Describing a scatter plot
The dots are on the plot. What does the whole cloud say about how the two measurements go together, and what can it not tell you?
Where you are 2 in Scatter plots
When you reach for it
Plotting paired data: explanatory and response variables turns each individual into one dot. Once the dots are down, somebody wants to know what the plot shows: do thicker trees tend to be taller? Do colder months bring bigger heating bills? The answer is a few sentences about the two variables, in their own units, so that someone who never saw the plot knows what it showed.
You reach for this whenever a scatter plot of Bivariate data is in front of you, before drawing a line through it or claiming anything from it. The description has a fixed set of parts: the overall pattern (its direction, form and strength), then anything that breaks the pattern (clusters and outliers), then a limit on what it means: two variables that go together need not affect each other. The lesson takes two days, the overall pattern first and the rest second.
The procedure
Part 1: direction, form and strength. Two variables are associated when knowing one tells you something about the other. On a scatter plot that shows as a pattern in the cloud, and three words sum it up.
Direction: which way does the cloud run? Read from left to right. If the dots tend to climb, the association is positive: larger values of one variable go with larger values of the other. If they tend to fall, it is negative: larger values of one go with smaller values of the other. If they neither climb nor fall, there is no association: knowing the value across does not help you guess the value up. Direction belongs to the whole cloud, so a positive association allows a few pairs in which the dot further right is lower. That is what “tend” means.
Form: what path does the cloud follow? If the dots run roughly along a straight line, the form is linear: the variable up changes by about the same amount for each equal step across. If they follow a bend, the form is curved (nonlinear): the change for each step grows or shrinks as you move across. A ruler from the leftmost dots to the rightmost is a fair test. Dots on both sides of it all the way along mean straight; dots on one side all through the middle mean the path bends.
Strength: how closely do the dots follow that form? Picture a band drawn round the cloud, along its form. A thin band means the association is strong: knowing one variable lets you guess the other closely. A fat band means it is weak: the pattern is there, but a guess could be far off. In between is moderate. Strength is a judgment made by eye, not a measurement, and it is separate from direction: a negative association can be strong and a positive one weak.
Say it in context. The three words are a summary. The description is a sentence that names both variables: “trees with thicker trunks tend to be taller” says what “positive” means here. Quote a number from the plot, with its unit, when it sharpens the point.
Part 2: clusters, outliers, curves and causes.
Clusters. Sometimes the dots fall into separate groups with empty space between them. Each group is a cluster, and clusters usually mean the individuals are of different kinds. Say where each cluster sits and what it holds, and note that the plot shows nothing about the empty space between.
Outliers. A dot far from the pattern the rest follow is an outlier. It is judged against the pattern, not one variable at a time: a tree with an ordinary trunk and an ordinary height can be an outlier, because that height is unusual for that trunk. Say where it is and what it might mean in context, then leave it on the plot. An outlier is a real individual, and deleting it because it spoils the pattern would make the plot claim more than the data show. If it might be a recording slip, check the record, and change the dot only if a slip is found.
Curves. A curve is described like a straight pattern, plus how it bends: “wave height rises with wind speed, slowly in light wind and faster in strong wind”. A curve can be strong, since strength is about how closely the dots follow the curve. A path that climbs and then falls has no single direction, so describe each part.
Association is not causation. A scatter plot shows that two variables go together, never why. Among children aged 5 to 12, those with bigger feet tend to read at higher levels, but growing feet do not teach anyone to read: both grow with age. A third variable like age, which drives both plotted variables, can produce a strong association between two things that do not affect each other at all. Showing that one variable changes the other takes more than the plot: an experiment, or a known way in which one acts on the other (Correlation is not causation: lurking and confounding variables, Observational studies and experiments).
Worked examples
Examples 1 to 4 go with the first part of the lesson, Examples 5 to 7 with the second. All the data are made up.
Trunks and heights in a park
A park keeper measured ten trees: the circumference of each trunk (the distance round it, at chest height) and the height of the tree. As (circumference in cm, height in m) the trees are (60, 8), (75, 9), (85, 11), (100, 12), (110, 14), (120, 13), (135, 16), (150, 17), (160, 19) and (175, 20). Describe the association.
Direction. Reading left to right, the dots climb. The thinnest trunk, 60 cm, belongs to an 8 m tree, and the thickest, 175 cm, to a 20 m tree. Not every step climbs: the tree at 120 cm is 13 m tall, a meter shorter than the tree at 110 cm. One dip does not turn the cloud round, which is why we say thicker trunks tend to go with taller trees. The direction is positive.
Form. Compare two equal steps across. From 60 to 110 cm the height rises from 8 to 14 m, that is 6 m. From 110 to 160 cm, another 50 cm, it rises from 14 to 19 m, that is 5 m. Equal steps across give about equal rises, and a ruler from the first dot to the last has dots on both sides all along it. The form is linear.
Strength. Every dot lies within about 1.5 m of that ruler line, on a plot whose heights span 12 m. The band round the cloud is thin, so the association is strong.
In context: for these ten trees there is a strong, positive, linear association between trunk circumference and height. Trees with thicker trunks tend to be taller, and height climbs at a steady rate as the trunks get thicker.
Heating bills through the winter
A family recorded, for each month from October to May, the average outdoor temperature and the home’s heating bill for that month.
| Month | Temperature (^\circC) | Heating bill (dollars) |
|---|---|---|
| October | 12 | 70 |
| November | 6 | 125 |
| December | 1 | 165 |
| January | -4 | 205 |
| February | -2 | 190 |
| March | 3 | 145 |
| April | 8 | 105 |
| May | 14 | 50 |
Temperature goes across, since the heating responds to the weather. Describe the association.
The plot is arranged by temperature, not by month. January, the coldest month at -4\,^\circ\text{C}, is the leftmost dot, and May, at 14\,^\circ\text{C}, the rightmost.
Direction. From left to right the dots fall: the coldest month has the biggest bill, 205 dollars, and the warmest the smallest, 50 dollars. The direction is negative, which names a direction only, not a judgment that the pattern is bad.
Form. From -4 to 6\,^\circ\text{C}, a rise of 10 degrees, the bill falls from 205 dollars to 125 dollars, that is 80 dollars, or 8 dollars a degree. From 6 to 14\,^\circ\text{C}, 8 degrees, it falls from 125 dollars to 50 dollars, that is 75 dollars, or a little over 9 dollars a degree. About the same drop for each degree all the way across: the form is linear.
Strength. The dots sit almost exactly on a straight line. Strong.
In context: there is a strong, negative, linear association between the month’s outdoor temperature and the heating bill. The colder the month, the higher the bill, by roughly the same amount for each degree colder.
Screen size and price
A shop sells the thirteen TVs plotted below. Screen size is measured across the diagonal, in inches. Describe the association between screen size and price.
The sets come in pairs at 32, 40, 50, 55, 65 and 75 inches, with one more at 43 inches.
Direction. The smallest sets, at 32 inches, cost 250 and 500 dollars; the largest, at 75 inches, cost 850 and 1{,}300 dollars. The cloud climbs: positive.
Form. A ruler through the middle of the cloud, from about 375 dollars at 32 inches to about 1{,}075 dollars at 75 inches, has dots above and below it all the way along, with no bend. Linear.
Strength. At every size the two prices are far apart. At 50 inches they are 400 and 800 dollars, and the 40-inch set at 650 dollars costs more than the 50-inch set at 400 dollars. So the band round the cloud is wide. Yet the climb is still clear: no set under 45 inches costs more than 650 dollars, and no set of 65 inches or more costs less. A clear trend in a wide band is a moderate association.
In context: there is a moderate, positive, linear association between screen size and price. Bigger screens tend to cost more, but sets of the same size differ widely in price, so size explains only part of what a set costs.
Does it matter where a tree stands?
The park keeper of Example 1 also measured how far each of the same ten trees stands from the park gate. As (distance in m, height in m) they are (40, 14), (70, 16), (110, 19), (150, 12), (190, 8), (230, 17), (260, 11), (300, 9), (340, 13) and (380, 20). Describe the association between distance from the gate and height.
The dots are spread over the plot with no climb and no fall. The tallest tree, 20 m, is the farthest from the gate, but the next tallest, 19 m, is the third nearest, and the shortest, 8 m, stands in the middle. Split the trees into the five nearest the gate and the five farthest. The near five average
and the far five average
Almost the same. There is no association, and with no direction there is no form or strength to name.
In context: knowing how far a tree stands from the gate tells you nothing about how tall it is. These are the same ten heights as in Example 1, so whether there is an association depends on the pair of variables, not on the individuals alone.
A second shop: clusters and an outlier
A second shop sells the fourteen TVs plotted below. Describe the plot.
Clusters. The dots fall into two groups with empty space between them. Seven sets from 24 to 43 inches sit low on the left, priced from 120 to 340 dollars. Six sets from 65 to 85 inches sit high on the right, priced from 900 to 1{,}900 dollars. No set lies between 43 and 65 inches: the shop stocks small sets and large sets and nothing in between. The plot says nothing about the price of a 55-inch set, so it cannot show whether the pattern across the gap is straight or bent.
Direction. Within each cluster price climbs with size, and the large cluster sits far above the small one. The association is positive.
Outlier. One dot, a 32-inch set at 1{,}200 dollars, sits far above the small cluster. The other 32-inch sets cost 180 and 220 dollars, while 1{,}200 dollars lies between the prices of the 70- and 75-inch sets here. Its size is ordinary and so is its price, for this shop; the pair together is what is unusual. It might be built to stand outside in rain and sun, or it might be a typing slip for 120 dollars. The shop’s own list would settle which. Until then the dot stays on the plot, and the description says “apart from one 32-inch set priced at 1{,}200 dollars”.
In context: prices rise with screen size; the sets form a cluster of small, cheap sets and a cluster of large, expensive ones with no sizes between; and one small set is priced like a large one.
Wind and waves
A lifeguard recorded the wind speed and the height of the waves at noon on thirteen days. As (wind speed in km/h, wave height in m) the days are (5, 0.1), (8, 0.1), (12, 0.2), (15, 0.2), (18, 0.3), (22, 0.5), (25, 0.6), (28, 0.8), (32, 1.0), (35, 1.2), (38, 1.5), (42, 1.8) and (45, 2.0). Describe the association.
Direction. The dots climb from left to right: positive.
Form. Compare two equal steps of 20 km/h. From 5 to 25 km/h the waves grow from 0.1 to 0.6 m, that is 0.5 m. From 25 to 45 km/h they grow from 0.6 to 2.0 m, that is 1.4 m, nearly three times as much. The ruler test agrees: a ruler from the first dot (5, 0.1) to the last (45, 2.0) is halfway up, at 1.05 m, when the wind is 25 km/h, while the dot there is at 0.6 m. The middle dots sit below the ruler and the end dots on it. The form is curved, bending upward.
Strength. The dots lie along a smooth bend with hardly any scatter about it. Strong.
In context: there is a strong, positive, curved association between wind speed and wave height. Waves grow slowly in light wind and faster and faster as the wind strengthens. A straight line through this cloud would give waves too high in moderate wind and too low in light and strong wind.
Big feet, good readers?
A primary school recorded, for fourteen children aged 5 to 12, their shoe size (EU sizes) and their reading level on the school’s scale from 1 to 10, where a higher level means harder books. Describe the association, and say whether the plot shows that bigger feet make better readers.
Description. The dots climb from (27, 1) at the lower left to (38, 8) at the upper right, along a straight path in a thin band. There is a strong, positive, linear association: children with bigger shoe sizes tend to read at higher levels.
Cause. The plot shows only that the two go together. It cannot say why, and here a third variable explains it: age. A 12-year-old has had seven more years than a 5-year-old to grow, so has bigger feet, and seven more years of reading, so reads harder books. Age drives both, and the two rise together without either one acting on the other. Buying a child bigger shoes would not change their reading level at all.
A fairer look would compare children of the same age. If, among the 8-year-olds alone, feet and reading show no association, then age was the whole link. So the plot supports “bigger feet go with higher reading levels in this school”, and nothing about cause.
Boundaries
- Both variables must be numbers, and paired. Direction, form and strength need a scatter plot, which needs two numerical measurements on the same individuals (Plotting paired data: explanatory and response variables). Two categorical variables show an association in a different way (Association in a two way table and segmented bar graphs).
- Strength is judged by eye here. “Strong”, “moderate” and “weak” have no fixed cut-offs, and two careful people can differ on a cloud near the border. A number for the strength of a linear pattern comes later, with Correlation.
- Direction needs a pattern that keeps going one way. A path that climbs and then falls has no single direction, and calling it “no association” would hide a clear pattern.
- A description holds only over the values plotted. The heating bills of Example 2 fall steadily from -4 to 14\,^\circ\text{C}; nothing on that plot says what happens in a hot summer month, and nothing on the plot of Example 5 says what a 55-inch set costs.
- An outlier is judged against the pattern. A dot ordinary in each variable separately can still be far from the pattern. How an outlier can move a fitted line is the subject of Outliers, leverage and influential points.
- Association is not causation. No scatter plot, however strong, shows that one variable changes the other.
- A handful of dots is a weak basis. With four or five dots, one unusual individual can make a pattern look strong or hide it altogether.
Where this goes next
A strong linear pattern invites a line drawn through the cloud. That is the Line of best fit, which turns the description into predictions: a height for a trunk not yet measured, a bill for a month not yet come. Correlation and The correlation coefficient put a single number on the direction and strength of a linear association, and Outliers, leverage and influential points asks which unusual dots change that line. Curved patterns lead to Linear and nonlinear functions and to Modeling with linear relationships with other shapes. The warning about third variables grows into Correlation is not causation: lurking and confounding variables and the reason experiments are designed the way they are.
Exercises
Exercises 1 to 7 go with the first part of the lesson (Examples 1 to 4), Exercises 8 to 15 with the second (Examples 5 to 7), and the reasoning group draws on both. All data are made up.
Part 1: direction, form and strength
For Exercises 1 to 3, give the direction, form and strength of the plot named above. The plots have no context, so describe the pattern only.
- Plot A.
- Plot B.
- Plot C.
For Exercises 4 to 6, say whether you would expect a positive association, a negative association or none, and why.
- The age of each oak tree in a park, and the circumference of its trunk.
- The outdoor temperature on summer days, and the electricity a home uses that day to run its air conditioning.
- The height of each tree in a park, and the last digit of the number on its tag.
- In Example 1, find the pair of trees in which the tree with the thicker trunk is the shorter one. Does that pair change the direction of the association?
Part 2: clusters, outliers, curves and causes
- Describe Plot D: direction, form and strength, and how it bends.
The park keeper of Example 1 adds an eleventh tree to the plot.
- Which dot is the new tree? Give its circumference and height.
- Neither number is unusual on its own: another tree is also 9 m tall, and two others have trunks within 10 cm of this one. Explain why the tree is still an outlier, suggest what might explain it, and say whether the keeper should delete it from the plot.
- The family of Example 2 adds the summer months: June, 17\,^\circ\text{C} and 25 dollars; July, 21\,^\circ\text{C} and 15 dollars; August, 20\,^\circ\text{C} and 15 dollars; September, 16\,^\circ\text{C} and 30 dollars. Describe the association for all twelve months, using the drop per degree between 14 and 16\,^\circ\text{C} and between 20 and 21\,^\circ\text{C}.
- In Example 6, find how much the waves grow from 15 to 25 km/h and from 35 to 45 km/h. What do the two rises say about the form?
- In Example 1, trees with thicker trunks tend to be taller. Name a third variable that could drive both, and explain how.
- Does the plot of Example 6, on its own, show that wind makes waves? Explain.
- Suppose the school of Example 7 had measured only children aged 5 and 6 and children aged 11 and 12. Describe in words what the scatter plot would most likely look like.
Reasoning
- Explain why “positive, linear, strong” on its own is not a full description of a scatter plot.
- The dots of a plot rise in a clear arch to a peak in the middle and fall again, hugging the arch closely, and the dots at the two ends are at the same height. Explain why “no association” would be a poor description, and give a better one.
- Explain why the strength and the direction of an association are separate judgments, using one example from this lesson in which a strong association is negative.
- Rank Plots A, B and C from the strongest association to the weakest, and say what you looked at.
- The plot of Example 1 is redrawn with the height axis running from 0 to 100 m instead of from 0 to 22 m. The dots now look like a thin, nearly flat line. Has the association become stronger? Explain.
Answers
- Negative, linear, strong. The dots fall steadily in a thin straight band.
- Positive, linear, weak. The cloud rises a little, but in a wide band, so the value across says little about the value up.
- No association. The dots neither climb nor fall, so there is no form or strength to name.
- Positive. A tree adds wood every year, so older oaks tend to have thicker trunks.
- Positive. Hotter days need more cooling, so the air conditioning runs longer and uses more electricity.
- None. A tag number is given out without regard to height, so its last digit tells nothing about how tall the tree is.
- The trees at (110, 14) and (120, 13), the only such pair. It does not change the direction: the cloud as a whole climbs, and direction describes a tendency, not every pair.
- Negative, curved, strong. The dots fall steeply at first and then level off, along a smooth curve with little scatter: the same step across gives a big drop on the left and almost none on the right.
- The dot far below the others at the right: a circumference of 170 cm and a height of 9 m. Trees with trunks of 160 and 175 cm are 19 and 20 m tall.
- The combination is unusual: every other tree with a trunk near 170 cm is about 20 m tall, so a 9 m tree is far below the pattern. Perhaps its top broke off in a storm, or its height was written down wrongly. The keeper should check the record but keep the dot, and describe the plot “apart from one thick-trunked tree only 9 m tall”.
- Negative and curved. Between 14 and 16\,^\circ\text{C} the bill falls (50 - 30) \div 2 = 10 dollars a degree, much as it did all winter; between 20 and 21\,^\circ\text{C} it falls 0 dollars. The bill drops steadily as the months warm, then levels off at 15 dollars, when the heating is off and only the part of the bill that does not depend on the weather is left.
- From 15 to 25 km/h the waves grow 0.6 - 0.2 = 0.4 m; from 35 to 45 km/h they grow 2.0 - 1.2 = 0.8 m. The same 10 km/h step gives twice the rise in stronger wind, so the form is curved, bending upward.
- Age. A tree adds wood to its trunk and height to its crown every year, so older trees tend to be both thicker and taller. (How much light a tree gets could also drive both.)
- No. The plot shows only that wind speed and wave height go together. We do know that wind raises waves, but from how waves are made, by wind pushing on the water, not from the plot. A storm out at sea, for instance, could bring both strong wind and big waves on the same days.
- Two clusters: young children at the lower left, with small shoes and low reading levels, and older children at the upper right, with large shoes and high levels, with empty space between. Within each cluster there would most likely be little pattern.
- It never names the variables, so a reader cannot tell what goes with what, in which units, or for whom; “trees with thicker trunks tend to be taller” carries the meaning. It also says nothing about clusters or outliers.
- The dots follow a clear, strong pattern, so knowing the value across tells you a lot about the value up. Better: “The variable up rises and then falls in a strong, curved pattern, peaking in the middle”: positive on the left half, negative on the right.
- Direction says which way the cloud runs; strength says how closely the dots follow the pattern, whichever way it runs. Example 2 has dots almost exactly on a falling line: strong and negative. Plot B rises but loosely: weak and positive.
- A, then B, then C. The width of the band: A’s dots hug a thin straight band, B’s follow a trend in a wide band, and C shows no trend at all.
- No. The trees and their measurements are the same, so the association is the same. Squeezing the scale squeezes the band and the climb alike, so the cloud only looks thinner. Strength is judged on a plot whose scales fit the data, as in Example 1.