Math · Problem Solving and Data Analysis

Two-variable data: Models and scatterplots

Explanation

Two-variable (bivariate) data analysis focuses on studying the relationship between two different quantitative variables, usually plotted on a coordinate plane as a scatterplot to identify patterns, trends, and models.

Understanding Scatterplots and Trends

A scatterplot displays data points where the horizontal axis represents the independent variable ( $x$ ) and the vertical axis represents the dependent variable ( $y$ ).

Positive Association (Trend): As $x$ increases, $y$ tends to increase.
Example: Study hours ( $x$ ) vs. test scores ( $y$ ). As study time increases, test scores generally rise.
Negative Association (Trend): As $x$ increases, $y$ tends to decrease.
Example: Car age ( $x$ ) vs. resale value ( $y$ ). As a car gets older, its value generally goes down.
No Association: There is no clear pattern or relationship between $x$ and $y$ .
Example: Shoe size ( $x$ ) vs. score on a math test ( $y$ ). The points are scattered randomly.
Linear vs. Non-linear Patterns: Points can form a straight line pattern (linear) or a curve (such as quadratic or exponential).

Line of Best Fit (Linear Models)

A line of best fit (trend line) is a straight line drawn through a scatterplot to model the general relationship between $x$ and $y$ , usually written as $\hat{y} = mx + b$ .

Slope ( $m$ ): The predicted change in $y$ for every $1$ -unit increase in $x$ .
Example: For a model $y = 2.5x + 10$ tracking plant height ( $y$ , in cm) over weeks ( $x$ ), the plant grows by an estimated $2.5\text{ cm}$ each week.
$y$ -intercept ( $b$ ): The predicted starting value of $y$ when $x = 0$ .
Example: In $y = 2.5x + 10$ , the initial height of the plant at week $0$ is $10\text{ cm}$ .

Example 1

The scatterplot shows the relationship between two variables,

x

and

y

, for

9

data points. A line of best fit for the data is also shown. Which of the following is closest to the difference between the

y

-coordinate of the data point at

x = 3

and the

y

-value predicted by the line of best fit at

x = 3

1

2

3

8

Choice B is correct. The data point at

x = 3

has a

y

-coordinate of

8

. The line of best fit passes through

(0, 3)

and

(10, 13)

, so its slope is

\frac{13-3}{10-0} = 1

and its equation is

y = x + 3

. At

x = 3

, the line of best fit predicts

y = 3 + 3 = 6

. The difference between the observed value and the predicted value is

8 - 6 = 2

. Choice A is incorrect and may result from misreading the scatterplot by one unit. Choice C is incorrect and may result from misreading the line of best fit's slope. Choice D is incorrect; this is the

y

-coordinate of the data point itself, not the difference between it and the predicted value.

Example 2

A gardener models the height of a sunflower,

h

, in centimeters,

w

weeks after it was planted, using the equation

h = 8.5w + 12

. Which of the following is the best interpretation of

8.5

in this context?

The sunflower's height, in centimeters, when it was planted

The predicted increase in the sunflower's height, in centimeters, each week

The total number of weeks the sunflower was measured

The predicted height of the sunflower, in centimeters, after

8.5

weeks

Choice B is correct. The equation is in slope-intercept form,

h = mw + b

, where

m

is the slope. Since the slope is

8.5

, and

w

represents the number of weeks, an increase of

1

w

corresponds to a predicted increase of

8.5

centimeters in

h

. Choice A is incorrect; this describes

12

, the

y

-intercept, not

8.5

. Choice C is incorrect; the number of weeks is represented by the variable

w

, not by the constant

8.5

. Choice D is incorrect; this describes a specific predicted output value, which would require substituting

w = 8.5

into the equation, not the meaning of the coefficient itself.

Tips for solving

Interpret the slope in word problems by adding the phrase "for each additional unit of $x$ ."
Example: A scatterplot models cost $C$ based on weight $w$ with equation $C = 3.5w + 12$ . What does $3.5$ mean?
- Identify $x$ -variable: Weight ( $w$ , in pounds)
- Meaning: The cost increases by $\$3.50$ for each additional pound of weight.
Distinguish between model predictions and actual data points on graph questions.
Example: A scatterplot shows a point at $(5, 40)$ and a line of best fit passing through $(5, 32)$ at $x = 5$ .
- Actual value: $y = 40$ (the plotted point)
- Model prediction: $\hat{y} = 32$ (the line height)

Practice questions

00:00

Question 1 · easy

The number of visitors,

v

, to a museum

t

weeks after a new exhibit opened is modeled by the equation

v = 45t + 900

. What does the number

45

represent in this context?

Question 2 · easy

The number of visitors,

v

, to a museum

t

weeks after a new exhibit opened is modeled by the equation

v = 45t + 900

. What does the number

900

represent in this context?

Question 3 · easy

Week	Height (cm)
2	14
5	26

The table shows the height of a plant, in centimeters, recorded at two different weeks after it was planted. What is the average rate of change, in centimeters per week, in the height of the plant from week

2

to week

5

Question 4 · easy

The scatterplot shows the relationship between two variables,

x

and

y

, for a set of data points. A line of best fit for the data is also shown. Which of the following is closest to the difference between the

y

-coordinate of the data point at

x = 3

and the

y

-value predicted by the line of best fit at

x = 3

Question 5 · easy

x	y
0	5
1	8
2	11
3	14

The table shows values of

x

and

y

for four data points. Which type of equation best models the relationship between

x

and

y

Question 6 · easy

A company's monthly profit, in dollars, is modeled by the equation

p = 250x - 400

, where

x

is the number of units sold. What is the predicted profit, in dollars, when

30

units are sold?

Question 7 · easy

Hours, h	Distance, d (miles)
1	20
4	80

The table shows a car's distance from home, in miles, at two different times after it began a trip. What is the average rate of change, in miles per hour, of the car's distance from home between these two times?

Question 8 · easy

A model predicts that a company's sales,

s

, are related to the number of ads run,

n

, by the equation

s = 3n + 50

. On a day when

10

ads ran, actual sales were

92

. What is the difference between the actual sales and the sales predicted by the model (actual minus predicted)?

Question 9 · easy

x	y
0	100
1	80
2	64
3	51.2

The table shows values of

x

and

y

for four data points that follow the pattern

y = a(b)^x

. What is the value of

b

Question 10 · easy

A line of best fit passes through the points

(2, 15)

and

(6, 35)

. What is the slope of this line of best fit?

SAT Prep