Points on a Field
A scatterplot shows two measurements per subject at once, exposing the direction, form, and strength of any relationship between the variables. · 12 min
Until now every number has described one variable at a time — one column of data. But the interesting questions usually involve two. Does more studying go with higher scores? Does more screen time go with less sleep? To see a relationship you need both measurements for the same person, plotted together. The picture that does this is the scatterplot.
Guess before you learn
On a hot day, do you expect the points for temperature and hot-chocolate sales to slope upward or downward as temperature rises?
Warmer days, fewer hot chocolates: the cloud drifts downward. That downward drift is what a scatterplot calls a negative direction — one of three things you will learn to read off any scatter.
Every point on a scatterplot is one subject measured twice: its horizontal position is one variable, its vertical position the other. By convention the explanatory variable — the one you think does the influencing — goes across the bottom, and the response goes up the side. Once the cloud is drawn, you read it with three words: direction, form, and strength.
9–12
3–5
Each dot stands for one person and holds two facts at once — say, hours of sleep and how awake they felt. When the dots drift upward together, the two facts tend to rise together. When they drift down, one rises as the other falls.
6–8
A scatterplot plots two measurements per subject: one across, one up. You read it in three words. Direction: do the points rise or fall as you move right? Form: a line, a curve, or no pattern? Strength: how tightly do the points cluster around that pattern? A point far from all the rest is an outlier.
9–12
A scatterplot displays paired data — two quantitative variables measured on the same individuals — as points on a plane. Describe it in four parts: direction (positive if y tends to rise with x, negative if it falls), form (linear, curved, or none), strength (how little scatter there is about the overall pattern), and any outliers. The explanatory variable is conventionally horizontal and the response vertical, though the plot itself does not decide which variable causes the other.
K–2
Line up your friends. For each one, mark two things: how tall they are and how far they can jump. Put a dot for each friend. The cloud of dots shows if taller friends jump farther.
Undergrad
The scatterplot is the primary diagnostic for bivariate quantitative data, and you inspect it before computing any summary. It reveals structure that single numbers hide: nonlinearity, distinct clusters, heteroscedasticity (scatter that widens along x), and influential outliers. Fitting a correlation or a line to a cloud you have not looked at is how people summarize a curve with a straight number and never notice the mistake.
Postgrad
Regard the scatter as a sample from a joint distribution on the plane. Its cloud is an empirical estimate of that bivariate density; direction, form, and strength are informal readings of the conditional mean E[Y|X], its functional shape, and the conditional variance about it. Anscombe's quartet is the standing warning: four data sets with identical means, variances, correlation, and regression line, yet four qualitatively different scatterplots. Always plot before you summarize.
scatterplot
A plot of two quantitative variables, one point per subject, with one variable on each axis.
explanatory variable
The variable you treat as the influence, drawn on the horizontal axis; the response goes on the vertical axis.
Direction and form are quick to name. Strength takes a closer look: it asks how much the points scatter around the overall pattern. Two clouds can both slope upward, yet one hugs a line while the other barely leans. Strength is what separates a reliable relationship from a vague tendency — and the next folio gives it a single number.
A scatterplot turns two columns of numbers into a shape you can judge at a glance. You can now name a relationship's direction, its form, and its strength. The next folio replaces the words strong and weak with a single measured number between −1 and 1.
Practice — new ink and old, interleaved
1.A histogram of house prices has a long tail stretching to the high end. What is its shape?
2.A survey answer runs from strongly disagree to strongly agree — an ordinal variable. Which measure of center is defensible?
3.A histogram is clearly left-skewed. Where is the mean relative to the median?
4.Order these three scatters from weakest to strongest relationship.
- a wide, loose band
- a moderate oval cloud
- points nearly on a line
5.Without looking back: what does each point on a scatterplot represent?
One subject measured on two variables — its position across is one variable, its position up is the other.
How close were you? Grade yourself honestly — it sets your review date.
6.For the sorted data 3, 5, 6, 8, 10, 11, 14, compute the IQR (Q3 − Q1).
7.A scatterplot needs two quantitative variables. Which of these is quantitative?
8.You roll a fair die 6000 times and make a histogram of the results 1 to 6. What shape do you expect?
9.As the temperature climbs, winter-coat sales fall. What is the direction of that scatter?