Free SAT Math Problem-Solving and Data Analysis

SAT Math · Problem-Solving and Data Analysis · free lesson

Scatterplots and Line of Best Fit

The idea, then worked examples solved step by step, then practice questions whose numbers change every time you try them.

The topic in one minute

Press Play: the idea and a worked example, explained aloud (about 1:29).

Scatterplots and Line of Best Fit

Each dot is one student: hours studied across, quiz score up.

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Read what is said

A scatterplot shows two measurements of the same things at once. Here eight students recorded hours studied and a quiz score. The dots have a direction, and reading that direction is a skill the SAT tests often.
The dots rise from left to right, so the association is positive. A line of best fit runs through the middle of the dots. It does not have to touch any dot, and about as many dots lie above it as below.
Now use the line as a model. To predict, put the x value in. Nobody in the data studied nine hours, so this prediction comes from the line itself, and the green dot sits right on it.
A dot above the line did better than the line predicts. The student who studied six hours scored nineteen, and the line gives less. The dashed segment on the picture is the gap between them.
Here is the trap with slope. On this picture one square across is one hour, but one square up is five points, so the line looks flatter than it is. Take two points on the line and divide the change in y by the change in x.
Now a used car question. The line of best fit for price against age is negative two x plus twenty-two. For a car five years old, go up to the line, not to the dot, and read across. The dot at five years is lower, so that car sold for less than predicted.
To remember: the line is a model, so a prediction means reading the line, never the nearest dot. Read the numbers on both axes before you find a slope. Now practice it with new numbers.

The idea

One dot for each pair

A scatterplot shows two measurements of the same things at once. Each dot is one pair: its place across is the first measurement and its height is the second. Eight students wrote down how many hours they studied for a 25-point quiz and what they scored:

Hours studied	1	2	3	4	5	6	7	8
Quiz score	8	8	10	14	13	19	21	19

The dots do not lie on one line, but they have a direction: more hours go with higher scores. That direction is called the association.

Positive: the dots rise from left to right. As one quantity grows, so does the other.
Negative: the dots fall from left to right. As one grows, the other shrinks.
None: the dots show no direction at all.

The line of best fit

A line of best fit is the straight line that runs through the middle of the dots, as close to all of them as a line can be. It does not have to pass through any dot. About as many dots lie above it as below it. The engine finds it from the data:

Try it by eye first. Drag the sliders until the line runs through the middle of the dots, and watch the count of dots above and below it. The best line has slope 2 and starts at 5.

What the line is for

The line is a model, $y = 2x + 5$ , and you use it like any linear model. To predict, put the $x$ in. The slope says how much the predicted $y$ changes for each 1 more of $x$ : here 2 more points for each extra hour. The number standing alone is the prediction when $x$ is 0.

\begin{aligned} predicted &= 2 \times 9 + 5 \\ &= 23 \end{aligned}

A student who studies 9 hours is predicted to score $23$ . No student in the data studied 9 hours: the prediction comes from the line.

A dot above the line did better than the line predicts, and a dot below it did worse. The student who studied 6 hours scored 19. The line gives less:

\begin{aligned} online &= 2 \times 6 + 5 \\ &= 17 \\ gap &= 19 - online \\ &= 19 - 17 \\ &= 2 \end{aligned}

That dot is $2$ above the line. In the picture the dashed segment is that gap, and the green dot is the prediction for 9 hours: it sits on the line.

Warning.

Two things go wrong here. First, when a question asks what the line predicts, read the line at that $x$ , not the nearest dot. Second, look at the numbers on the axes before you find a slope. In this picture one square across is 1 hour but one square up is 5 points, so the line looks flatter than its slope of 2. Take two points on the line, such as $(2, 9)$ and $(8, 21)$ , and divide the change in $y$ by the change in $x$ :

\begin{aligned} slope &= \frac{21 - 9}{8 - 2} \\ &= 2 \end{aligned}

Not every cloud is a line

Some data bends. In the next picture the dots rise slowly at first and then faster and faster. No straight line runs through the middle of them all the way: the dashed line is too high in the middle and too low at both ends. The blue curve, which grows by the same factor each step, fits better. When the test asks which model fits, look at the shape of the cloud before anything else.

Note.

On the SAT a scatterplot question shows the plot, often with the line drawn, and asks one of these: which equation fits (check the sign of the slope and where the line meets the $y$ -axis), what the line predicts at some $x$ , the slope of the line, how many dots lie above or below the line, or which kind of curve fits a bent cloud.

Worked examples

Each example is one of the kinds of question the test asks about a scatterplot. The numbers in the working are found by the engine.

Example.

1. Which equation fits? The scatterplot shows the price $y$ , in thousands of dollars, of eight used cars of one model against their age $x$ , in years. Which of these could be the line of best fit: $y = 2x + 22$ , $y = -2x + 22$ , $y = 2x - 22$ or $y = -2x - 22$ ?

The four options have the same two numbers and differ only in their signs, so two looks at the picture settle it. The dots fall from left to right: the slope is negative. Followed back to the left edge, the dots would meet the $y$ -axis high up, above 20: the number standing alone is positive. Only $y = -2x + 22$ has both. The engine's own fit of the eight dots agrees:

Example.

2. What the line predicts. The line of best fit for the cars is $y = -2x + 22$ . What price does it predict for a car that is 5 years old?

Put 5 in for $x$ . On the picture, go up from 5 to the line, not to the dot, and read across.

\begin{aligned} predicted &= -2 \times 5 + 22 \\ &= 12 \end{aligned}

The line predicts $12$ thousand dollars. The dot at 5 years is at 10: that one car sold for less than the line predicts. A question that asks for the prediction wants the line's value.

Example.

3. The slope, when the two axes have different scales. A shop recorded the number of ads $x$ it ran in each of eight weeks and its sales $y$ , in dollars, that week. The line of best fit is drawn. What is its slope, and what does the slope mean?

Pick two points on the line that are easy to read. The black marks are at $(2, 50)$ and $(8, 140)$ . Divide the change in $y$ by the change in $x$ .

\begin{aligned} slope &= \frac{140 - 50}{8 - 2} \\ &= 15 \end{aligned}

The slope is $15$ : each extra ad goes with about $15$ more dollars of sales. Counting squares gives the wrong number here, because one square across is 1 ad and one square up is 10 dollars. Always read the numbers on the axes.

Example.

4. Dots above and below the line. For the shop of example 3 the line is $y = 15x + 20$ . In how many of the eight weeks were the sales greater than the line predicts?

A dot above the line is a week whose sales beat the prediction. On the picture you can count them. The engine works out the prediction for each week and takes it from the real sales:

The last list is the sales minus the prediction, week by week. It is positive in the weeks with 1, 4, 6 and 7 ads, so 4 dots lie above the line. The other 4 lie below it.

Example.

5. The average rate of change between two dots. In example 3, the week with 1 ad had sales of 40 dollars and the week with 6 ads had sales of 120 dollars. What is the average rate of change of the sales between these two weeks, in dollars per ad?

This question is about two dots, not about the line. Divide the change in sales by the change in ads.

\begin{aligned} rate &= \frac{120 - 40}{6 - 1} \\ &= 16 \end{aligned}

Between those two weeks the sales rose by $16$ dollars per ad on average. This is close to the slope of the line of best fit, 15, but it need not be equal to it: it uses only two of the eight dots.

Example.

6. Every height is multiplied by a number. A second shop ran the same numbers of ads, and each week its sales were exactly 2 times the first shop's. The first shop's line of best fit is $y = 15x + 20$ . What is the second shop's?

Doubling every $y$ doubles every height on the picture: the place where the line starts and also the amount it rises for each ad. So both numbers of the equation are multiplied by 2, not only one of them.

\begin{aligned} slope &= 2 \times 15 \\ &= 30 \\ start &= 2 \times 20 \\ &= 40 \end{aligned}

The second shop's line is $y$ $=$ $30$ $x$ + $40$ . The engine's fit of the doubled data agrees, and the picture shows the two sets of dots with their lines.

Practice

Answer SAT-style questions on scatterplots and line of best fit. Each answer is checked, with the full solution, and every try brings new numbers. Practice needs a free account: it keeps your progress.

Practise scatterplots and line of best fit with new numbers