Bivariate Data and Scatter Plots
Bivariate data consists of pairs of values measured on the same individuals, such as height and weight, and a scatter plot shows each pair as a point to reveal a possible relationship. In brief: the points can rise from left to right (positive association), fall (negative association) or show no pattern. For example, study hours and exam scores of students usually form an upward-sloping cloud of points.
Bivariate statistics asks whether two quantities are related, how strongly and in what way. The first step is always to draw a scatter plot, because it shows the direction, the form, the strength and any unusual points at a glance. This guide explains how to prepare the data, how to construct and read a scatter plot, how to describe an association and why association does not prove causation.
What is bivariate data?
Univariate data has one value per individual, such as a height. Bivariate data has two values per individual: a pair (x, y). Usually one variable is the explanatory (or independent) variable x, which may help to explain or predict the other, the response (or dependent) variable y. In a study of temperature and ice cream sales, temperature is x and sales are y.
Both variables can be numerical, such as age and income. For two categorical variables, such as gender and favorite sport, a two-way table is used instead of a scatter plot. This article concentrates on two numerical variables.
How do you construct a scatter plot step by step?
- Decide which variable is x (horizontal axis) and which is y (vertical axis).
- Choose scales that cover the smallest and largest values with equal steps. The axes need not start at 0 but should be labelled with names and units.
- Plot one point for each pair (x, y). Do not connect the points.
- Add a title. If points overlap, mark them differently or use a slightly larger dot.
How do you describe a scatter plot?
Four features should be mentioned.
- Direction: positive (y tends to increase with x), negative (y tends to decrease) or none.
- Form: linear (points near a straight line) or curved.
- Strength: strong if points are close to a clear pattern, weak if widely scattered.
- Unusual features: outliers, points that are far from the pattern, and clusters.
Worked example with a figure
Eight students recorded their weekly study hours (x) and their test scores (y):
| Student | Hours x | Score y |
|---|---|---|
| A | 1 | 52 |
| B | 2 | 58 |
| C | 3 | 61 |
| D | 4 | 68 |
| E | 5 | 70 |
| F | 6 | 77 |
| G | 7 | 79 |
| H | 8 | 85 |
The points rise from left to right, so the association is positive. The pattern is close to a straight line, so the form is linear, and the strength is strong. There are no outliers. Each extra hour of study is associated with an increase of about 4.7 points in the score: from 52 to 85 over 7 hours is 33/7 ≈ 4.7.
Types of association at a glance
| Pattern | Description | Example |
|---|---|---|
| Rising line | positive linear | height and arm length |
| Falling line | negative linear | age of a car and its value |
| Shapeless cloud | no association | shoe size and test score |
| U-shape or arc | nonlinear | height of a thrown ball and time |
| Rising line with one far point | linear with an outlier | a data entry error |
Does association mean causation?
No. Two variables can move together because one causes the other, because a third variable influences both or by coincidence. Ice cream sales and the number of people who swim in the sea are both higher in summer, but ice cream does not cause swimming. Both depend on temperature, which is a lurking variable. Only a carefully designed experiment, where the explanatory variable is assigned at random, can show cause and effect.
Even in the study example, it is plausible that studying improves scores, but students who are more motivated may both study more and score higher for other reasons.
Where do scatter plots come from?
The idea of plotting pairs of measurements goes back to the early nineteenth century. John Herschel used a graph of the positions of double stars in 1833, which is often cited as an early scatter plot. Francis Galton studied heights of parents and children in the 1880s, leading to the concepts of regression and correlation, developed further by Karl Pearson.
Where are they used today?
Scientists use scatter plots to look for relationships in experimental data, doctors to compare a dose with a response, and economists to relate income and spending. In data science, a scatter plot is the first look at any pair of variables. Software draws them in seconds, but the interpretation of direction, form and strength still depends on the analyst.
Which mistakes are common?
Joining the points with lines
A scatter plot shows individual observations. Connecting them suggests an order in time that may not exist.
Swapping the variables
Place the explanatory variable on the horizontal axis and the response on the vertical axis.
Distorting scales
Uneven or truncated axes can make a weak pattern look strong. State the scales clearly.
Claiming causation
Say “associated with”, not “causes”, unless the data come from a randomised experiment.