Why you must plot the data

Summary statistics can agree and still lie

Imagine two analysts each hand you a one-line description of a dataset. Both say: “Average around 7.5, spread about 2, and a strong upward relationship.” You would naturally assume the two datasets look alike. That assumption is the trap this whole guide is built to disarm. Two collections of numbers can share every headline statistic and yet describe completely different worlds — a calm trend, a sharp curve, or a single rogue point doing all the work.

First, the plain words. A summary statistic is a single number that stands in for a whole column of data — the average (the mean), the middle value (the median), or a measure of how spread out the values are (the standard deviation). The correlation coefficient is one more: a number from −1 to +1 that says how tightly two variables move together. These numbers are genuinely useful. The problem is what they are by design: a compression. They take dozens, thousands, or millions of values and squeeze them into a handful.

Here is the smallest example that bites. Two neighborhoods both report an average household income of $60,000. In the first, almost everyone earns close to $60,000. In the second, half the households earn $20,000 and half earn $100,000 — and not a single family actually lives near the “average.” Same mean, opposite realities. The mean alone hid the spread. Your instinct might be: fine, just report the spread too. But Anscombe's famous example, next, shows that even the mean and the standard deviation and the correlation, all matching, are not enough.

The mean in symbols: add up all n values and divide by n. It is one number standing in for the whole column — useful, but blind to shape.

Anscombe's quartet

In 1973 the statistician Francis Anscombe published four tiny datasets, now called Anscombe's quartet. Each has just eleven points, each point a pair of numbers (x, y). He built them so that, to many decimal places, they share almost every summary statistic a textbook would compute. And yet, when you plot them, they could not look more different.

Here is what all four share. The average x is 9.0; the average y is 7.5. The spread of x and of y match. The correlation between x and y is about 0.816 in every set — “strong and positive.” Fit a straight line through each (the line of best fit, which the next guides in this track explain) and you get the same line every time: y ≈ 3.0 + 0.5·x, explaining about 67% of the variation. By the numbers, the four datasets are identical twins.

The one regression line and one correlation that all four datasets produce — introduced here in words above, shown as symbols so you can see how completely they agree.

Now look at what the pictures reveal. Dataset I is the honest case: a loose, roughly linear cloud where the line genuinely describes the trend. Dataset II is a smooth curve — the relationship is real but bent, so a straight line is simply the wrong model. Dataset III is almost perfectly linear except for one outlier sitting far above the rest; that single point drags the line up and weakens the correlation. Dataset IV is the most unsettling: every point shares the same x-value except one lone point far to the right — and that single point alone determines the slope. Remove it and there is no relationship at all.

What your eyes catch that numbers miss

Why is the human eye so good here? Because a scatter plot keeps every data point on the page, while a summary throws almost all of them away. Your visual system is a pattern-detector refined over millions of years: it spots curves, clusters, gaps, and lonely points in a fraction of a second, long before you could compute anything. A plot lets you ask questions you did not know you had.

Here is a concrete list of things a picture surfaces immediately and a summary will quietly hide.

1. Non-linearity: the relationship bends (like Anscombe II). A straight-line model would be confidently wrong.
2. Outliers and influential points: one or two extreme values that drag an average or tilt a line (Anscombe III and IV).
3. Clusters and subgroups: two or three separate clouds that an overall average describes for nobody.
4. Skew and gaps: a long tail on one side, or an empty band where you expected values — signs the mean is the wrong summary.
5. Changing spread: points fan out as x grows (the noise is not constant), which quietly breaks many models' assumptions.
6. Plain errors: impossible values, ages of 999, prices of 0, a wall of duplicates — bugs that no average will ever flag.

The most dangerous of these is the influential point. In Anscombe IV, deleting one observation erases the entire finding. In real work this happens constantly: a single mistyped value, one corporate mega-customer in a table of individuals, one day of broken logging. A model trained on that data inherits the lie. The widget below lets you feel it: drag a point and watch the best-fit line swing toward it. One stubborn outlier can rewrite the “relationship.”

None of these patterns — the curve, the cluster, the influential point, the bug — show up in the mean, the standard deviation, or the correlation. They only show up in the picture. That is the whole argument of this guide in one sentence: the numbers tell you a value; the picture tells you whether to trust the number.

Plotting as the first step of analysis

Because of all this, experienced analysts treat plotting not as the pretty final step but as the very first thing they do — before any average, any test, any model. This habit has a name: exploratory data analysis (EDA), a practice championed by the statistician John Tukey, who argued that the first job of analysis is simply to look. You explore the data with quick, ugly, throwaway plots whose only purpose is to let you see what you are dealing with.

The order of operations matters. The wrong order is: load the data, compute the averages, run the model, report the result, and then maybe make a polished chart for the slide deck. The right order puts the eyes first: load, plot the raw data, then summarize and model what the plot showed you was sensible. If the scatter is a curve, you now know not to fit a straight line. If there is a glaring outlier, you investigate it before it poisons everything downstream.

import pandas as pd

df = pd.read_csv("sales.csv")

df.describe()                      # the summary numbers -- necessary, not sufficient
df.hist(bins=30)                    # ALWAYS look at each column's shape first
df.plot.scatter(x="ad_spend", y="revenue")   # and look at the relationship

Be honest about the failure mode this prevents. A model fit to unplotted data can be confidently, precisely, beautifully wrong. It will still report a slope, a correlation, an R-squared to three decimals — Anscombe proved all of those can look perfect on data that the model completely misreads. The numbers never warn you. Only the plot does. Skipping the plot is not a small shortcut; it is choosing to fly blind.

Distributions: always look before you summarize

Anscombe was about the relationship between two variables. But the same warning applies to a single variable on its own. Before you ever report the average of a column, look at its distribution — the full picture of which values appear and how often. The shape of that picture decides whether the average even means anything.

The workhorse tool is the histogram: chop the range of a numeric variable into equal-width bins and draw a bar for how many values land in each bin. A histogram shows the shape at a glance — is it a single symmetric hump, a long tail stretching to the right, or two separate peaks? A close cousin is the boxplot, which draws a box from the 25th to the 75th percentile (the middle half of the data), a line at the median, and whiskers out to the typical range, marking far-out points as candidate outliers. The boxplot is compact and great for comparing many groups side by side.

Shape changes which summary you should even use. Salaries, response times, and house prices are usually right-skewed: most values are modest but a few are enormous, so the mean gets dragged upward and describes almost nobody — the median is the fairer summary. A bimodal distribution, with two peaks, is worse: the average lands in the empty valley between them, describing a person who does not exist (imagine averaging the heights of a kindergarten class and a basketball team). And a distribution with outliers needs you to ask whether each extreme value is a real, important event or a data-entry error — a question the summary alone can never raise.

Building the plot-first habit

Knowing you should plot is not the same as doing it under deadline pressure. The fix is to turn plotting into a reflex — a short routine you run on every new dataset before you let yourself draw any conclusion. It does not have to be pretty. These are diagnostic plots for an audience of one: you.

1. Plot every numeric column as a histogram. You are checking shape, range, and obviously impossible values.
2. Plot every pair you care about as a scatter plot. You are checking for curves, clusters, and influential points before trusting any correlation.
3. Break plots down by an important group (region, device, cohort) using small multiples — a grid of the same chart, one panel per group — so a subgroup pattern can't hide inside the overall average.
4. Plot the data again AFTER cleaning, not just before, to confirm your fixes did what you intended and introduced nothing new.
5. Only now compute the summaries, run the test, or fit the model — on the parts of the data the pictures showed were sound.

A word on small multiples, since they are the habit's secret weapon. When you suspect different groups behave differently, do not overlay everything on one busy chart. Draw the same simple chart once per group and lay the panels in a grid. Patterns that an overall number averages into mush — one region falling while three rise, one device with a broken funnel — leap out the instant they sit side by side. This single technique is the cure for the most common real-world version of Anscombe: a healthy-looking aggregate hiding a sick subgroup.

What good looks like

Pull it together. A summary statistic is a compression, and compressions lose information; many different datasets can wear the same numbers. Anscombe's quartet proves it with four datasets that share their mean, spread, correlation, and best-fit line yet look nothing alike. Your eyes catch what those numbers miss — curves, clusters, outliers, influential points, and outright bugs. So the disciplined move is to plot first: explore the raw data and its distributions by eye, then summarize and model only what the pictures showed was sound.

Concretely, good data visualization practice in this first sense looks like this: a notebook whose top is a wall of quick histograms and scatter plots; a habit of breaking charts down by group with small multiples; a second look after cleaning; and a refusal to report any average, correlation, or model until you have seen the shape it came from. The reward is not just fewer mistakes — it is the discoveries that only ever come from looking: the unexpected cluster, the seasonal dip, the segment behaving backwards.

If you remember one sentence from this guide, make it this: never trust a number you have not seen the picture of. Compute the summary, by all means — but plot the data first, every single time. It is the cheapest, most reliable way to avoid being confidently, beautifully wrong.