Scatter Graphs And Correlation

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Scatter Graphs

Scatter graphs (or scatter diagrams) plot pairs of data, e.g. Maths grade vs Physics grade.
Points are plotted as crosses (×) and are not joined up.
The vertical and horizontal axes represent the two quantities being measured.
The general shape formed by the points shows the type of correlation.

Positive correlation

Types of Correlation

Positive correlation: as one quantity increases, the other also increases (points go from bottom left to top right).
Negative correlation: as one quantity increases, the other decreases (points go from top left to bottom right).
No (zero) correlation: no apparent relationship (points form a cloud).
Correlations can be weak or strong – the stronger the correlation, the closer the points lie to a straight line.

Correlation vs Causation

Correlation does not imply causation – a relationship does not mean one variable causes the other.
Example: sunflower height and puppy weight both increase over time, but neither causes the other.
Both quantities may be increasing due to a third factor (e.g. time).

Line of Best Fit

A line of best fit is a single ruled straight line drawn by eye on a scatter graph when correlation exists.
It should extend across the full data set, with roughly equal numbers of points on either side.
Ignore outliers (extreme values that don't fit the pattern) when drawing the line.
Use a transparent ruler to help position the line.

Sophie's scatter diagram

Using the Line of Best Fit

The line can be used to predict values (e.g. estimate y when x is given).
Interpolation: predicting within the range of the data – reliable.
Extrapolation: predicting beyond the data range – unreliable and should be avoided.
Example: from a line of best fit, estimate the price of a computer that takes 3.4 seconds to run a program.

Using the line of best fit

Common Exam Questions

Read values from a scatter graph (e.g. number of phones when 42 people are in a building).
Identify the type of correlation from a scatter diagram.
Plot additional points from a table onto an existing scatter graph.
Draw a line of best fit and use it to estimate a value.
Describe the relationship between two variables (e.g. age and value of cars).

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1.What type of correlation is shown when one quantity increases and the other quantity also increases?
Easy
- APositive correlation
- BNegative correlation
- CNo correlation
- DWeak correlation
2.A scatter diagram shows points that go from top left to bottom right. What type of correlation does this show?
Easy
- APositive correlation
- BNegative correlation
- CNo correlation
- DStrong correlation
3.The scatter diagram shows the number of people and the number of phones in each of 8 buildings. One of the buildings contains 42 people. How many phones are in that building?
Medium
- A18
- B22
- C26
- D30
4.What type of correlation is described by the statement: 'We eat more ice cream as the temperature rises'?
Medium
- APositive correlation
- BNegative correlation
- CNo correlation
- DCausation
5.In a scatter graph showing the age and value of cars, the points go from top left to bottom right. Describe the relationship between age and value.
Easy
- AAs age increases, value increases
- BAs age increases, value decreases
- CNo relationship
- DAs age decreases, value decreases
6.A scatter graph shows the distance from the city centre and monthly rent of apartments. Points go from top left to bottom right. What does this tell you?
Easy
- AThe further from the centre, the higher the rent
- BThe further from the centre, the lower the rent
- CNo relationship
- DRent is constant
7.In a scatter diagram of physics marks and chemistry marks, the points show a positive correlation. Which statement is correct?
Medium
- AHigher physics marks tend to have higher chemistry marks
- BHigher physics marks tend to have lower chemistry marks
- CPhysics marks cause chemistry marks
- DThere is no relationship
8.A line of best fit is drawn on a scatter diagram. For a given x-value, how do you estimate the corresponding y-value?
Medium
- ARead up from x to the line, then across to y
- BRead across from y to the line, then down to x
- CFind the mean of all y-values
- DUse the closest data point

更多題目

9.A scatter graph shows the distance driven and time taken for deliveries. The points show a positive correlation. What does this mean?
Easy
- ALonger distances take more time
- BLonger distances take less time
- CDistance and time are unrelated
- DTime causes distance
10.The value of houses in 1996 and their value in 2016 were recorded. The line of best fit through the data passes through the points (10, 20) and (90, 180), where both axes are in thousands of dollars. A house had value $70 000 in 1996. Use the line of best fit to estimate its value in 2016.
Medium
- A$100 000
- B$120 000
- C$140 000
- D$160 000
11.A scatter diagram shows the number of documents checked and the number of errors. The points show a negative correlation. What does this suggest?
Medium
- AMore documents checked leads to more errors
- BMore documents checked leads to fewer errors
- CNo relationship
- DErrors cause documents to be checked
12.A scatter graph shows distance run (km) and average speed (km/h). The line of best fit passes through the points (2, 15) and (10, 7). Estimate the average speed for a run of 6 km.
Medium
- A10 km/h
- B11 km/h
- C12 km/h
- D13 km/h
13.A scatter graph shows the duration (minutes) and cost (£) of flights. The line of best fit passes through the points (60, 30) and (240, 120). One flight cost £90. Estimate its duration.
Medium
- A120 minutes
- B150 minutes
- C180 minutes
- D210 minutes
14.Why should you not use a line of best fit to predict the number of units of electricity used when the outside temperature is 30 °C, if data only goes up to 20 °C?
Hard
- A30 °C is an outlier
- BIt is extrapolation, which is unreliable
- CCorrelation does not imply causation
- DThe line of best fit is only for interpolation
15.A scatter graph shows a positive correlation between the height of a sunflower and the weight of a puppy over time. What can you conclude?
Medium
- ASunflower height causes puppy weight
- BPuppy weight causes sunflower height
- CBoth are increasing due to time
- DThere is no relationship
16.What type of correlation does the scatter diagram show?
Easy
- APositive correlation
- BNegative correlation
- CNo correlation
- DCannot be determined
17."We eat more ice cream as the temperature rises." What type of correlation is this?
Medium
- APositive correlation
- BNegative correlation
- CNo correlation
- DCausation
18.The scatter graph shows the age (years) and the value (£) of cars. Describe the relationship between the age and the value of the cars.
Easy
- APositive correlation
- BNegative correlation
- CNo correlation
- DNon-linear correlation
19.The scatter graph shows the distance from the city centre and the monthly rent of apartments. Describe the relationship between the distance from the city centre and the monthly rent.
Easy
- APositive correlation
- BNegative correlation
- CNo correlation
- DNon-linear correlation
20.The scatter diagram shows the marks of students in two tests. Hamish scored 40 on test 1 but was absent for test 2. Use a line of best fit to estimate his test 2 mark.
Medium
- A30
- B35
- C40
- D45
21.A publisher checks documents for errors. The table shows the number of documents checked and the total number of errors found, for 5 days. | Documents checked | 20 | 40 | 60 | 80 | 100 | |---|---|---|---|---|---| | Errors found | 18 | 16 | 14 | 12 | 10 | Describe the correlation shown.
Easy
- APositive correlation
- BNegative correlation
- CNo correlation
- DNon-linear correlation
22.The table shows the number of documents checked and the total number of errors found, for 5 days. | Documents checked | 20 | 40 | 60 | 80 | 100 | |---|---|---|---|---|---| | Errors found | 18 | 16 | 14 | 12 | 10 | The line of best fit through this data has gradient -0.1 and passes through (20, 18). One day 110 documents are checked. Use the line of best fit to estimate the total number of errors.
Medium
- A9
- B12
- C14
- D16
23.A delivery driver records distance driven (km) and time taken (minutes). The line of best fit passes through the points (4, 15) and (20, 55). For a 10 km delivery, estimate the time taken.
Medium
- A25 minutes
- B30 minutes
- C35 minutes
- D40 minutes
24.Outside temperature and units of electricity used for heating were recorded. The line of best fit passes through the points (5, 30) and (15, 10). Molly says, "On average the number of units of electricity used for heating decreases by 4 units for each °C increase in outside temperature." Is Molly right?
Medium
- AYes, because the line of best fit has gradient -4
- BNo, because the gradient is about -2
- CNo, because the data does not show a linear relationship
- DYes, because the correlation is negative
25.A scatter diagram shows the physics marks and chemistry marks of 12 students. The points show a positive correlation. Which of the following statements is correct?
Easy
- AStudents who score high in physics tend to score high in chemistry.
- BStudents who score high in physics tend to score low in chemistry.
- CThere is no relationship between physics and chemistry marks.
- DPhysics marks cause chemistry marks to increase.
26.A scatter graph shows the distance from the city centre (km) and monthly rent (£) of apartments. The points go from top left to bottom right. What does this tell you?
Easy
- AAs distance from the city centre increases, rent tends to decrease.
- BAs distance from the city centre increases, rent tends to increase.
- CThere is no correlation between distance and rent.
- DRent is highest for apartments closest to the city centre.
27.The scatter diagram shows the age and value of ten cars of the same model. By drawing a line of best fit, estimate the value of a car that is 6 years old.
Medium
- A$4000
- B$5000
- C$6000
- D$7000
28.What type of correlation is shown when one quantity increases and the other quantity decreases?
Easy
- APositive correlation
- BNegative correlation
- CNo correlation
- DPerfect correlation
29.A scatter graph shows the height (cm) and weight (kg) of 10 people. The points show a positive correlation. Which of the following is the best description?
Easy
- ATaller people tend to weigh more.
- BTaller people tend to weigh less.
- CHeight and weight are not related.
- DBeing tall causes you to weigh more.
30.In a scatter diagram, the points are scattered randomly with no pattern. What type of correlation does this show?
Easy
- APositive correlation
- BNegative correlation
- CNo correlation
- DWeak positive correlation
31.A scatter graph shows the number of hours studied and the exam score for 15 students. The line of best fit passes through (2, 50) and (6, 70). What is the estimated exam score for a student who studies 4 hours?
Medium
- A60
- B55
- C65
- D70
32.A researcher records the temperature (°C) and the number of units of electricity used for heating. The scatter diagram shows a negative correlation. The line of best fit passes through (5, 30) and (15, 10). Using the line, estimate the number of units when the temperature is 10 °C.
Medium
- A20
- B15
- C25
- D10
33.What type of correlation is shown when one quantity decreases as the other quantity increases?
Easy
- APositive correlation
- BNegative correlation
- CNo correlation
- DPerfect correlation
34.A scatter diagram shows points that go from bottom left to top right. What type of correlation does this show?
Easy
- APositive
- BNegative
- CZero
- DWeak
35.A line of best fit must pass through the origin.
Easy
True or false?
36.Which of the following are true about a line of best fit? (Select all that apply)
Medium
- AIt must be drawn with a ruler
- BIt must pass through the origin
- CIt should have roughly the same number of points on each side
- DIt can be used to predict values within the range of data
37.Match each type of correlation with its description.
Medium
- Positive correlation
- Negative correlation
- No correlation
- Points go from bottom left to top right
- Points are scattered randomly
- Points go from top left to bottom right
38.Put the following steps for using a line of best fit to estimate a y-value for a given x in the correct order.
Medium
- Draw a vertical line from the x-value to the line of best fit
- Read off the y-value at that point
- Draw a horizontal line from the intersection to the y-axis
- Locate the given x-value on the x-axis
39.A scatter graph shows the number of hours spent on social media per day and the exam score (%). The line of best fit has equation y = -2.5x + 80. What does the gradient tell you?
Medium
- AEach extra hour on social media is associated with a 2.5% lower exam score
- BEach extra hour on social media is associated with a 2.5% higher exam score
- CThe exam score is 80% when no time is spent on social media
- DThe line of best fit passes through (0, 2.5)
40.Match each description of a scatter graph to the type of correlation it shows.
Easy
- Points go from top left to bottom right
- Points are scattered randomly with no pattern
- Points go from bottom left to top right
- Positive correlation
- Negative correlation
- No correlation
41.Which of the following statements about a line of best fit are correct? (Select all that apply)
Hard
- AIt must pass through the origin
- BIt should have roughly the same number of points above and below it
- CIt can be used to estimate values within the range of the data
- DIt must be a straight line drawn with a ruler
- EIt should be drawn to pass through as many points as possible
42.Complete the sentence.
Easy
If a scatter graph shows a positive correlation, as one quantity increases, the other quantity ____.
43.What is the name given to a point on a scatter graph that does not fit the general pattern?
Easy
44.The line of best fit on a scatter diagram passes through the points (2, 10) and (6, 22). Estimate the value of y when x = 4.
Medium
45.Explain what the phrase 'correlation does not imply causation' means in the context of a scatter graph showing a positive correlation between the number of ice creams sold and the number of sunburns reported.
Medium
46.The line of best fit on a scatter diagram passes through the points (2, 10) and (6, 22). Estimate the value of y when x = 4.
Medium
- A15
- B18
- C16
- D17

複習筆記

## Scatter Graphs & Correlation ### Correlation - **Positive correlation**: as one quantity increases, the other increases (e.g., temperature → ice cream sales). Points go from bottom left to top right. - **Negative correlation**: as one quantity increases, the other decreases (e.g., age of car → value). Points go from top left to bottom right. - **No (zero) correlation**: no apparent relationship; points form a cloud. - Correlation can be **weak** or **strong** (closer to a straight line = stronger). - **Correlation does not imply causation**: two variables may correlate due to a third factor (e.g., time). ### Scatter Graphs - Plot pairs of data as **crosses (×)**; do not join points. - Axes represent the two quantities. ### Lines of Best Fit - Drawn **by eye** as a single straight line using a ruler. - Extend across the full data set; roughly equal points on each side. - Ignore **outliers** (extreme values not fitting the pattern). - Use to **predict** values: - **Interpolation**: predict within the data range (reliable). - **Extrapolation**: predict beyond the data range (unreliable). - To estimate: read off from the line at a given x (or y). ### Key Tips - Use a transparent ruler to see all points. - Always describe correlation with context (e.g., "negative correlation: as distance increases, rent decreases"). - When asked "estimate", use your drawn line of best fit. - Do not use line of best fit to predict far outside the data range (extrapolation).

Positive correlation: When one quantity increases, the other also increases; points slope upward from left to right.
Negative correlation: When one quantity increases, the other decreases; points slope downward from left to right.
No (zero) correlation: No apparent relationship between the two quantities; points are scattered randomly.
Line of best fit: A straight line drawn on a scatter graph to represent the trend, used for predictions.
Interpolation: Predicting a value within the range of the given data; reliable.
Extrapolation: Predicting a value outside the range of the given data; unreliable.
Outlier: An extreme data point that does not fit the general pattern; ignored when drawing a line of best fit.

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