Scatter Graphs And Correlation

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課程筆記

Scatter Graphs

Scatter graphs (or scatter diagrams) plot pairs of data using crosses (×).
The vertical and horizontal axes represent the two quantities being measured.
Points are not joined up; the overall shape shows the type of correlation.
Positive correlation: points go from bottom left to top right (positive gradient).
Negative correlation: points go from top left to bottom right (negative gradient).
No correlation: points form a random cloud with no apparent pattern.
Correlation can be weak or strong depending on how close points are to a straight line.

Positive correlation

Types of Correlation

Positive correlation: as one quantity increases, the other also increases (e.g., temperature and ice cream sales).
Negative correlation: as one quantity increases, the other decreases (e.g., car age and value).
Zero correlation: no relationship between the two quantities (e.g., snail mass and exam score).
Correlation does not imply causation – a relationship may be due to a third factor (e.g., time).

Lines of Best Fit

A line of best fit is a single straight line drawn by eye on a scatter graph.
It should extend across the full range of data, with roughly equal numbers of points on either side.
Ignore any outliers (extreme values that do not fit the pattern) when drawing the line.
Use a transparent ruler to help position the line accurately.

Sophie's scatter diagram

Using Lines of Best Fit for Predictions

To estimate a value, read off the line at the given x or y coordinate.
Interpolation: predicting within the range of the data – reliable.
Extrapolation: predicting beyond the data range – unreliable and should be avoided.
Always use the line of best fit, not individual points, for predictions.

Using the line of best fit

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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
- DZero correlation
2.A scatter diagram shows points going from top left to bottom right. What type of correlation is this?
Easy
- APositive correlation
- BNegative correlation
- CNo correlation
- DWeak correlation
3.Which of the following best describes 'no correlation' in a scatter graph?
Easy
- APoints form a straight line
- BPoints form a cloud with no pattern
- CPoints go from bottom left to top right
- DPoints go from top left to bottom right
4.The average speed of a train and the time taken for a journey: what type of correlation would you expect?
Easy
- APositive correlation
- BNegative correlation
- CNo correlation
- DWeak correlation
5.The distance travelled by a car and the amount of fuel used: what type of correlation would you expect?
Easy
- APositive correlation
- BNegative correlation
- CNo correlation
- DZero correlation
6.A scatter diagram shows points with a clear upward trend from left to right. Which statement is correct?
Easy
- AThere is a negative correlation
- BThere is a positive correlation
- CThere is no correlation
- DThe line of best fit has a negative gradient
7.In a scatter diagram, a line of best fit should be:
Medium
- AA curved line passing through all points
- BA single ruled straight line with roughly equal points on each side
- CA line connecting the first and last points
- DA line that passes through the origin
8.When using a line of best fit to estimate a value within the range of the data, this is called:
Medium
- AExtrapolation
- BInterpolation
- CCorrelation
- DCausation

更多題目

9.A scatter graph shows a strong negative correlation. Which of the following is a possible line of best fit?
Medium
- AA horizontal line
- BA vertical line
- CA line sloping downwards from left to right
- DA line sloping upwards from left to right
10.What does the phrase 'correlation does not imply causation' mean?
Medium
- AIf two variables are correlated, one must cause the other
- BCorrelation always implies causation
- CTwo variables can be correlated without one causing the other
- DCausation is the same as correlation
11.A scatter diagram has an outlier that does not fit the general pattern. When drawing a line of best fit, you should:
Medium
- AInclude the outlier and make the line pass through it
- BIgnore the outlier and draw the line through the remaining points
- CRemove the outlier and draw a curved line
- DDraw two lines of best fit
12.Using a line of best fit to predict a value far outside the range of the data is called:
Medium
- AInterpolation
- BExtrapolation
- CEstimation
- DCorrelation
13.Which of the following statements about a line of best fit is correct?
Medium
- AIt must pass through the origin
- BIt must pass through at least two data points
- CIt should have roughly equal numbers of points on either side
- DIt should be drawn as a dashed line
14.A scatter graph shows a positive correlation between the number of hours studied and exam score. Which conclusion is justified?
Medium
- AStudying more causes higher exam scores
- BThere is a relationship but not necessarily causation
- CExam scores cause more studying
- DThere is no relationship
15.In a scatter diagram, points are plotted as:
Easy
- ADots
- BCircles
- CCrosses
- DTriangles
16.Which of the following pairs is likely to show a positive correlation?
Medium
- AAge of a car and its value
- BOutside temperature and sales of ice cream
- CNumber of umbrellas sold and amount of sunshine
- DDistance run and time taken at constant speed
17.A scatter diagram shows points going from bottom left to top right. What type of correlation is this?
Easy
- APositive correlation
- BNegative correlation
- CNo correlation
- DWeak correlation
18.On a scatter diagram, points are plotted as:
Easy
- ACrosses (×)
- BDots (•)
- CCircles (○)
- DTriangles (▲)
19.A scatter diagram shows a strong negative correlation. Which of the following is a possible line of best fit?
Medium
- AA line sloping downwards from left to right
- BA line sloping upwards from left to right
- CA horizontal line
- DA vertical line
20.A scatter diagram shows points with no clear pattern. What type of correlation is this?
Easy
- ANo correlation
- BPositive correlation
- CNegative correlation
- DStrong correlation
21.On a scatter diagram, what does a line of best fit with a negative gradient indicate?
Easy
- ANegative correlation
- BPositive correlation
- CNo correlation
- DWeak correlation
22.What does a scatter graph show?
Easy
- APairs of data plotted as points
- BA single set of data plotted over time
- CCategorical data in bars
- DData in a pie chart
23.On a scatter diagram, points are plotted using which symbol?
Easy
- A×
- B•
- C○
- D▲
24.What type of correlation is shown when points go from top left to bottom right?
Easy
- ANegative correlation
- BPositive correlation
- CNo correlation
- DStrong correlation
25.Which of the following pairs is likely to show a negative correlation?
Medium
- AAge of a car and its value
- BTemperature and sales of cold drinks
- CNumber of hours studied and exam score
- DHeight and weight of a person
26.What does it mean if a scatter graph shows 'no correlation'?
Easy
- AThere is no apparent relationship between the two quantities
- BOne quantity increases as the other increases
- COne quantity decreases as the other increases
- DThe points lie exactly on a straight line
27.A scatter diagram shows a strong positive correlation. Which statement is correct?
Easy
- AThe points lie close to a straight line with a positive gradient
- BThe points are widely scattered with no pattern
- CThe points lie close to a straight line with a negative gradient
- DThe points form a curve
28.When drawing a line of best fit on a scatter graph, you should:
Medium
- AMake sure there are roughly as many points on either side of the line
- BMake the line pass through the first and last points
- CMake the line pass through as many points as possible
- DMake the line start at the origin
29.What is the term for using a line of best fit to estimate a value within the range of the data?
Easy
- AInterpolation
- BExtrapolation
- CCorrelation
- DEstimation
30.A scatter graph shows a negative correlation between the number of hours spent watching TV and exam scores. Which conclusion is justified?
Medium
- AThere is a correlation, but it does not prove that watching TV causes lower exam scores
- BWatching TV causes lower exam scores
- CLower exam scores cause more TV watching
- DThere is no relationship between TV watching and exam scores
31.On a scatter diagram, what should you do with an outlier when drawing a line of best fit?
Medium
- AIgnore the outlier
- BInclude the outlier and make the line pass through it
- CRemove the outlier from the data set entirely
- DDraw two separate lines of best fit
32.The line of best fit on a scatter graph should be:
Easy
- AA single ruled straight line
- BA curve that passes through all points
- CA dashed line connecting the highest and lowest points
- DA line that starts at the origin
33.A scatter diagram shows points with a clear downward trend from left to right. Which of the following is a possible line of best fit?
Easy
- AA line with a negative gradient
- BA line with a positive gradient
- CA horizontal line
- DA vertical line
34.A line of best fit for the number of people and number of phones in a set of buildings passes through (10, 6) and (50, 30). Estimate the number of phones in a building with 42 people.
Medium
- A15
- B20
- C25
- D30
35.As the maximum power of a car increases, the time it takes to accelerate from 0 to 100 km/h decreases. What type of correlation is this?
Medium
- APositive correlation
- BNegative correlation
- CNo correlation
- DNon-linear correlation
36.A line of best fit for the test 1 and test 2 marks of a group of students passes through (20, 15) and (60, 51). Hamish scored 40 on test 1 but was absent for test 2. Use the line of best fit to estimate his test 2 mark.
Medium
- A27
- B30
- C33
- D36
37.A line of best fit for the age and value of cars of the same model passes through (2, $7000) and (8, $3400). Estimate the value of a car that is 6 years old.
Medium
- A$4000
- B$4600
- C$5200
- D$5800
38.As the height of a goat increases, its mass also tends to increase. What type of correlation does this describe?
Medium
- APositive correlation
- BNegative correlation
- CNo correlation
- DNon-linear correlation
39.As the number of hours a student studies increases, their exam score also tends to increase. Which statement best describes this relationship?
Medium
- AThere is no correlation between hours studied and exam score.
- BThere is a positive correlation: more study is associated with higher scores.
- CThere is a negative correlation: more study is associated with lower scores.
- DThe correlation is non-linear.
40.A sample of cars with maximum power between 30 kW and 90 kW was used to draw a line of best fit relating power to acceleration time. Two additional cars have powers of 63 kW and 140 kW. Which estimate is more reliable?
Hard
- AThe estimate for 63 kW is more reliable because it is interpolation.
- BThe estimate for 140 kW is more reliable because it is extrapolation.
- CBoth are equally reliable.
- DNeither is reliable because the correlation is weak.
41.What type of correlation is shown when the points on a scatter graph go from bottom left to top right?
Easy
- APositive correlation
- BNegative correlation
- CNo correlation
- DWeak correlation
42.A scatter graph shows a strong correlation if the points lie close to a straight line.
Easy
True or false?
43.On a scatter graph, points are plotted using which symbol?
Easy
- ACross (×)
- BDot (•)
- CCircle (○)
- DTriangle (△)
44.A scatter diagram shows points going from top left to bottom right. Which of the following best describes the correlation?
Easy
- APositive correlation
- BNegative correlation
- CNo correlation
- DZero correlation
45.A scatter graph shows the age of a car (in years) and its value (in £). What type of correlation would you expect?
Easy
- APositive correlation
- BNegative correlation
- CNo correlation
- DPerfect positive correlation
46.Which of the following statements about a line of best fit are correct? (Select all that apply)
Medium
- AIt must pass through the origin.
- BIt should have roughly the same number of points on each side of the line.
- CIt should be drawn with a ruler.
- DIt should extend beyond the data points to allow extrapolation.
- EIt must pass through at least two data points.
47.A scatter diagram shows a strong positive correlation between ice cream sales and number of sunburns. Which conclusion is justified?
Medium
- AEating ice cream causes sunburn.
- BGetting sunburn makes people buy ice cream.
- CA third factor, such as sunny weather, may cause both to increase.
- DThere is no relationship between ice cream sales and sunburns.
48.A scatter graph has an outlier that clearly does not fit the general pattern. When drawing a line of best fit, you should:
Medium
- AInclude the outlier and draw the line through it.
- BIgnore the outlier when drawing the line.
- CDraw two lines, one with and one without the outlier.
- DRemove the outlier from the graph entirely.
49.Arrange the following steps for using a line of best fit to estimate a value in the correct order.
Medium
- Draw a line of best fit on the scatter graph.
- Locate the given value on the appropriate axis.
- Draw a horizontal or vertical line from that value to the line of best fit.
- Read off the corresponding value from the other axis.
50.Match each description of a scatter graph to the correct type of correlation.
Medium
- Points go from bottom left to top right
- Points go from top left to bottom right
- Points show no clear pattern
- Positive correlation
- Negative correlation
- No correlation
51.A scatter graph shows the number of hours studied and the exam score. A line of best fit passes through (2, 40) and (6, 80). Estimate the exam score for a student who studied 4 hours.
Medium
52.Complete the sentences about interpolation and extrapolation.
Easy
Using a line of best fit to predict a value within the range of the data is called ____. Predicting a value outside the range of the data is called ____.
53.A scatter graph shows the number of hours studied and the exam score. A line of best fit passes through (2, 40) and (6, 80). Estimate the exam score for a student who studied 4 hours.
Medium
- A66
- B72
- C60
- D54

複習筆記

## Scatter Graphs & Correlation ### Correlation - **Positive correlation**: as one quantity increases, the other increases (e.g., temperature and ice cream sales). Points go from bottom left to top right. - **Negative correlation**: as one quantity increases, the other decreases (e.g., car age and value). Points go from top left to bottom right. - **No correlation**: no apparent relationship; points form a cloud. - **Strength**: strong if points lie close to a straight line; weak if scattered. - **Correlation ≠ causation**: a relationship does not mean one causes the other (e.g., sunflower height and puppy weight both increase with time). ### Scatter Graphs - Plot pairs of data as crosses (×). - Axes represent the two quantities. - Do not join points. ### Lines of Best Fit - Draw a single straight line by eye through the data. - Extend across the full data set. - Roughly equal numbers of points on each side of the line. - Ignore outliers (extreme values that don't fit the pattern). ### Using the Line of Best Fit - **Interpolation**: estimate values within the range of data (reliable). - **Extrapolation**: estimating beyond the data (unreliable). - To estimate: read off the y-value for a given x (or vice versa) using the line. ### Exam Tips - Use a transparent ruler to position the line of best fit. - Always draw a line of best fit when asked. - State correlation type (positive, negative, or none).

Correlation: A relationship between two quantities; can be positive, negative, or zero.
Positive correlation: As one quantity increases, the other increases.
Negative correlation: As one quantity increases, the other decreases.
No correlation: No apparent relationship between two quantities.
Line of best fit: A straight line drawn on a scatter graph to show the trend; used for predictions.
Interpolation: Estimating a value within the range of known data.
Extrapolation: Estimating a value outside the range of known data; often unreliable.
Outlier: A data point that does not fit the general pattern; may be ignored when drawing a line of best fit.

字卡

What is correlation?

Correlation describes how two quantities are related to each other.

What is positive correlation?

Positive correlation is when one quantity increases and the other quantity also increases.

What is negative correlation?

Negative correlation is when one quantity decreases while the other quantity increases.

What is no (zero) correlation?

No correlation is when there is no apparent relationship between the two quantities.

What does 'correlation does not imply causation' mean?

If two quantities correlate, it does not mean that the first causes the second. Both may be influenced by another factor.

What are scatter graphs used for?

Scatter graphs (or scatter diagrams) are used to plot pairs of data, e.g., students' Maths grades against their Physics grades.

How are points plotted on a scatter graph?

Points are plotted as crosses (×) and are not joined up.

What does a positive correlation look like on a scatter graph?

Points go from bottom left to top right (positive gradient).

What does a negative correlation look like on a scatter graph?

Points go from top left to bottom right (negative gradient).

What does no correlation look like on a scatter graph?

Points appear as a cloud with no clear pattern.

What is a line of best fit?

A line of best fit is a straight line drawn on a scatter graph to show the trend when there is positive or negative correlation.

How do you draw a line of best fit?

Draw a single ruled straight line by eye that extends across the full data set, with roughly as many points on either side.

How do you handle an outlier when drawing a line of best fit?

Ignore the outlier when drawing the line of best fit.

What is interpolation?

Interpolation is using a line of best fit to predict values within the region covered by the data points.

What is extrapolation and why is it unreliable?

Extrapolation is predicting values outside the data range; it is unreliable because the trend may not continue.

How do you use a line of best fit to estimate a value?

Read off the y-value from the line at a given x-value, or vice versa, by drawing horizontal/vertical lines to the axes.

What type of correlation is shown if points go from top left to bottom right?

Negative correlation.

What type of correlation is expected between average speed of a train and time taken for a journey?

Negative correlation (higher speed means less time).

What type of correlation is expected between distance travelled by a car and amount of fuel used?

Positive correlation (more distance uses more fuel).

What type of correlation is 'we eat more ice cream as temperature rises'?

Positive correlation.

What is the range of a data set?

The range is the difference between the highest and lowest values.

How do you find the median?

Arrange values in order and find the middle value; if even number, average the two middle values.

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