Construct a Scatter Plot from Paired Values | Grade 8 Math | LevelKite

Some data sets record two measurements for each item. For example, for each player on a basketball team, you might record the hours the player practiced in a week and the number of free throws the player made out of 10 tries. Data like this is called bivariate data (bi means two). The two numbers for one item form a pair. They describe the same player, so they must always stay together.

A scatter plot is a graph that shows bivariate data as points on a coordinate plane. In the last lesson, you described positive, negative and no association in scatter plots. Now you'll construct one yourself.

Key concept

One item, one point

Pick one quantity for the horizontal axis and call its values $x$ . The other quantity goes on the vertical axis, and its values are $y$ . Each item's pair becomes one ordered pair $(x, y)$ , horizontal value first, and you plot it as one point. Each axis also needs a name with a unit, such as practice (hours), and a scale. A scale is a set of numbers marked in equal steps, so equal distances always stand for equal amounts. That is what lets a point's position show its two values.

Weekly practice and free throws made by six players

Player	Practice (hours per week)	Free throws made (out of 10)
A	$1$	$2$
B	$2$	$5$
C	$4$	$4$
D	$5$	$7$
E	$7$	$6$
F	$8$	$9$

Worked example

Six players, six points

Construct a scatter plot of the six players' data in the table, with practice on the horizontal axis.

Define the variables so each position has a meaning. Let $x$ be a player's weekly practice in hours. Let $y$ be the number of free throws the player made out of $10$ . Here both are whole numbers. The horizontal axis shows $x$ and the vertical axis shows $y$ . Either quantity could go on the horizontal axis, but once you choose, every pair must follow that choice.
Choose a scale for each axis. The $x$ -values run from $1$ to $8$ and the $y$ -values run from $2$ to $9$ . Marking both axes from $0$ to $10$ in steps of $1$ covers every value with room to spare.
$1 \le x \le 8, \qquad 2 \le y \le 9$
Write each row as an ordered pair, practice first. The pair keeps one player's two numbers together, and its first number gives the horizontal position.
$A(1, 2),\ B(2, 5),\ C(4, 4),\ D(5, 7),\ E(7, 6),\ F(8, 9)$
Plot each pair as one dot. For Player B, start at the origin $(0, 0)$ . Move right $2$ for $2$ hours, then up $5$ for $5$ free throws. Plot the other five pairs the same way. Leave the dots unconnected, because each dot is a separate player, not a stop along a path.
$B(2, 5):\ \text{right } 2,\ \text{up } 5$
Check the plot. No two players share the same pair, so six rows must give six separate points. Read each point's coordinates back and match them to a row of the table. Checking coordinates is more reliable than judging by eye.
$6 \text{ rows} \;\to\; 6 \text{ points}$

Answer

The scatter plot has six points, at $(1, 2)$ , $(2, 5)$ , $(4, 4)$ , $(5, 7)$ , $(7, 6)$ and $(8, 9)$ , with practice in hours on the horizontal axis and free throws made on the vertical axis.

Points A to F stand for Players A to F. Point B is at

(2, 5)

for

2

hours of practice and

5

free throws made.

In the figure, the horizontal axis is practice in hours and the vertical axis is free throws made out of 10. Each point is labelled with its player's letter. Every point matches its row. For example, Point F is $8$ units right of the origin and $9$ units up, which is the pair $(8, 9)$ .

With the data drawn, you can see that the points tend to rise from left to right. This is the kind of positive association you described in the last lesson. The plot shows this pattern, but it does not prove that practice causes more free throws made. Something else, such as years of playing, could affect both.

The point

(2, 5)

is Player B plotted correctly. The point

(5, 2)

is the same two numbers plotted backward. The unlabelled points are the other five players.

The next data set includes temperatures below zero, so choosing a scale takes more thought. First find the least and greatest value of each quantity. Then mark each axis in equal steps, such as steps of $2$ or $5$ . The marks should start at or below the least value and reach at or above the greatest value. Keep the same step on the negative side of $0$ , so $-2$ is exactly as far from $0$ as $2$ is, on the opposite side. The two axes may use different steps when their quantities have very different sizes. Here, steps of $2$ suit both.

Noon temperature and cups of cocoa sold at a school snack stand on six winter days

Day	Noon temperature (°C)	Cups of cocoa sold
1	$2$	$10$
2	$-8$	$19$
3	$6$	$11$
4	$-4$	$17$
5	$0$	$14$
6	$8$	$5$

Worked example

Temperatures below zero

Construct a scatter plot of the cocoa data in the table, with noon temperature on the horizontal axis.

Define the variables. Let $x$ be the noon temperature in degrees Celsius. Let $y$ be the number of cups sold that day. Temperatures below $0$ degrees are negative, so $x$ can be negative. The value of $y$ is a whole number of cups.
Choose the $x$ -scale. The temperatures run from $-8$ to $8$ , a spread of $16$ degrees. Steps of $2$ cover that spread in $8$ steps, which is a manageable number of marks. Mark the axis from $-10$ to $10$ to leave room at each end.
$8 - (-8) = 8 + 8 = 16, \qquad 16 \div 2 = 8$
Choose the $y$ -scale. The cup counts run from $5$ to $19$ , so mark the axis from $0$ to $20$ in steps of $2$ . That takes $10$ steps. The marks fall on even numbers, so an odd count such as $19$ sits exactly halfway between two marks, here $18$ and $20$ .
$20 \div 2 = 10, \qquad \frac{18 + 20}{2} = \frac{38}{2} = 19$
Write the pairs with temperature first, then plot them. A negative $x$ means you move left from the origin instead of right. For Day 2, move left $8$ , then up $19$ , halfway between the marks $18$ and $20$ . Day 5 has $x = 0$ , so you move neither left nor right, and its point lands on the vertical axis.
$(2, 10),\ (-8, 19),\ (6, 11),\ (-4, 17),\ (0, 14),\ (8, 5)$
Check the plot. No two days share a pair, so six days give six points. Match each point's coordinates to its row in the table.
$6 \text{ days} \;\to\; 6 \text{ points}$

Answer

The scatter plot has six points, at $(2, 10)$ , $(-8, 19)$ , $(6, 11)$ , $(-4, 17)$ , $(0, 14)$ and $(8, 5)$ , with noon temperature in °C on the horizontal axis and cups sold on the vertical axis.

Day 2 is

(-8, 19)

, left of the vertical axis, and Day 5 is

(0, 14)

, on it. The other four points are

(-4, 17)

,

(2, 10)

,

(6, 11)

and

(8, 5)

.

Takeaway: To construct a scatter plot, name each axis with its quantity and unit. Mark each axis with an evenly spaced scale that covers every value. Then plot each item's pair as one point $(x, y)$ , horizontal value first. Next, you'll use scatter plots like these to identify linear and nonlinear association, describe associations between two variables and make predictions.