Scatter Plots

Two measurements of each person

Five students are learning the piano. Each one records how many hours they practiced in a week, and at the end of the week their teacher gives their playing a score out of 10. So every student has two measurements: hours practiced and score.

A scatter plot shows the two measurements together. It has one axis for each measurement, and each student is drawn as one dot, placed across by their hours and up by their score. The dot for a student is the point (hours, score).

Each row of the table is one student and becomes one dot. Ben practiced for 3 hours and scored 4.

Plotting the points

Take one row at a time. Ben practiced for 3 hours and scored 4, so go 3 across the horizontal axis and 4 up, and mark a dot at (3, 4). Do the same for every row: Ana at (1, 2), Cal at (5, 5), Dee at (7, 8) and Eli at (9, 9). Five students give five dots.

Put the measurement that might explain the other on the horizontal axis. Practice may help the score, so hours go across and score goes up the side. Label both axes with what they measure, because a dot on its own is only two numbers.

Do not join the dots. Each dot is a different student, so a line from one to the next would connect two people, not show one thing changing. A scatter plot is not a line graph.

Ben’s dot is 3 squares across and 4 squares up. The other four students are plotted the same way.

The pattern in the dots

Now look at all the dots together. They drift upward from left to right: the students who practiced more tended to score higher. When one measurement tends to increase as the other increases, the two have a positive association. The pattern is a tendency, not a rule for each student: it says what happens in general.

The drift is what the scatter plot shows that the table does not show at a glance. A dashed line along the drift makes it easy to see, but the pattern belongs to the dots, not to the line.

The dots drift upward together, from Ana at the lower left to Eli at the upper right: a positive association.

Falling together, or no pattern at all

A scatter plot can show two other patterns. When one measurement tends to decrease as the other increases, the dots drift downward from left to right, and the association is negative. The age of a used car and its price are like this: older cars tend to cost less.

When the dots are scattered with no drift either way, there is no association: knowing one measurement tells you nothing about the other. A person’s height and the day of the month they were born on are like this. The next lesson, Correlation, gives these three patterns their names and describes how strong they are.

Eight used cars: the older a car, the lower its price tends to be. The dots drift downward, a negative association.

Nine people: the day of the month they were born on says nothing about their height. The dots show no drift, so there is no association.

An outlier on a scatter plot

A sixth student, Fay, practiced for 8 hours but scored only 2. Her dot, at (8, 2), sits far below the drift of the other five. A point that lies far from the pattern of the rest is an outlier.

Neither of Fay’s measurements is unusual on its own: Eli practiced for longer, and Ana also scored 2. It is the pair that does not fit. So an outlier on a scatter plot is found by looking at the dots, not by checking each measurement separately.

An outlier deserves a second look. Perhaps Fay was ill on the day she played, or the score was written down wrongly. If there is a reason like that, the point can be left out when describing the pattern. If there is no reason, it is real data and stays in, and the description should mention it.

Fay’s dot, at 8 hours and a score of 2, lies far below the upward drift of the other five: it is an outlier.

The mean point and a line of best fit

The application below also draws a line of best fit, which is the subject of a later lesson, Line of Best Fit. A line of best fit is one straight line drawn through the middle of the dots, following their drift, with about as many dots above it as below it.

It is drawn through the mean point, whose coordinates are the mean of the x-values and the mean of the y-values. For the five piano students, the mean of the hours is (1 + 3 + 5 + 7 + 9) ÷ 5 = 25 ÷ 5 = 5 and the mean of the scores is (2 + 4 + 5 + 8 + 9) ÷ 5 = 28 ÷ 5 = 5.6, so the mean point is (5, 5.6). A line of best fit through it is y = 0.9x + 1.1, since 0.9 × 5 + 1.1 = 5.6.

An estimate is read from the line, not from the nearest dot, because the line uses every student. A student who practiced for 6 hours would be expected to score about 0.9 × 6 + 1.1 = 6.5.

The line y = 0.9x + 1.1 passes through the mean point, (5, 5.6). Two dots lie above it, two below it, and Ana’s dot lies on it.