Scatter Plots
A scatter plot shows pairs of data as dots. The pattern of the dots tells you how the two things are linked.
What you'll be able to do
- Pick a scale and plot data as a scatter plot.
- Say if an association is positive, negative or none.
- Say if it is linear or not, and strong or weak.
- Find outliers and clusters.
The big idea
One dot for each pair. Each pair of data, like age and price, is one dot. x goes across and y goes up.
Direction. If y tends to go up as x goes up, the association is positive. If y tends to go down, it is negative.
No association. If the dots are spread out with no pattern, there is no association.
Form. If the dots lie near a line, the association is linear. If they follow a curve, it is nonlinear.
Strength. Dots close to a line or curve show a strong association. Dots spread wide show a weak one.
Outliers. An outlier is a dot far from the pattern of the others. It can be inside the range of the data.
Clusters. A cluster is a group of dots close together. A gap comes before the next group.
Pick a scale. Divide the biggest value by the number of squares. Round up to a step like 2, 5, 10, 25 or 50.
Words you need
- Scatter plot
- A graph of data pairs shown as dots.
- Association
- A link between two sets of data. As one changes, the other tends to change.
- Positive association
- y tends to go up as x goes up.
- Negative association
- y tends to go down as x goes up.
- Outlier
- A dot far from the pattern of the other dots.
- Cluster
- A group of dots close together.
- Linear
- Close to a straight line.
Ways to describe a scatter plot
| Question | Words to use |
|---|---|
| Which way? | Positive, negative or none |
| What shape? | Linear or nonlinear |
| How close? | Strong or weak |
| Anything odd? | Outliers and clusters |
Worked examples
Example 1
x is hours of sun a day. y is a tomato plant's height in cm. Data: (4, 30), (5, 36), (6, 41), (7, 47), (8, 55), (9, 58). Describe it.
- As x goes up, y goes up. So the association is positive.
- The dots lie close to a line. So it is linear and strong.
- No dot is far from the others. There is no outlier.
Answer: Positive, linear and strong
Example 2
x is minutes walked. y is steps. Data: (10, 1100), (15, 1650), (20, 2100), (30, 3300), (35, 3700), (40, 4400). The x-axis has 8 squares and the y-axis 10. Make a scatter plot.
- x: 40 ÷ 8 = 5, so each square across is 5 minutes.
- y: 4400 ÷ 10 = 440. Round up to 500 steps a square.
- Plot each pair. (10, 1100) is 2 squares across and 2.2 squares up.
- The dots rise along a line: positive and linear.
Answer: x: start at 0, 5 a square. y: start at 0, 500 a square. The association is positive, linear and strong
Example 3
x is a sneaker's price in dollars. y is pairs sold in a week. Data: (40, 52), (50, 45), (60, 39), (70, 30), (80, 26), (90, 60), (100, 14). Find the outlier.
- The pattern is negative: pricier pairs sell less.
- (90, 60) is far above the line the other dots make.
- So (90, 60) is the outlier.
Answer: Negative, linear and strong, with an outlier at (90, 60)
Common mistakes
Thinking positive means the numbers are positive. Positive means y tends to go up as x goes up. The numbers can be below zero.
Calling the biggest value the outlier. An outlier is far from the pattern, not just big.
Calling a curve linear. Check if the dots bend. A curve is nonlinear.
Reading shape from an uneven axis. The steps on an axis must be equal, like 0, 2, 4, 6.
Practice: 22 questions
17 of 22 are word problems. For a word problem, sketch it first and mark what you know.
Tier 1 Foundations Use the rule directly
- The plot shows used scooters. x is age in years. y is price in dollars. Describe the association. Word problem
- The plot shows shoe size x and quiz score y for 8 students. Is there an association? Word problem
- The plot shows arm span x and height y in cm for 8 people. Which point is the outlier? Word problem
- The plot shows 9 eruptions of a geyser. x is the eruption's length in minutes. y is the wait for the next one. Describe the clusters. Word problem
- A freezer door is left open. x is minutes. y is the temperature inside in °C. Gia says the association is negative, because the temperatures are below zero. Find her mistake. Word problem
- x is hours of a train trip. y is km from home. Data: (1, 70), (2, 150), (3, 210), (4, 290), (5, 360). The y-axis has 10 squares. Which scale fits: 20, 25 or 50 km a square? Plot the points and describe them. Word problem
Tier 2 Intermediate One more step to think through
- The plots A, B and C show three data sets. Which shows a negative nonlinear association? Describe the other two.
- x is a car's speed in km/h. y is its stopping distance in m. The plot shows 7 cars. Describe the association. Word problem
- One grid shows two kinds of saplings. x is months since planting. y is height in cm. Oaks are navy dots. Pines are orange dots. Which association is stronger? Word problem
- Ravi's x-axis is labeled 0, 2, 4, 8, 16 with equal spaces. He plots (2, 5), (4, 9), (8, 17), (16, 33). He says the points curve, so the association is nonlinear. Find his mistake.
- x is a dog's mass in kg. y is food in grams a day. Data: (12, 240), (18, 330), (25, 420), (30, 490), (40, 610), (45, 660). The x-axis has 8 squares. The y-axis has 10. Pick a start and a step for each. Word problem
- x is a town's distance from the coast in km. y is its yearly rain in cm. Data: (120, 60), (20, 95), (75, 72), (5, 102), (150, 51), (40, 88), (100, 66). Is the association positive or negative? Word problem
Tier 3 Advanced Several steps - find the hidden structure
- x is the day's high in °C. y is swimmers at a pool. The plot shows 9 days. Describe the plot fully. Word problem
- x is km driven on a car tire. y is tread depth in mm. Data: (0, 8.0), (1000, 7.2), (2000, 6.5), (3000, 5.6), (4000, 4.9), (6000, 3.3). Which reading at 5000 km fits the pattern: 3.5, 4.1 or 4.7 mm? Word problem
- x is hours worked a week by teens. y is money earned. Ming says there is no association because there are two groups. Find her mistake. Word problem
- x is a pizza's width in inches. y is its price in dollars. The plot shows 7 pizzas. Remove the outlier. Is the association stronger or weaker? Word problem
- Write 5 data points that show a negative, nonlinear association. Say what x and y could be. Here is one try: tea cooling, with x in minutes and y in °C. Check (0, 90), (5, 72), (10, 59), (15, 50), (20, 43). Word problem
Tier 4 Challenge Stretch problems, some above grade level
- x is the hour of the day. y is the temperature in °C. The plot shows 8 hours. Kim says there is no association. Is she right? Word problem
- x is km from home to school. y is minutes to get there. Ola says there is no outlier, because every value is inside the others' range. Is she right? Word problem
- The points (1, 9), (2, 7), (4, 6), (5, 3), (7, 2) show a negative association. Swap x and y in every point. What is the new association?
- Plot A and Plot B show the same 5 points. Plot B is drawn twice as tall. Which shows the stronger association?
- The number r goes from −1 to 1. Near 1 is strong positive. Near −1 is strong negative. Near 0 is none. Match plots A, B and C to r = 0.95, r = −0.6 and r = 0.05. Above grade level
Answers
Tier 1 Foundations
- Negative, linear and strong. As x goes up, y tends to go down. The points lie close to a line.
- No association. As x goes up, y goes up and down with no pattern.
- The outlier is (168, 152). As x goes up, y tends to go up. The points lie close to a line. (168, 152) is far from the trend of the other points.
- Two clusters. Short eruptions, near 2 minutes, have waits near 55. Long ones, near 4 minutes, have waits near 80. As x goes up, y tends to go up. The points lie close to a line. A big gap in x splits the points into groups.
- Positive, linear and strong. Positive is about the pattern, not the sign of the numbers. The temperature rises as the minutes go up. As x goes up, y tends to go up. The points lie close to a line.
- 50 a square. The association is positive, linear and strong. 10 squares × 50 = 500, which reaches 360. 10 × 20 = 200, 10 × 25 = 250 are too short.
Tier 2 Intermediate
- A: positive, linear and strong. B: negative, nonlinear and strong. C: no association. A: As x goes up, y tends to go up. The points lie close to a line. B: As x goes up, y tends to go down. The points follow a curve, not a line. C: As x goes up, y goes up and down with no pattern.
- Positive, nonlinear and strong. As x goes up, y tends to go up. The points follow a curve, not a line.
- Oaks. The points for Oaks sit closer to a line. Both are positive.
- Positive, linear and strong. His axis steps are 2, 2, 4 and 8, not equal. On an even axis the points lie on a line. As x goes up, y tends to go up. The points lie close to a line.
- x: start at 10, 5 a square. y: start at 200, 50 a square. The association is positive, linear and strong. x: 10 + 8 × 5 = 50, which covers 12 to 45. y: 200 + 10 × 50 = 700, which covers 240 to 660.
- Negative, linear and strong. Sort by x first. Then the rain goes 102, 95, 88, 72, 66, 60, 51: down as the distance goes up.
Tier 3 Advanced
- Positive, linear and strong, with an outlier at (30, 30). As x goes up, y tends to go up. The points lie close to a line. (30, 30) is far from the trend of the other points.
- 4.1. The trend of the other points gives about 4.1 at 5000. 4.1 is closest.
- Positive, linear and strong, in 2 clusters. Two clusters can still show an association. Across both groups, more hours bring more money. As x goes up, y tends to go up. The points lie close to a line. A big gap in x splits the points into groups.
- The outlier is (18, 12). Without it, the association is stronger. The other points lie close to a line, so removing the point far from it makes the pattern clearer.
- The tea data work: they fall fast, then more slowly. Many answers work. As x goes up, y tends to go down. The points follow a curve, not a line.
Tier 4 Challenge
- A strong nonlinear association: it rises, then falls. It is neither positive nor negative. No association means no pattern. These dots follow a clear curve. The dots rise, then fall, close to a curve. So the association is strong but nonlinear. It is neither positive nor negative.
- No. (2.5, 24) is an outlier. An outlier is far from the pattern, not outside the range. (2.5, 24) is a short trip that took a long time.
- Still negative. Swapping keeps each pair together. When one value is big, the other is still small.
- Neither: the association is the same. The data are the same, so the association is the same. A taller picture only looks steeper.
- A is r = 0.95. B is r = −0.6. C is r = 0.05. A: r ≈ 0.97. B: r ≈ −0.59. C: r ≈ 0.06. The closer to 1 or −1, the closer the points are to a line.
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