the start of the work, not the end
When a result says "without 12.74 the skewness would be 0.202", the next step is to find out what 12.74 is. It might be a typing error, a different kind of observation, or the most important point in the study. The numbers only say where to look.
Starting over, in the right order
Run this project again and the order changes. First, skewness and kurtosis for every column, X and Y, because they take seconds and they name the values that matter. Second, the correlation, with an eye on the Spearman figure and on any note about a single pair. Third, when r and n are all a report gives you, the second mode of the Pearson calculator, which puts the number in context: a correlation of 0.8 from 30 pairs has a 95% interval from 0.618 to 0.901, a far tighter statement than 0.816 from eleven. And last, the step Anscombe's quartet was built to force: plot the points before the sentence goes into the report.
For set I, that sentence can say r = 0.816. For the other three, it should say something else, and now there is evidence for what.
Tools discussed in this article
- Pearson correlation calculator: r with its t statistic, p-value and Fisher interval, Spearman beside it, the most influential pair, and a second mode that interprets an r you already have from r and n alone.
- Skewness calculator: G1 as Excel SKEW, g1 as SKEW.P and b1, D'Agostino's test from eight values, and how much one extreme value drives the result.
- Kurtosis calculator: excess kurtosis as Excel KURT, g2 and Pearson's b2, the Anscombe-Glynn and D'Agostino-Pearson tests from twenty values, and the nearest benchmark distribution.
More statistics tools
Interquartile range (outlier fences for a single column) · Coefficient of variation (spread as a share of the mean) · Percentile (where one value sits in a list) · Mode (the most frequent value) · Wilcoxon test (rank tests when the tails are heavy) · Fisher's exact test (small 2x2 tables of counts) · Standard deviation (the spread each column is scaled by) · Median (the middle value, untouched by one extreme reading) · Average (arithmetic, weighted and geometric means)