11.2 Scatter Plots

Aligned outcomes:

Learning Objectives

By the end of this section, you will be able to:

In this section, you will learn to:

construct a scatter plot from a table of paired data, and read one that has already been drawn;
describe the direction of a relationship shown by a scatter plot;
judge the strength of a relationship from how closely the points follow a line;
identify the overall pattern and any deviations from it;
decide whether a regression line is appropriate for a given pair of variables.

Before we take up the discussion of linear regression and correlation, we need to examine a way to display the relation between two variables $x$ and $y$. The most common and easiest way is a scatter plot. Section 11.1 gave you an equation that describes a perfectly straight relationship; real paired data never lands exactly on a line. A scatter plot is how you find out whether a line is even worth drawing — and, just as importantly, when it is not.

11.2.1 Displaying Paired Data

Definition 11.2.1: Scatter Plot

A scatter plot displays the relation between two variables by plotting each observation as a single point, with the independent variable $x$ on the horizontal axis and the dependent variable $y$ on the vertical axis.

Each row of a two-column data table becomes exactly one dot. Nothing is summarised and nothing is averaged, which is what makes the plot honest: every observation you collected is visible, including the ones that do not fit.

Definition 11.2.1 — Scatter Plot: A scatter plot built from Table 11.2.1: six dots, one per row of the two-column table, plotted with practice hours on the horizontal axis and....

Try It Now 11.2.1

Amelia plays basketball for her high school. She wants to improve to play at the college level. She notices that the number of points she scores in a game goes up in response to the number of hours she practices her jump shot each week. She records the data in Table 11.2.1. Construct a scatter plot and state whether what Amelia thinks appears to be true.

**Table 11.2.1** — Jump-shot practice and points scored.
X (hours practicing jump shot)	Y (points scored in a game)
5	15
7	22
9	28
10	31
11	33
12	36

Solution — plotting six pairs and reading the result

Step 1 — Assign the axes. Amelia believes points scored respond to practice hours, so practice hours is the independent variable on the horizontal axis and points scored is the dependent variable on the vertical axis.

Step 2 — Plot one point per row. The six points are $(5,15)$, $(7,22)$, $(9,28)$, $(10,31)$, $(11,33)$ and $(12,36)$.

Step 3 — Describe the direction. Reading left to right, the points climb steadily: the lowest practice hours pair with the fewest points and the highest with the most. Nothing reverses that trend.

Step 4 — Describe the strength. The points lie very close to a straight path, with no dot far off the trend.

Step 5 — Answer Amelia's question. What she thinks does appear to be true — more practice hours go with more points scored.

Step 6 — Add the honest caution. The plot shows the two go together; it does not by itself prove that practising causes the scoring. Six games is also a small record.

Answer: A strong positive linear pattern. Amelia's belief appears to be supported by her data.

Example 11.2.1: Vocabulary size and age

An educational researcher collects data on the vocabulary size of children as a function of age. The data is shown in Table 11.2.2. Is there a relationship between age and vocabulary size for young children? Construct a scatter plot. Let $x$ = Child's Age, and let $y$ = Vocabulary Size.

**Table 11.2.2** — Child's age and vocabulary size.
Age (years)	Vocabulary Size (number of words)
3	655
4	1098
6	2463
7	3195

Solution — four points, one clear direction

Step 1 — Assign the axes. Let $x$ = Child's Age on the horizontal axis, and $y$ = Vocabulary Size on the vertical axis.

Step 2 — Plot the four pairs. They are $(3, 655)$, $(4, 1098)$, $(6, 2463)$ and $(7, 3195)$.

Step 3 — Read the direction. As age increases, vocabulary size increases. Low values of $x$ occur with low values of $y$, and high with high.

Step 4 — Read the strength. The four points fall close to a straight rising path.

Step 5 — Answer the question. There does appear to be a relationship between age and vocabulary size for young children, and it looks linear over this age range.

Answer: Yes — the scatter plot shows a strong positive relationship between age and vocabulary size.

Using the TI-83, 83+, 84, 84+ Calculator

To create a scatter plot:

Four points is not much to go on

The vocabulary data covers ages 3 to 7 with a single child's-worth of information at each age. The pattern is clear, but a pattern drawn through four points is fragile, and it says nothing about what happens outside that age range — vocabulary growth certainly does not keep climbing in a straight line forever. Reading a scatter plot always includes noticing how many points drew it and how wide a span they cover.

Enter your X data into list L1 and your Y data into list L2.
Press 2nd STATPLOT ENTER to use Plot 1. On the input screen for PLOT 1, highlight On and press ENTER. Make sure the other plots are OFF.
For TYPE: highlight the very first icon, which is the scatter plot, and press ENTER.
For Xlist: enter L1 ENTER and for Ylist: enter L2 ENTER.
For Mark: it does not matter which symbol you highlight, but the square is the easiest to see. Press ENTER.
Make sure there are no other equations that could be plotted. Press Y = and clear any equations out.
Press the ZOOM key and then the number 9 (for menu item ZoomStat); the calculator will fit the window to the data. You can press WINDOW to see the scaling of the axes.

11.2.2 The Direction of a Relationship

A scatter plot shows the direction of a relationship between the variables. A clear direction happens when there is either:

High values of one variable occurring with high values of the other variable, or low values of one variable occurring with low values of the other variable; or
High values of one variable occurring with low values of the other variable.

The first case is a positive relationship and the second is a negative one. Notice that "negative" is a statement about direction only — it carries no suggestion that the relationship is weaker or less useful.

Try It Now 11.2.2

For each pair of variables, say whether you would expect the scatter plot to show a positive direction, a negative direction, or no clear direction.

a. Hours studied and exam score. b. Outside temperature and heating cost. c. A student's shoe size and their exam score.

Solution — matching the high/low pattern

Step 1 — Consider part a. More hours studied generally goes with a higher score: high with high, low with low. That is a positive direction.

Step 2 — Consider part b. As the outside temperature rises, heating cost falls. High values of one occur with low values of the other, so the direction is negative.

Step 3 — Consider part c. There is no reason for shoe size to travel with exam score in either direction. Expect no clear direction — a formless cloud of points.

Step 4 — Note the distinction that matters. Part b is negative, not weak. A negative relationship can be extremely strong; the sign describes which way the points run, not how tightly they follow the trend.

Answer: (a) positive, (b) negative, (c) no clear direction.

11.2.3 Strength, and the One Exception

You can determine the strength of the relationship by looking at the scatter plot and seeing how close the points are to a line, a power function, an exponential function, or to some other type of function.

For a linear relationship there is an exception. Consider a scatter plot where all the points fall on a horizontal line providing a "perfect fit." The horizontal line would in fact show no relationship.

The perfect fit that means nothing

This is the most counterintuitive idea in the section, and it follows directly from Section 11.1. A horizontal line is $y = a + 0x$ — slope zero. Every point sits exactly on it, so by the "how close to a line" test the fit is flawless. But look at what the line says: whatever $x$ you pick, $y$ is the same. Knowing $x$ tells you nothing whatsoever about $y$, which is the definition of no relationship. Closeness to a line measures strength only once the line has a non-zero slope.

Try It Now 11.2.3

A scatter plot of daily temperature against the price of a fixed-price lunch special shows every point lying exactly on a horizontal line at $12. A student says the relationship is "perfectly strong because the fit is exact." Explain what is wrong with that.

Solution — a flawless fit to a flat line

Step 1 — Describe the line. Every point lies on $y = 12$, which in the form $y = a + bx$ is $y = 12 + 0x$, with slope $b = 0$.

Step 2 — Ask what the line predicts. For any temperature at all, the predicted price is $12. The prediction never changes.

Step 3 — Ask what knowing $x$ buys you. Nothing. You could predict the price just as well without ever looking at the temperature.

Step 4 — State the conclusion. A horizontal line providing a perfect fit shows no relationship. The student has measured closeness to the line and mistaken it for strength of relationship; the two only agree when the slope is non-zero.

Answer: The fit is exact but the relationship is absent. A slope of zero means $x$ carries no information about $y$, however tightly the points sit on the line.

Example 11.2.2: Judging strength from two plots

Two scatter plots both show a negative direction. In the first, the points sit tightly along a downward path. In the second, the points drift downward overall but are widely scattered, with several sitting far from any line you might draw. Describe each.

Solution — same direction, different strength

Step 1 — Describe the first plot. The direction is negative and the points are close to a straight line, so this is a strong negative linear pattern.

Step 2 — Describe the second plot. The direction is also negative, but the points are far from any single line, so this is a weak negative linear pattern.

Step 3 — Separate the two ideas. Direction and strength are independent descriptions. Both plots answer "which way?" the same way and answer "how tightly?" very differently.

Step 4 — Note the practical consequence. A line fitted to the first plot would predict well. A line fitted to the second would capture the general drift but would be wrong by a lot on individual points.

Answer: The first is a strong negative linear pattern, the second a weak negative linear pattern; they share a direction and differ in strength.

11.2.4 Pattern and Deviation

When you look at a scatterplot, you want to notice the overall pattern and any deviations from the pattern.

Not every pattern is a straight line. A scatter plot may show exponential growth, some other curve, or no pattern at all — a shapeless cloud. Naming the overall pattern comes first, because it decides whether a straight line is the right tool before any line gets drawn.

**Table 11.2.3** — Patterns a scatter plot can show.
Pattern	What it looks like
Positive linear, strong	points rise left to right, tightly along a line
Linear with one deviation	a clear line, plus a single point well off it
Negative linear, strong	points fall left to right, tightly along a line
Negative linear, weak	points drift downward but scatter widely
Exponential growth	points nearly flat at first, then curving sharply upward
No pattern	a formless cloud with no direction

Try It Now 11.2.4

A scatter plot shows six points rising in a tight straight line and a seventh point sitting far above the others at the left-hand end. Describe the overall pattern and the deviation, and say what you should do about the odd point.

Solution — naming both parts of the description

Step 1 — Name the overall pattern. Six of the seven points form a strong positive linear pattern.

Step 2 — Name the deviation. One point sits far from that line, high on the left where the trend is low. It is a deviation from the pattern, not part of it.

Step 3 — Say what to do about it. Notice it and report it. A single far-off point can drag a fitted line noticeably, so it must not be quietly ignored.

Step 4 — Say what NOT to do. Do not delete it merely because it is inconvenient. It may be a recording error, or it may be a real and interesting observation; deciding which requires going back to how the data were collected, not looking harder at the plot.

Answer: A strong positive linear pattern with one deviation. The odd point should be reported and investigated, not discarded on sight.

Example 11.2.3: A pattern that is not a line

A scatter plot of seven points shows the first four sitting essentially flat near zero, after which the points curve sharply upward, the last one far higher than the rest. What pattern is this, and is a straight line the right description?

Solution — recognising exponential growth

Step 1 — Describe the shape. The points are nearly flat at first and then rise more and more steeply. The rate of increase is itself increasing.

Step 2 — Name the pattern. This is an exponential growth pattern, not a linear one.

Step 3 — Answer the question about the line. A straight line is not the right description. Fitting one would run above the data at both ends and below it in the middle, and would predict badly everywhere.

Step 4 — State the general rule this illustrates. Judge the shape of the pattern before fitting anything. Strength is only meaningful once you have chosen the right kind of curve to measure closeness to.

Answer: An exponential growth pattern; a straight line would misdescribe it and should not be fitted.

11.2.5 When a Regression Line Is Appropriate

In this chapter, we are interested in scatter plots that show a linear pattern. Linear patterns are quite common. The linear relationship is strong if the points are close to a straight line, except in the case of a horizontal line where there is no relationship.

If we think that the points show a linear relationship, we would like to draw a line on the scatter plot. This line can be calculated through a process called linear regression. However, we only calculate a regression line if one of the variables helps to explain or predict the other variable. If $x$ is the independent variable and $y$ the dependent variable, then we can use a regression line to predict $y$ for a given value of $x$.

Two conditions, and the second is not visual

A regression line needs a linear pattern — that you can see. It also needs one variable to explain or predict the other — and that you cannot see, because it is a claim about the subject matter rather than about the dots. Software will happily fit a line to any two columns of numbers, including two that have no business being paired. The plot tells you whether a line fits; only knowing what the variables are tells you whether a line means anything.

Try It Now 11.2.5

For each situation, say whether calculating a regression line is appropriate, and why.

a. A shop records the number of umbrellas sold and the rainfall that day, and wants to predict sales from rainfall.

b. A researcher plots the number of letters in 50 people's surnames against their annual income, and the points form a shapeless cloud.

c. A plot of a city's population against year shows a clear curve that bends sharply upward.

Solution — checking both conditions each time

Step 1 — Check part a. Rainfall plausibly explains umbrella sales, so one variable helps predict the other, and the relationship would be expected to be roughly linear over an ordinary range. A regression line is appropriate, with rainfall as $x$ and sales as $y$.

Step 2 — Check part b. The points show no pattern, so there is no linear relationship to describe. Neither does surname length explain income. Both conditions fail, and a regression line is not appropriate.

Step 3 — Check part c. Year does help explain population, so the second condition holds — but the pattern is a curve, not a line. A straight-line fit is not appropriate for these data as they stand.

Step 4 — Summarise the rule. Both conditions must hold: a linear pattern, and one variable genuinely explaining or predicting the other.

Answer: (a) appropriate; (b) not appropriate, no pattern and no explanatory link; (c) not appropriate as a straight line, because the pattern is curved.

Problem Set 11.2

Problem 1. Explain what a scatter plot displays and which variable goes on which axis.

Solution

Step 1 — Define the display: A scatter plot displays the relation between two variables by plotting each observation as a single point — one dot per row of paired data, with nothing summarised or averaged.

Step 2 — Assign the axes: The independent variable $x$ goes on the horizontal axis and the dependent variable $y$ on the vertical axis.

Answer: A scatter plot shows each paired observation as one point, with $x$ (the independent variable) horizontal and $y$ (the dependent variable) vertical.

Problem 2. State the two ways a scatter plot can show a clear direction.

Solution

Step 1 — State the first way: High values of one variable occur with high values of the other, and low values occur with low values. This is a positive direction.

Step 2 — State the second way: High values of one variable occur with low values of the other. This is a negative direction.

Answer: A clear direction appears either as a positive relationship (high with high, low with low) or as a negative relationship (high of one with low of the other).

Problem 3. A scatter plot's points rise steeply and lie very close to a straight line. Describe the direction and strength.

Solution

Step 1 — Read the direction: The points rise from left to right, so high values of $x$ go with high values of $y$. That is a positive direction.

Step 2 — Read the strength: The points lie very close to a straight line, so the linear relationship is strong.

Answer: The scatter plot shows a strong positive linear relationship.

Problem 4. Explain why a horizontal line with a perfect fit shows no relationship.

Solution

Step 1 — Write the line in slope form: A horizontal line is $y = a + bx$ with slope $b = 0$, so it is $y = a + 0x$.

Step 2 — Ask what the line predicts: For any value of $x$, the predicted value of $y$ is the same constant $a$. Knowing $x$ changes nothing about the prediction.

Step 3 — Draw the conclusion: Since knowing $x$ tells you nothing about $y$, there is no relationship between them — even though every point sits exactly on the line. Closeness to a line measures strength only when the line has non-zero slope.

Answer: A perfect fit to a horizontal line shows no relationship because the slope is zero: whatever $x$ you pick, $y$ is the same, so $x$ carries no information about $y$.

Problem 5. Give an example, other than one in this section, of two variables you would expect to show a negative direction.

Solution

Step 1 — Recall what negative direction means: High values of one variable must occur with low values of the other — as one rises, the other tends to fall.

Step 2 — Construct an example: Consider hours spent watching television per week and grade-point average: more TV time would generally be expected to go with lower grades.

Step 3 — Check the example: As TV hours increase, GPA tends to decrease — high with low — so this fits the definition of a negative direction.

Answer: Example: weekly hours of television watched and GPA — as viewing hours increase, GPA tends to fall, giving a negative direction. (Any sensible pair such as exercise amount and resting heart rate also works.)

Problem 6. A scatter plot shows a tight upward line with one point far below it. Describe the overall pattern and the deviation.

Solution

Step 1 — Name the overall pattern: Most of the points form a tight upward trend, so the overall pattern is a strong positive linear pattern.

Step 2 — Name the deviation: One point sits far below that line, so it departs noticeably from the pattern. It is a deviation, not part of the pattern.

Step 3 — Say what to do about it: Report it and investigate why it is off the trend — it may be a recording error or a genuinely unusual observation. Do not delete it merely because it is inconvenient, since a far-off point can drag a fitted line noticeably.

Answer: A strong positive linear pattern with one deviation: a single point lying well below the line, which should be reported and investigated rather than discarded.

Problem 7. State the two conditions that must both hold before a regression line should be calculated.

Solution

Step 1 — State the visual condition: The scatter plot must show a linear pattern — points reasonably close to a straight line (excluding the horizontal-line case, which shows no relationship).

Step 2 — State the explanatory condition: One of the variables must help explain or predict the other; if $x$ is independent and $y$ dependent, the line can predict $y$ from a given $x$.

Step 3 — Note why both are needed: The plot can only tell you whether a line fits; only knowledge of the variables tells you whether the line means anything. Both conditions must hold together.

Answer: A regression line should be calculated only if (1) the scatter plot shows a linear pattern, and (2) one variable helps explain or predict the other.

Problem 8. A scatter plot of two variables shows a formless cloud of points. What does this tell you about using $x$ to predict $y$?

Solution

Step 1 — Describe the cloud: A formless cloud has no direction and no shape: high values of $x$ occur with both high and low values of $y$ indiscriminately.

Step 2 — Ask what prediction is possible: Since $y$ does not systematically change with $x$, knowing $x$ gives no information about $y$. Any predicted value would be no better than guessing the average of all the $y$-values.

Step 3 — Conclude about regression: There is no linear relationship to describe, so using $x$ to predict $y$ is pointless and a regression line is not appropriate.

Answer: A formless cloud means $x$ carries essentially no information about $y$; predictions from $x$ would be no better than ignoring $x$ entirely, so a regression line should not be used.

Problem 9. Plot the points $(2, 9)$, $(4, 7)$, $(6, 6)$, $(8, 4)$ and $(10, 2)$. Describe the direction and the strength.

Solution

Step 1 — Assign the axes: Let $x$ be the first coordinate of each pair (horizontal axis) and $y$ the second (vertical axis).

Step 2 — Plot the five pairs: The points are $(2, 9)$, $(4, 7)$, $(6, 6)$, $(8, 4)$ and $(10, 2)$. Each row of the data becomes exactly one dot.

Step 3 — Read the direction: Reading left to right, the points fall steadily: as $x$ increases from 2 to 10, $y$ decreases from 9 to 2. High values of $x$ go with low values of $y$, so the direction is negative.

Step 4 — Read the strength: The points lie very close to a straight downward path — indeed they are nearly perfectly collinear — so the relationship is strong.

Answer: Plotting $(2, 9)$, $(4, 7)$, $(6, 6)$, $(8, 4)$, $(10, 2)$ gives a strong negative linear pattern: as $x$ increases, $y$ decreases closely along a straight line.

Problem 10. Explain why software fitting a line to two columns of numbers is not by itself evidence that a regression line is appropriate.

Solution

Step 1 — Recall what software does: Statistical software will happily compute and draw a regression line for any two columns of numbers, whether or not the variables are meaningfully related.

Step 2 — Separate fitting from meaning: A computed line only addresses whether a line fits the plotted points. Whether a line means anything depends on the subject matter: one variable must genuinely help explain or predict the other.

Step 3 — Apply to arbitrary columns: Two columns with no real connection (for example, shoe size and exam score) may still yield a fitted line, but it describes no relationship and predicts nothing useful.

Answer: Software output alone is not evidence because a program will fit a line to any pair of columns regardless of subject matter. A regression line is appropriate only when the scatter plot shows a linear pattern AND one variable plausibly explains or predicts the other — a claim about the variables themselves, not about the computation.

Key Terms

scatter plot — a display of the relation between two variables, plotting each paired observation as one point with $x$ horizontal and $y$ vertical.

direction — whether high values of one variable occur with high values of the other (positive) or with low values of the other (negative).

strength — how close the points lie to a line or other curve; strong means tightly clustered around it.

overall pattern — the general shape the points form: linear, exponential, some other curve, or none.

deviation — a point that departs noticeably from the overall pattern; to be reported and investigated, not discarded.

linear regression — the process that calculates a line for a scatter plot showing a linear pattern.

the horizontal-line exception — a perfect fit to a horizontal line shows NO relationship, because the slope is zero and $x$ carries no information about $y$.