Scatter Plots and Correlation

A school wants to compare how long students study with their test scores. Recording only the study times gives one list. Recording the score for each of those same students gives paired data: two values that belong together.

Keep each pair together

Data involving two variables are called bivariate data. A variable is something whose value can differ between people or objects. Study time and test score are both numerical variables: their numbers measure or count something. Favorite sport is categorical because its answers name groups. Giving sports number codes would not turn them into numerical measurements.

Both values must describe the same individual. If Maya studied 4 hours and scored 66, her pair is $(4,66)$. The height of one student paired with the shoe size of another does not describe either student. Keep a row for each individual as you collect the measurements. Sorting one column alone would destroy those pairings.

Decide what the graph will show

A scatter plot, also called a scatter graph or scatter diagram, places one variable on each axis. Each pair becomes a point with coordinates $(x,y)$. The first coordinate sets its horizontal position; the second sets its vertical position.

When you use one variable to explain or predict another, put the explanatory variable on the horizontal $x$-axis. Put the response variable, which you want to predict, on the vertical $y$-axis. These are also called the independent and dependent variables.

For a question about predicting test score from study time, study time is explanatory and score is the response. For predicting free throws made from practice hours, practice hours go across and free throws made go up. Calling something explanatory describes its role in the graph. Choosing that axis does not establish a cause.

Sometimes neither variable has a clear explanatory role. If you are comparing height and arm span, either can go across. Label the axes so the choice is clear, and keep that choice throughout your calculations.

Read a point as a sentence

On the study plot, move down from $(4,66)$ to read 4 hours. Move across to read a score of 66. Write: “One student studied 4 hours and scored 66.” The pair $(66,4)$ would mean 66 hours and a score of 4 on these axes. The coordinate order changes the meaning.

Ten students supply ten observations. Their points stay separate: joining them would suggest a path from one student to the next. Two students can have the same horizontal value but different scores, so their points sit above one another. If two pairs are identical, they share one plotted position; a small count label can show both observations.

A useful scatter plot lets someone read the measurements without guessing the units or the scale. Begin with a paired-data table, then choose a grid that leaves room for every pair.

Choose the axes and scales

These ten pairs give hours studied and score out of 100:

Put study hours across and score up the page. Hours run from 1 to 6, so an axis from 0 to 7 in steps of 1 works well. Scores run from 55 to 81. An axis from 50 to 85 in steps of 5 uses more of the grid than one from 0 to 100.

An axis can start above zero. Show a small zigzag break near the origin when skipping the unused range. After that break, equal distances on the axis must represent equal changes. The space from 50 to 55 must match the space from 80 to 85. The two axes can use different scales; each needs to be consistent along its own length.

Label the axes “Hours studied” and “Test score (points out of 100).” A title such as “Study time and test scores for 10 students” tells the reader whose data the graph shows.

Place every pair

For $(2,61)$, find 2 hours on the horizontal axis. Move straight up to a height of 61 points and draw a small dot or cross. On the score scale, 61 is one fifth of the gap from 60 to 65. Likewise, 66 is one fifth of the gap from 65 to 70. Do not place either halfway between the gridlines.

Work through the table one row at a time. At 6 hours, plot three points at 76, 79, and 81. They belong to three students, so none replaces another. Two identical pairs would overlap; mark their shared position with a count such as “2.”

Count the observations you have represented and compare them with the table. A count label matters when the number of visible marks is smaller than the number of people. Keep the points unjoined so that no row order becomes an invented path.

Read the graph back

To read a point, go down to the horizontal scale and across to the vertical scale. Include the variable names or units in your answer. A point at $(5,71)$ means 5 hours and 71 points, not a score of 5.

Estimate values between gridlines to the accuracy the scale supports. A thick dot near 71 cannot justify a reading of 71.036. When checking a finished graph, look for missing labels, uneven steps, reversed coordinates, and points plotted from two different rows.

Look across a scatter plot from left to right. The overall movement of the points describes the direction of a correlation. Their scatter around that movement describes its strength.

Describe the direction

A positive correlation rises from lower left to upper right: larger values of one variable tend to go with larger values of the other. In the study data, students who studied longer tended to score higher.

A negative correlation falls from upper left to lower right: larger values of one variable tend to go with smaller values of the other. A runner's finishing time may tend to fall as the number of training weeks rises. “Negative” describes the direction, not whether the outcome is good or bad.

With no correlation, there is no upward or downward straight-line trend. For example, someone's birthday day number gives no useful prediction of their height. Look for a curve before deciding there is no relationship at all: a curved pattern can still link the two measurements closely.

Correlation describes a tendency across the group. It need not hold between every two points. In the study data, one student studied 2 hours and scored 61; another studied 3 hours and scored 60. That small reversal fits within the scatter of a positive trend.

Judge strength by the scatter

In a strong correlation, points lie close to a straight line. In a weak correlation, they are widely scattered but a trend remains visible. Moderate describes the middle ground. These judgments are made by eye here; a full description should explain what you see.

If every point lies exactly on an upward-sloping line, the correlation is perfect positive. If every point lies exactly on a downward-sloping line, it is perfect negative. A horizontal row of points has no upward or downward trend: the response stays constant as the other variable changes.

A steep line does not mean a strong correlation. A gentle slope with points tightly grouped around it is strong. A steep slope with points far from it is weak. Slope measures a rate of change; strength describes how consistently the points follow the pattern.

Changing an axis scale can make the same data look tall, flat, crowded, or spread out. It changes the picture, not the strength of the relationship. When comparing plots, read the scales and compare the scatter around the trend with the overall spread of the data.

Put the description in context

“There is a strong positive correlation: students who studied for more hours tended to get higher scores” gives direction, strength, and meaning. Saying “the graph goes up” misses which quantities are changing. Saying “every student scores higher after more study” claims more than a group pattern shows.

Two plots can both rise yet have very different shapes. The form of a pattern tells you whether a straight line describes it or whether the rate of change bends.

Follow the whole shape

A linear pattern follows a straight line. The points may sit on either side of it, but there is no clear bend through the cloud. A nonlinear pattern follows a curve. For example, a puppy's mass rises quickly at first and then more slowly. The pattern increases, but its changing rate makes it curved.

A thrown ball's height rises and then falls as time passes. The points form an arch. Calling the whole pattern positive would ignore the falling part; calling it negative would ignore the rising part.

Consider these pairs:

The vertical values fall toward zero, then rise again. The result is a U shape. One straight line cannot follow both arms. Yet every listed horizontal value has a clearly matched vertical value, so the absence of an upward or downward straight-line trend does not remove the relationship.

Use words that fit the whole plot: “The pattern curves downward and then upward,” or “The values increase quickly at first, then level off.” A line fitted to only one part can hide what happens elsewhere.

Look for groups and empty ranges

A cluster is a group of points close together, with a gap separating it from other points. On a height-and-mass plot containing 6-year-olds and 14-year-olds, two clusters may reflect the two age groups. Ask what the groups represent before explaining an overall trend.

Two clusters can create the appearance of a positive trend just because one lies higher and farther right. Within either cluster, the points might show little clear trend. Describe both the whole plot and the pattern within each group. An overall relationship need not describe comparisons among members of one group.

A gap is an empty range with no observations. It gives you less evidence about what happens there. A model may connect the regions on either side, but there are no measurements in the gap to check that connection.

Build a full description

Start with direction, then form and strength. Add unusual features with their locations. For example: “The points form two groups. The group farther right also lies higher, giving an overall positive pattern. Within each group there is no clear upward or downward trend.” This is more informative than a single label for the whole cloud.

Choose a straight-line model only after inspecting the form. A clear curve calls for a description of that curve and, where useful, a smooth curve of best fit.

An outlier is a point far from the overall pattern. Look at its position compared with the rest of the cloud, then ask what its two measurements mean together.

Judge the pair, not just one value

The main study pattern suggests a score of about 53 for 1 hour of study. Now add one student with the pair $(1,78)$. A score of 78 is ordinary among the higher scores, and 1 hour is ordinary among the shorter times. Together, though, they sit far above the trend. That unusual combination makes the point an outlier.

A point at the edge of the plot is not necessarily an outlier. A student who studied 8 hours and scored 88 would continue the same upward pattern. The horizontal value is beyond the original range, but the pair follows the trend. Distance from the pattern matters more than distance from the middle of an axis.

To describe an outlier, name its values and explain its position: “One student studied 1 hour but scored 78, well above the scores suggested by the main trend.” This tells someone which point to inspect and why it stands out.

Check what happened

An unusual point may come from a recording error, a measurement problem, an unusual individual, or a different group. A score could have been mistyped; a student could have been ill on the test day. These are possible explanations to check, not facts supplied by the graph.

First compare the entry with the original record and check how it was measured. Correct a confirmed error. If the point is genuine, keep it as part of the recorded data. Its difference may reveal something useful about the group or the way the measurements were collected.

You may leave an outlier out when placing a line through the main pattern. Say which point you left out of the fit and why. Keep this decision separate from deleting the observation: the reader still needs to know that the point exists.

Consider its effect

An outlier can make the pattern look weaker. It can also pull a fitted line toward itself, moving the predictions away from the rest of the points. Compare the main cloud with the unusual point before deciding where a line should go.

The original ten study times add to 40 hours and the scores add to 680. Their means are 4 hours and 68 points. Adding $(1,78)$ gives eleven observations, so the new means are:

The mean point moves left and upward, toward the added point. One unusual observation has changed the anchor for a line, even though the other ten pairs stayed the same.

When points follow a roughly straight-line trend, a line of best fit gives a simple model of that trend. It is also called a line of fit or trend line. It runs through the middle of the cloud, allowing the data points to lie on either side.

Place one balanced line

Use a ruler and follow the overall direction of the points. Place the line so that there are roughly as many points above as below, at reasonably similar distances along its length. Look near both ends as well as the middle. Extend the line across the range of the data.

The line need not pass through any particular observation. Joining the first and last points lets just two individuals decide the fit. Forcing a line through the origin is also a mistake unless the data support that position. A zigzag joining every point follows the separate observations, not the overall trend.

Balancing counts is useful, but count alone is not enough. Five points just above a line and five far below it would still leave a poor balance of distances. Compare how far the points lie from the line in the vertical direction.

Use the mean point

The mean point combines the mean of the horizontal values with the mean of the vertical values:

For the ten study pairs, the hours add to 40 and the scores add to 680. Divide each total by 10:

Plot that point as an anchor. A well-placed line passes through or very close to it. When a question gives the mean point or asks you to calculate it, draw the line through it.

For the study data, a line through $(4,68)$ can also pass through $(0,48)$ and $(6,78)$. It has five data points above and five below. The point $(0,48)$ is a point on the line, even though it is not one of the ten student records.

Check vertical errors

A residual is the actual vertical value minus the value predicted by the line. At 1 hour the line predicts 53, but the student scored 55. The residual is $55-53=2$ points, so the point is above the line. At 2 hours, the score 56 has residual $56-58=-2$, so it is below.

For this line the ten residuals are $2,-2,3,-3,-2,2,-2,1,-2,3$. They add to zero and are small on the score scale. When choosing between candidate lines, prefer the one with smaller, better-balanced vertical errors.

Lines drawn by eye can differ slightly while still fitting reasonably. Their predictions are estimates. Check an outlier before letting it change the line's position, and explain if you fit only the main cloud.

Match the model to the form

A straight line suits a linear pattern. With no trend it tells you little, and with a clear curve it misses the bend. A smooth curve of best fit can follow a curved pattern, using the same idea of balancing points on either side.

Once you have drawn a line of best fit, its equation makes predictions easier to calculate.

Choose points on the line

Pick two points that lie on the fitted line, far apart and easy to read. They do not have to be observed data points. Taking two observations away from the line would give a different line's equation.

For the study line, choose $(0,48)$ and $(6,78)$. Find the change in score and divide by the change in hours:

The line crosses the vertical axis at 48 when $x=0$, so $b=48$. In slope-intercept form:

Here $x$ is study time in hours and $y$ is the predicted score in points. The slope of 5 means each extra hour is associated with about 5 more predicted points. It describes the group trend; individual scores can fall above or below that prediction.

Interpret a negative slope

A runner records these pairs of training weeks and finishing time in minutes: $(1,34)$, $(2,33)$, $(3,31)$, $(4,30)$, $(5,29)$, $(6,28)$, $(7,27)$, $(8,25)$.

A fitted line passes through $(0,35)$ and $(10,23)$. Its slope is:

The equation is $y=-1.2x+35$. Each additional training week is associated with a finishing time about 1.2 minutes shorter. That change is 72 seconds. Keep the units with the slope: minutes per week, not weeks per minute.

The line is an estimate by eye. At the mean training time, 4.5 weeks, it predicts $-1.2(4.5)+35=29.6$ minutes. This is very close to the mean recorded time, 29.625 minutes, which supports its placement.

Find and interpret the intercept

The y-intercept is the predicted response at $x=0$. For the runner, 35 minutes estimates a finishing time before training, just outside the observed weeks. For a model predicting adults' shoe size from height, a height of zero has no useful meaning.

If the horizontal axis begins at 30, the line's crossing of the drawn vertical axis gives its value at $x=30$. That is not the y-intercept. Find $b$ by substituting one line point into $y=mx+b$ and solving for it.

Read values, not squares

On axes with 1 hour per horizontal square and 5 points per vertical square, one square up for one square across means 5 points per hour. Counting equal-looking squares would give a wrong slope of 1. Always subtract the actual coordinate values, keeping the same order in the numerator and denominator.

A line of best fit predicts a typical response for a chosen explanatory value. Use the graph or equation, then judge the estimate using the data range, scatter, and situation.

Read from the line

To predict a vertical value from a horizontal value, start at the horizontal axis. Move straight up to the line, then across to the vertical axis. Read the line's value, not the nearest observation: the observation describes one individual.

To estimate the horizontal value for a chosen response, reverse the route. Start on the vertical axis, move across to the line, and then down to the horizontal axis. Check the units at the end so a score does not become a number of hours.

Calculate in either direction

The study model is $y=5x+48$. At 3.5 hours:

The predicted score is about 66 points. The equation's decimal result is exact for that line, but the line is only a model of the observations.

For a predicted score of 70, put 70 in place of $y$ and solve:

The line suggests about 4.4 hours. Substituting 70 for $x$ would treat a score as a study time and answer a different question.

Check the data range

The observed study times run from 1 to 6 hours. Predicting at 3.5 hours is interpolation, because it stays inside that range. Predicting at 12 hours is extrapolation, because it goes outside.

At 12 hours the study line gives $5(12)+48=108$ points. The test is out of 100, so that prediction is impossible. The runner's line gives $-1.2(30)+35=-1$ minute at week 30, which is also impossible. These results show where continuing the line stops making sense.

Judge reliability

Interpolation is usually more dependable when the points show a strong correlation and there are many relevant observations. Wide scatter means individuals vary greatly around the line, even within the observed range. A large empty gap also leaves that part of the pattern unchecked.

Check who or what the data describe. A line based on one group may work poorly for a different group. More observations help when they are relevant to the prediction; a large count cannot fix using the wrong population.

Report a sensible rounded estimate with units and a word such as “about.” Give a reason when judging reliability: “The input is inside the observed range, and the points lie close to the line” is more useful than saying an answer is reliable just because a calculation produced it.

A scatter plot can show that two measurements tend to change together. Explaining why they change is a different question. Correlation describes an association; causation means that changing one variable produces a change in the other.

Consider a third variable

Suppose ice cream sales and sunburn cases both rise on hot, sunny days. The plot may show a positive correlation. Hot, sunny weather can explain both: more people buy ice cream, and more people spend time in strong sunlight. Buying ice cream is not the cause of sunburn.

A third factor that helps explain both measurements is called a lurking variable or confounding variable. A correlation can remain strong even when its apparent explanation is wrong.

The claim “Towns with more libraries have more crime, so libraries cause crime” overlooks town size. Larger towns have more people, more libraries, and more crimes. Comparing raw counts across towns mixes the relationship of interest with a difference in population.

Check the direction of a possible cause

More tutoring may be associated with lower grades. One explanation is that students with lower grades receive more tutoring. The need for help comes first. Treating that pattern as evidence that tutoring lowers grades reverses this possible direction of cause and effect.

This is called reverse causation. Putting tutoring on the horizontal axis does not settle which way a cause runs. An explanatory axis names the variable used for prediction, not a proven cause.

Allow for chance and hidden patterns

A few points can happen to line up. An apparent trend in four or five observations may disappear when more data are collected. A small sample gives less evidence that a pattern holds more widely.

Likewise, a weak straight-line trend does not settle whether there is any relationship. The pattern could curve, another variable could hide it, or the sample could be too small to reveal it. Inspect the graph and ask what was measured before explaining its cause.

Match the claim to the evidence

Correlation is useful for prediction and for suggesting questions to investigate. A causal claim needs stronger evidence. In a fair experiment, one variable is changed on purpose while other factors are kept the same. This helps separate the effect of that change from competing explanations. A sensible account of how the change could work also matters.

For an observed study pattern, write “Students who studied more tended to score higher” or “Study time was associated with score.” Those sentences describe the measurements. “Each extra hour makes every student gain 5 points” adds both a causal claim and an individual guarantee that the plot cannot support.

A bivariate investigation starts with a question about two numerical variables in a named group. The measurements, graph, and conclusion should all answer that same question.

Ask a question you can measure

“Do students in our grade who sleep more on school nights tend to have faster reaction times?” names both measurements and the group. “Is sleep good?” is too broad for a single scatter plot. It leaves “good” undefined and does not say who will be measured.

Plan how to record both values for each person. Use the same measurement method for everyone and keep the units in the table. For the sleep question, record hours and reaction time for each participating student in the same row. Decide how students will be chosen so that you can describe whose results you have.

Collect enough observations to see whether a pattern is consistent. Four or five points can line up by chance. Measuring only a few friends may also leave out important differences within the grade. Record who participated and how they were chosen; that information shapes the conclusion.

Analyze the paired data

Choose the axes, draw the scatter plot, and describe direction, form, strength, and unusual features. Check suspicious points against the original records. If you leave an outlier out of a fitted line, record that decision and its reason.

Draw a line by eye when the pattern is linear. Use it to estimate a value within the range and explain what the estimate describes. If the pattern bends, describe the curve instead of forcing a straight line through it.

A spreadsheet can speed up the plotting, but inspect its result. Make sure it used a scatter plot, kept each row paired, and assigned the intended horizontal and vertical columns. Check the axis scales and units before interpreting an automatically drawn picture.

Write a conclusion with evidence

For the ten-student study example, a conclusion could read:

“For these 10 students, longer study times were associated with higher test scores. The plot shows a strong positive linear correlation. A fitted line, score ≈ 5 × hours + 48, predicts about 66 points for 3.5 hours of study.”

Then state the limits that matter:

“The recorded times cover 1 to 6 hours. Predictions beyond that range are less reliable. The small group may not represent all students, and other differences between students could help explain the pattern.”

The first part answers the question and uses the graph. The second tells a reader how far the findings reach. Include any outlier treatment and avoid claiming cause from an observed association.

Read other people's claims

When a report presents a scatter plot, check the axis labels, scales, and source of the data. Ask which people or objects were measured and whether the headline matches the pattern. If it claims cause and effect, look for evidence beyond the correlation.