Reading a scatter plot

What you will be able to do

• draw a scatter plot, with the explanatory variable across and the response variable up;
• describe a pattern by its direction, its form and its strength;
• spot an outlier and the centre of the cloud, the point of the means.

Two numbers for each student

A teacher asks ten students in a made-up class two questions. How many hours a week do you practise typing? How many words a minute can you type? Here are the answers.

In First Steps you looked at one variable at a time: its average and its spread. Now each student gives two numbers. A table of pairs is hard to read. So we draw each student as one dot: across for the hours, up for the speed. This picture is a scatter plot.

What do you see? The dots climb from the lower left to the upper right. Students who practise more tend to type faster. That word "tend" matters. Students C and D break the order: C practises less than D but types faster. A pattern in data is a tendency, not a rule.

Which variable goes across?

Ask which variable might explain the other. Practice may help to explain speed. So the hours are the explanatory variable, written $x$ and drawn across. The speed is the response variable, written $y$ and drawn up. Other books say independent and dependent variable, or predictor and outcome. They mean the same.

Sometimes neither variable explains the other, such as the heights and arm spans of the same students. Then either can go across. Choosing $x$ and $y$ does not prove that one causes the other. We come back to that in the course on correlation and cause.

Four things to look for

Describe every scatter plot with four words.

1. Direction. The dots rise to the right (a positive association), fall to the right (a negative association), or do neither.
2. Form. The dots follow a straight line (linear), a curve, or fall into separate clusters.
3. Strength. The dots sit close to the pattern (strong) or spread loosely around it (weak).
4. Outliers. A dot lies far from the rest of the pattern.

Here is a second example. Eight made-up weather stations stand along one mountain road. The higher the station, the cooler its mean July temperature: $31.2\ ^\circ\text{C}$ at $200\ \text{m}$ above sea level, but $19.6\ ^\circ\text{C}$ at $2100\ \text{m}$. The dots fall to the right, they follow a straight line, and they sit very close to it. That is a strong, negative, linear association with no outliers.

The centre of the cloud

Find the mean of each variable. For the typing survey, the mean practice is $\bar{x} = 4.2$ hours and the mean speed is $\bar{y} = 32.6$ words per minute. The point $(\bar{x}, \bar{y}) = (4.2, 32.6)$ is the point of the means. It sits in the middle of the cloud, and the next lesson uses it to measure the pattern.

Your turn: new numbers

Try it in Python

The program draws the scatter plot of the typing survey and marks the point of the means with a red cross.

The usual mistakes

• Joining the dots. A scatter plot shows separate people or places. Do not join the dots with a line as in a line graph.
• Reading a tendency as a rule. "Students who practise more type faster" is true on average, not for every pair of students.
• Putting the response across. Put the variable that may explain the other on the $x$ axis.
• Forgetting the units. Label each axis with its quantity and its unit, such as hours a week.

Remember these

• A scatter plot shows one dot for each individual, with $x$ across and $y$ up.
• Describe it by direction, form, strength and outliers.
• The point of the means $(\bar{x}, \bar{y})$ sits in the middle of the cloud.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. A made-up study records the mass of each new-born calf and its mass a year later. Which is the explanatory variable?

2. In a scatter plot of the age of a used phone (years) against its resale price (rupees), the dots fall from the upper left to the lower right. Describe the direction.

3. Why is a line graph the wrong picture for the typing survey?

4. The points of a scatter plot lie close to a curve that rises and then falls. Which of the four words is not "linear" here?

5. Four made-up students have $(x, y)$ values $(2, 10)$, $(4, 14)$, $(6, 15)$ and $(8, 21)$. Find the point of the means.

References

• OpenStax, Introductory Statistics, Chapter 12 (Linear Regression and Correlation), the section on scatter plots.