Linear and Nonlinear Associations: Predict and Describe | Grade 8 Math | LevelKite

You already know how to spot clusters and outliers in a scatter plot. Now you will study the overall pattern. Two variables have an association when their values tend to change together, so the points of their scatter plot form a pattern.

In this lesson you will answer three questions about an association. Does it follow a straight line or a curve? How can you use it to make a prediction? How do you describe it in words?

An association is linear when the points lie close to a straight line, though not necessarily on it. It is nonlinear when the points follow a curve instead.

A straight line changes $y$ by the same amount for every equal step in $x$ . That steady change is what makes it straight. So when the $x$ -values of your data go up in equal steps, you can test the form with numbers instead of judging by eye. Find the change in $y$ from each point to the next. Data scatter, so these changes are almost never exactly equal. Look at all of them together. Changes that stay near one value, going up and down with no steady trend, suggest a linear association. Changes that steadily grow or shrink suggest a nonlinear one.

Here is a first example. In a science lab, a class drops a ball from different heights and measures its first bounce. Let $x$ be the drop height and $y$ the bounce height, both in centimeters (cm). The drop heights go from 40 cm to 180 cm in steps of 20 cm. The measured points are $(40, 25)$ , $(60, 35)$ , $(80, 49)$ , $(100, 62)$ , $(120, 72)$ , $(140, 83)$ , $(160, 97)$ and $(180, 107)$ .

In the figure, the horizontal axis shows drop height and the vertical axis shows bounce height. The points rise from left to right and stay close to Line A, a straight line. Some points sit just above it and some just below it. That look suggests a linear association. You will check it with numbers soon.

Each point is one drop from the list, from

(40, 25)

to

(180, 107)

. Line A is a straight line that runs close to all eight points.

Here is a second example. A mug of hot cocoa cools on a table in a room at about 20 °C. Let $x$ be the time since the cocoa was poured, in minutes, from 0 to 60. Let $y$ be its temperature in degrees Celsius (°C). Measured every 10 minutes, the points are $(0, 80)$ , $(10, 62)$ , $(20, 50)$ , $(30, 41)$ , $(40, 35)$ , $(50, 31)$ and $(60, 28)$ .

In the figure, the horizontal axis shows time and the vertical axis shows temperature. The points fall quickly at first and then more and more slowly, so they bend along a curve. Line B, $y = 80 - 1.8x$ , passes through the first two points. It keeps dropping 18 °C every 10 minutes, as the cocoa did at first. The later points sit farther and farther above it. At 30 minutes, for example, Line B gives $80 - 1.8(30) = 80 - 54 = 26$ , but the cocoa measured 41 °C.

The seven cocoa points from the text. Line B passes through

(0, 80)

and

(10, 62)

. At 30 minutes it is at

(30, 26)

, far below the measured

(30, 41)

.

Worked example

Linear or nonlinear?

Use the ball and cocoa points listed above. Decide whether each association is linear or nonlinear by comparing the changes in $y$ .

For the ball, the drop heights go up in equal steps of 20 cm, so the changes in bounce height are fair to compare. Subtract each bounce height from the next one.
$35-25=10,\; 49-35=14,\; 62-49=13,\; 72-62=10,\; 83-72=11,\; 97-83=14,\; 107-97=10$
The changes stay between 10 cm and 14 cm. They go up and down with no steady trend. Points scattered close to a straight line give changes like these, so this is evidence of a linear association.
For the cocoa, the times go up in equal steps of 10 minutes. Do the same subtraction with the temperatures.
$62-80=-18,\; 50-62=-12,\; 41-50=-9,\; 35-41=-6,\; 31-35=-4,\; 28-31=-3$
The negative signs show that the temperature falls every time. Each drop is smaller than the one before, from 18 degrees down to 3, so the changes shrink steadily instead of staying near one value. The points bend along a curve, so this is evidence of a nonlinear association.

Answer

The ball data show a linear association, and the cocoa data show a nonlinear association.

A model line is a straight line chosen to stay close to the points of a linear association, so it can stand in for the overall pattern. Line A was chosen for you, and its equation is exactly $y = 0.6x$ . Its fit to the bounces is only approximate. It passes through just one data point, $(120, 72)$ . Four points lie above it and three lie below it, and none is more than 2 cm above or below it. Judge a model line by how close it stays to the points overall, not by whether it hits every point. Having points on both sides is a useful rough guide. A good line does not need exactly as many points above as below.

To make a prediction, substitute an $x$ -value into the model line's equation. The result is exact for the line. The data fit the line only approximately, so the result is only an estimate for the real quantity. Use a straight model line only for a linear association. Its predictions are most trustworthy for $x$ -values inside the range of the data, here 40 cm to 180 cm. Far outside that range, you would be assuming the pattern continues where no one measured.

Worked example

Predicting a bounce

Use Line A, $y = 0.6x$ , to predict the bounce height for a drop from $150$ cm.

Check that Line A suits this prediction. The ball data have a linear association, and 150 cm is inside the measured drop heights, so you are not stretching the pattern past the data.
$40 \le 150 \le 180$
Check the fit at (100, 62), the data point farthest from Line A. Substitute x = 100 into the equation and compare the line's value with the measured bounce. The gap is only 2 cm, which is small next to a bounce of about 60 cm, so Line A stays close to the data.
$0.6(100) = 60, \quad 62 - 60 = 2$
Substitute x = 150. This is valid because the equation gives the y-value of the point on Line A for every x.
$y = 0.6(150)$
Multiply. The decimal 0.6 means six tenths, so find six tenths of 150.
$0.6 \times 150 = \frac{6 \times 150}{10} = \frac{900}{10} = 90$
Interpret the result. The point (150, 90) is exactly on Line A. The measured bounces were up to 2 cm from the line, so treat 90 cm as an estimate. It also fits between the measured bounces of 83 cm for a 140 cm drop and 97 cm for a 160 cm drop.
$83 < 90 < 97$

Answer

Line A predicts that a ball dropped from $150$ cm will bounce about $90$ cm high.

Key concept

Describing and using an association

Describe an association by its direction and its form, in words that name both variables and their units. The direction is positive when $y$ tends to increase as $x$ increases, and negative when $y$ tends to decrease as $x$ increases. The form is linear or nonlinear. Mention any clusters or outliers too. The ball data show a positive linear association. The cocoa data show a negative nonlinear one.

A straight model line suits only a linear form. Its equation is exact, but its fit to the data is approximate, so its predictions are estimates. They are most trustworthy inside the range of the data. An association shows how two variables change together, not why. By itself, it does not show that one variable causes the other.

Describing an association carefully matters most when a conclusion is tempting. A town studied nine house fires. For each fire, let $x$ be the number of firefighters sent, from 5 to 40. Let $y$ be the damage in thousands of dollars, so $y = 15$ means 15,000 dollars. The points are $(5, 4)$ , $(10, 9)$ , $(15, 15)$ , $(20, 19)$ , $(25, 24)$ , $(30, 30)$ , $(35, 34)$ , $(40, 40)$ and $(30, 6)$ . In the figure, the horizontal axis shows firefighters sent and the vertical axis shows damage.

The nine fires from the text. The point labeled

(30, 6)

sits far below the other fire with 30 firefighters, at

(30, 30)

.

Worked example

Describing the fire data

Describe the association between the number of firefighters sent and the damage for these nine fires. Then decide whether the data show that sending more firefighters causes more damage.

Look first for a point that breaks the pattern. The fire at (30, 6) had 24 thousand dollars less damage than the other fire with 30 firefighters, and it sits far from the pattern of the other points. It is an outlier. Describe the main pattern from the other eight fires and mention the outlier separately.
$30 - 6 = 24$
Find the direction. For the other eight fires, the number of firefighters goes up in steps of 5, and the damage rises at every step. So the association is positive.
$4 < 9 < 15 < 19 < 24 < 30 < 34 < 40$
Find the form. The steps in x are equal, so compare the changes in damage. They stay between 4 and 6 with no steady trend, which is evidence of a linear association.
$9-4=5,\; 15-9=6,\; 19-15=4,\; 24-19=5,\; 30-24=6,\; 34-30=4,\; 40-34=6$
Put the description into words with both variables and their units, so it makes sense without the plot. Fires with more firefighters tended to have more damage, in thousands of dollars. The pattern is roughly a straight line, apart from one outlier.
Check for a cause claim. The plot shows only that the two variables tend to rise together, not why. A likely explanation is a third variable, the size of the fire. A bigger fire brings more firefighters and also causes more damage. So these data do not show that firefighters cause damage.

Answer

The data show a positive linear association between the number of firefighters sent and the damage, in thousands of dollars, with one outlier at $(30, 6)$ . This association alone does not show that sending more firefighters causes more damage.

Takeaway: The form of an association decides how you can use it. When the points stay close to a straight line, a model line with an exact equation gives approximate predictions. Those predictions are most trustworthy inside the range of the data. When the points curve, a straight line misleads. In every case, describe the direction and form in terms of the real variables. An association shows how two variables change together, not that one causes the other.