Scatter graphs and correlation

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수업 노트

What is a scatter graph?

A scatter graph (also called a scatter plot, scatter chart, scattergram or scatter diagram) shows values for two variables for a set of data.
Each point on the graph has one variable's value on the horizontal axis and the other variable's value on the vertical axis.
The points are plotted using Cartesian coordinates, so each dot represents one item in the data set.
Scatter graphs are used to look for a relationship between two variables.
A scatter diagram is one of the seven basic tools of quality control.

Choosing axes

If one variable is controlled or changed by the experimenter, it is the independent variable and is usually plotted on the horizontal axis.
The dependent variable is the one being measured, and is usually plotted on the vertical axis.
If there is no dependent variable, either variable can go on either axis.
Example: to study lung capacity and breath-holding time, put lung capacity on the horizontal axis and time holding breath on the vertical axis.
A person with lung capacity 400 cl who held their breath for 21.7 s is plotted at the point (400, 21.7).

Types of correlation

Positive correlation: as one variable increases, the other tends to increase. The pattern of dots slopes from lower left to upper right.
Negative correlation: as one variable increases, the other tends to decrease. The pattern of dots slopes from upper left to lower right.
No correlation (null or uncorrelated): there is no clear pattern in the dots.
Correlation can be described by its direction (positive or negative) and by its strength (strong or weak).
A scatter graph can suggest the kind of correlation between two variables.

Outliers

An outlier is a point that does not fit the overall pattern of the other points.
Outliers can affect the line of best fit and the strength of the correlation.
Always check for outliers before deciding how strong the correlation is.

Line of best fit

A line of best fit (also called a trendline) is drawn to show the general relationship between the two variables.
It should follow the direction of the points and have roughly as many points above it as below it.
The line of best fit can be used to estimate values that are not in the data set.
For a linear correlation, the best-fit procedure is called linear regression.

Using a line of best fit to estimate

To estimate a value, read across from the known value on one axis to the line of best fit, then read down or across to the other axis.
Estimates made within the range of the data are called interpolation and are usually reliable.
Estimates made outside the range of the data are called extrapolation and are risky.
Extrapolating too far beyond the data can give values that do not make sense.

Correlation and causation

Correlation does not prove causation: a relationship between two variables does not mean one causes the other.
There may be a third factor (a hidden variable) that explains the relationship.
The relationship could also be a coincidence.
Scatter graphs show whether variables are related, but they cannot prove why.

Scatter plot matrices

For a set of variables X₁, X₂, ..., X_k, a scatter plot matrix shows all pairwise scatter plots in one view.
The matrix has k rows and k columns.
The plot at the intersection of row i and column j is a scatter plot of X_i against X_j.
Each row and column represents one variable, and each cell plots two variables.

슬라이드

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연습 문제

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1.A scatter graph shows bivariate data. What does 'bivariate' mean?
Easy
- AThere are two variables for each item of data
- BThere are two points plotted on the graph
- CThe graph has two axes
- DThe data has been collected twice
2.On a scatter graph, which variable is customarily plotted on the horizontal axis?
Easy
- AThe independent (control) variable
- BThe dependent (measured) variable
- CThe variable with the largest values
- DEither variable, chosen at random
3.A scatter graph shows points sloping from upper left to lower right. What type of correlation is this?
Easy
- ANegative correlation
- BPositive correlation
- CNo correlation
- DZero correlation
4.A student measures the lung capacity and breath-holding time of 20 people. They plot lung capacity on the x-axis and breath-holding time on the y-axis. One person has lung capacity 400 cl and holds their breath for 21.7 s. Which point represents this person?
Medium
- A(400, 21.7)
- B(21.7, 400)
- C(400, 400)
- D(21.7, 21.7)
5.Which of the following are other names for a scatter graph? (select all that apply)
Medium
- AScatterplot
- BScatter chart
- CScattergram
- DScatter diagram
- ELine graph
6.A scatter graph can be used to show the relationship between two variables.
Easy
True or false?
7.Match each pattern on a scatter graph to the type of correlation it shows.
Medium
- Points slope from lower left to upper right
- Points slope from upper left to lower right
- Points show no clear pattern
- Positive correlation
- Negative correlation
- No correlation
8.Place these correlation strengths in order from strongest negative to strongest positive.
Medium
- Strong negative correlation
- Weak negative correlation
- No correlation
- Weak positive correlation
- Strong positive correlation

더 많은 문제

9.Which of the following are alternative names for a scatter graph? (select all that apply)
Easy
- Ascatterplot
- Bscatter chart
- Cscattergram
- Dscatter diagram
- Eline graph
10.Match each term to its definition.
Medium
- bivariate data
- positive correlation
- negative correlation
- no correlation
- data involving two variables
- as one variable increases, the other tends to increase
- as one variable increases, the other tends to decrease
- there is no clear relationship between the variables
11.A scatter graph shows points that slope from lower left to upper right. What type of correlation does this indicate?
Medium
- APositive correlation
- BNegative correlation
- CNo correlation
- DCannot be determined
12.A line of best fit can be used to estimate values for data points that lie outside the range of the plotted data.
Easy
True or false?
13.A scatter graph shows the relationship between hours studied and test scores. The line of best fit has equation y = 5x + 40. What does the gradient of 5 represent?
Medium
- AFor each additional hour studied, the test score increases by 5 on average
- BThe test score when no hours are studied
- CThe total number of hours studied
- DThe maximum possible test score
14.Which of the following statements about a line of best fit are true? (select all that apply)
Medium
- AIt can be used to estimate values within the range of the data.
- BIt must pass through every data point.
- CIt shows the general trend of the data.
- DIt can be used to predict values outside the data range without any risk.
15.A scatter graph shows a strong positive correlation between two variables. Which of the following can be concluded?
Medium
- AAs one variable increases, the other tends to increase.
- BOne variable causes the other to increase.
- CThere is no relationship between the variables.
- DThe variables are independent.
16.Match each correlation pattern to its description.
Medium
- Positive correlation
- Negative correlation
- No correlation
- Outlier
- Points slope from lower left to upper right
- Points slope from upper left to lower right
- Points show no clear pattern
- A point that does not fit the general pattern
17.Place these stages in order when drawing a line of best fit on a scatter graph.
Medium
- Plot the data points
- Draw a line that follows the general trend
- Ensure roughly equal numbers of points above and below the line
- Use the line to estimate values
18.A scatter graph shows the relationship between engine size and fuel consumption for 15 cars. The line of best fit has equation y = 0.8x + 2, where x is engine size in litres and y is fuel consumption in litres per 100 km. Estimate the fuel consumption for a car with engine size 3 litres.
Medium
- A4.4 litres per 100 km
- B2.4 litres per 100 km
- C3.8 litres per 100 km
- D6.4 litres per 100 km
19.A scatter graph shows the relationship between hours of exercise per week and resting heart rate. The line of best fit is y = -0.5x + 80, where x is hours of exercise and y is resting heart rate in beats per minute. Estimate the resting heart rate for someone who exercises 10 hours per week.
Medium
- A75 bpm
- B85 bpm
- C70 bpm
- D80 bpm
20.A line of best fit on a scatter graph always passes through every data point.
Easy
True or false?
21.On a scatter graph, what is an outlier?
Easy
- AA point that lies far away from the general pattern of the other points
- BA point that lies exactly on the line of best fit
- CThe point with the largest x-value
- DThe point with the smallest y-value
22.Which of the following statements about the line of best fit are true? (select all that apply)
Medium
- AIt can be used to estimate values within the range of the data.
- BIt must pass through the origin (0,0).
- CIt should have roughly equal numbers of points above and below it.
- DIt is always a straight line, even if the points show a clear curve.
23.Match each term to its correct description.
Medium
- Positive correlation
- Negative correlation
- No correlation
- Outlier
- As one variable increases, the other tends to increase.
- As one variable increases, the other tends to decrease.
- There is no clear relationship between the variables.
- A point that does not fit the general pattern.
24.Place these steps in order to draw a line of best fit on a scatter graph.
Medium
- Plot all the data points accurately.
- Look at the overall pattern of the points.
- Draw a straight line that best follows the trend.
- Check that there are roughly equal numbers of points above and below the line.
25.A scatter graph shows a strong positive correlation between the number of ice creams sold and the number of sunglasses sold. Which of the following is the most likely explanation?
Medium
- AEating ice cream makes people buy sunglasses.
- BBoth increase because of a third factor, such as hot weather.
- CSunglasses cause people to buy ice cream.
- DThe correlation is due to chance.
26.A scatter graph shows the relationship between hours studied (x) and test score (y). The line of best fit is y = 4x + 50. Use the line to estimate the test score for a student who studies for 5 hours.
Medium
- A70
- B54
- C90
- D45
27.A scatter graph has a line of best fit with equation y = 3x − 1. Use the line to estimate the value of y when x = 4.
Medium
- A11
- B13
- C12
- D7
28.A scatter graph shows the relationship between engine size in litres (x) and fuel consumption in miles per gallon (y). The line of best fit is y = 0.8x + 2. Estimate the fuel consumption when the engine size is 3.5 litres.
Medium
- A4.8 mpg
- B4.4 mpg
- C5.2 mpg
- D3.8 mpg
29.A scatter graph shows the relationship between temperature in °C (x) and ice cream sales in pounds (y). The line of best fit is y = 2x + 10. Use the line to estimate the temperature when sales are £50.
Hard
- A20 °C
- B25 °C
- C30 °C
- D15 °C
30.A scatter graph shows the relationship between hours of exercise per week (x) and resting heart rate in beats per minute (y). The line of best fit is y = -0.5x + 80. Use the line to estimate the number of hours of exercise for a person with a resting heart rate of 70 bpm.
Hard
- A20 hours
- B15 hours
- C25 hours
- D10 hours
31.Which of the following are names for a scatter graph? (select all that apply)
Easy
- Ascatter plot
- Bscattergram
- Cscatter diagram
- Dpie chart
- Ebar chart
32.A scatter graph can be used to display values for two variables for a set of data.
Easy
True or false?
33.A scatter graph shows the relationship between hours studied (x) and test score (y). The line of best fit is y = 4x + 50. Use the line to estimate the test score for a student who studied for 5 hours.
Medium
- A70
- B54
- C200
- D20
34.Match each correlation pattern on a scatter graph to the correct description.
Medium
- Points slope from lower left to upper right
- Points slope from upper left to lower right
- Points are scattered with no clear pattern
- Positive correlation
- Negative correlation
- No correlation
35.Place these correlation strengths in order from weakest (least strong) to strongest.
Medium
- No correlation
- Weak positive correlation
- Moderate positive correlation
- Strong positive correlation
36.A scatter graph has points that slope from lower left to upper right. What type of correlation does this show?
Easy
- APositive correlation
- BNegative correlation
- CNo correlation
- DZero correlation
37.Which of the following statements about correlation and causation are true? (select all that apply)
Medium
- ACorrelation does not prove causation.
- BA strong correlation means one variable definitely causes the other.
- CTwo variables may be correlated because a third factor affects both.
- DIf there is no correlation, there is no relationship at all.
38.Match each term to its correct meaning.
Medium
- Bivariate data
- Outlier
- Line of best fit
- Data involving two variables
- A point that does not fit the general pattern
- A line drawn to show the trend of the points
39.A scatter graph shows bivariate data. On which axis is the measured or dependent variable customarily plotted?
Medium
- AVertical axis
- BHorizontal axis
- CEither axis
- DNeither axis
40.A scatter graph shows the relationship between the age of a car in years (x) and its value in thousands of pounds (y). The line of best fit is y = -1.5x + 20. Estimate the value of a car that is 6 years old.
Medium
- A11 thousand pounds
- B12 thousand pounds
- C29 thousand pounds
- D9 thousand pounds
41.What is the name of the line drawn on a scatter graph to study the relationship between two variables?
Medium
42.A scatter graph has a line of best fit with equation y = 2x + 3. Use the line to estimate the value of y when x = 10.
Medium
43.Complete the sentence about correlation and causation.
Easy
Correlation does not prove ____.
44.A scatter graph shows the relationship between daily temperature (°C) and ice cream sales. The line of best fit is y = 2x + 10, where x is temperature in °C and y is sales in pounds. Estimate the sales when the temperature is 15 °C.
Medium
45.A scatter graph shows the relationship between hours studied and test score. The line of best fit is y = 4x + 50, where x is hours studied and y is test score. Estimate the test score for a student who studies for 5 hours.
Medium
46.A scatter graph shows the relationship between temperature in °C (x) and ice cream sales in pounds (y). The line of best fit is y = 2x + 10. Use the line to estimate the sales when the temperature is 15 °C.
Medium
47.A scatter graph shows the relationship between hours of exercise per week (x) and resting heart rate in beats per minute (y). The line of best fit is y = -0.5x + 80. Estimate the resting heart rate for someone who exercises for 10 hours per week.
Medium
48.A scatter graph shows the relationship between engine size in litres (x) and fuel consumption in miles per gallon (y). The line of best fit is y = 0.8x + 2. Estimate the fuel consumption for an engine size of 2.5 litres.
Medium
49.A scatter graph has a line of best fit with equation y = 3x - 1. Use the line to estimate the value of y when x = 4.
Medium
50.Complete the sentence about the line of best fit.
Medium
The line of best fit is drawn to show the overall ____ between the two variables.
51.Complete the sentence about extrapolation.
Medium
Using a line of best fit to estimate values outside the range of the data is called ____ and can be unreliable.
52.Complete the sentence about outliers.
Easy
A point that does not fit the general pattern of a scatter graph is called an ____.
53.Explain why it can be dangerous to use a line of best fit to estimate values outside the range of the data.
Hard
54.Explain why a line of best fit should not be used to estimate values outside the range of the plotted data.
Medium
55.Explain why it is dangerous to use a line of best fit to estimate values outside the range of the plotted data.
Hard
56.A scatter graph shows a strong positive correlation between the number of ice creams sold and the number of sunglasses sold. Explain why this does not prove that buying ice creams causes people to buy sunglasses.
Hard
57.A scatter graph shows a strong positive correlation between the number of hours spent training and the number of points scored in a game. Explain why this does not prove that training causes more points to be scored.
Hard
58.A scatter graph has a line of best fit with equation y = 2x + 3. Use the line to estimate the value of y when x = 10.
Medium
- A25
- B27
- C23
- D21
59.A scatter graph shows the relationship between daily temperature (°C) and ice cream sales. The line of best fit is y = 2x + 10, where x is temperature in °C and y is sales in pounds. Estimate the sales when the temperature is 15 °C.
Medium
- A44 pounds
- B48 pounds
- C36 pounds
- D40 pounds
60.A scatter graph shows the relationship between hours studied and test score. The line of best fit is y = 4x + 50, where x is hours studied and y is test score. Estimate the test score for a student who studies for 5 hours.
Medium
- A77 marks
- B84 marks
- C63 marks
- D70 marks
61.A scatter graph shows the relationship between temperature in °C (x) and ice cream sales in pounds (y). The line of best fit is y = 2x + 10. Use the line to estimate the sales when the temperature is 15 °C.
Medium
- A36 pounds
- B44 pounds
- C48 pounds
- D40 pounds
62.A scatter graph shows the relationship between hours of exercise per week (x) and resting heart rate in beats per minute (y). The line of best fit is y = -0.5x + 80. Estimate the resting heart rate for someone who exercises for 10 hours per week.
Medium
- A67 bpm
- B83 bpm
- C75 bpm
- D91 bpm
63.A scatter graph shows the relationship between engine size in litres (x) and fuel consumption in miles per gallon (y). The line of best fit is y = 0.8x + 2. Estimate the fuel consumption for an engine size of 2.5 litres.
Medium
- A5 mpg
- B3 mpg
- C6 mpg
- D4 mpg
64.A scatter graph has a line of best fit with equation y = 3x - 1. Use the line to estimate the value of y when x = 4.
Medium
- A11
- B10
- C13
- D12

복습 노트

## Scatter Graphs and Correlation A **scatter graph** (scatter plot, scatter diagram) shows values for **two variables** for a set of data. Each point has one variable's value on the horizontal axis and the other's on the vertical axis. ### Plotting bivariate data - Put the **independent (control) variable** on the horizontal axis. - Put the **dependent (measured) variable** on the vertical axis. - Example: lung capacity (horizontal) and time holding breath (vertical). A person with lung capacity 400 cl who held their breath 21.7 s is the point (400, 21.7). ### Types of correlation - **Positive correlation**: points slope from lower left to upper right (as one variable rises, the other tends to rise). - **Negative correlation**: points slope from upper left to lower right (as one rises, the other tends to fall). - **No correlation (null)**: no clear pattern. - **Strength**: points tightly clustered around a line show strong correlation; widely spread points show weak correlation. - **Outlier**: a point that does not fit the overall pattern. ### Line of best fit - A **line of best fit** (trendline) is drawn through the middle of the points to study the relationship. - Use it to **estimate values** for one variable from the other. - **Extrapolating** means estimating outside the range of the data. This is risky because the pattern may not continue. ### Correlation and causation - Correlation shows a relationship, **not** that one variable causes the other. Correlation does not prove causation. ### Key ideas to remember - Scatter graphs display **bivariate data** (two variables). - Describe correlation by **direction** (positive/negative/none) and **strength** (strong/weak). - Always label axes and give units. - A line of best fit should have roughly equal numbers of points above and below it.

Scatter graph: A plot using Cartesian coordinates to display values for two variables for a set of data; also called scatter plot, scatter diagram, scattergram or scatter chart.
Bivariate data: Data involving two variables, plotted on a scatter graph.
Positive correlation: The pattern of points slopes from lower left to upper right.
Negative correlation: The pattern of points slopes from upper left to lower right.
No correlation (null): There is no clear pattern between the two variables.
Outlier: A point that does not fit the overall pattern of the data.
Line of best fit: A line drawn through the middle of the points on a scatter graph to study the relationship and estimate values; also called a trendline.
Extrapolating: Estimating values outside the range of the data, which is dangerous because the pattern may not continue.
Correlation does not prove causation: A relationship between two variables does not mean that one causes the other.

플래시카드

What is a scatter graph?

A graph using Cartesian coordinates to display values for two variables for a set of data, with one variable on the horizontal axis and the other on the vertical axis. Each point represents one observation.

What are three alternative names for a scatter graph?

Scatter plot, scatter diagram, scatter chart.

In a scatter graph, which variable is usually plotted on the horizontal axis?

The independent variable (the control parameter) is customarily plotted on the horizontal axis.

In a scatter graph, which variable is usually plotted on the vertical axis?

The dependent variable (the measured variable) is customarily plotted on the vertical axis.

What does a pattern of dots sloping from lower left to upper right indicate?

A positive correlation between the two variables.

What does a pattern of dots sloping from upper left to lower right indicate?

A negative correlation between the two variables.

What does it mean if a scatter graph shows no clear pattern?

There is no correlation (null correlation) between the two variables.

What is a line of best fit?

A line drawn on a scatter graph to show the general trend of the data. It is also called a trendline.

What is an outlier on a scatter graph?

A point that does not fit the general pattern of the other points.

How can a line of best fit be used?

It can be used to estimate values of one variable from the other, within the range of the data.

Why is extrapolating beyond the range of the data dangerous?

The relationship may not continue outside the observed range, so predictions can be unreliable.

What is the difference between correlation and causation?

Correlation means two variables are related in a pattern; causation means one variable directly causes the other. Correlation does not prove causation.

What does a scatter graph show about two variables?

It shows the relationship between them, including the type and strength of any correlation.

What is a scatter plot matrix?

A grid showing all pairwise scatter plots of several variables. For k variables, it has k rows and k columns.

In a scatter plot matrix, what does the cell at row i and column j show?

A scatter plot of variable Xi against variable Xj.

Give an example of two variables that might be plotted on a scatter graph.

Lung capacity and time holding breath: lung capacity on the horizontal axis, time holding breath on the vertical axis.

What does a single point on a scatter graph represent?

One observation, with its coordinates giving the values of the two variables for that case.

What is the name of the statistical procedure that finds the best-fit line for a linear correlation?

Linear regression.

What does a scatter graph NOT prove about two variables?

It does not prove that one variable causes the other.

What is the strength of correlation?

How closely the points follow a clear pattern or line; a tight pattern indicates strong correlation, a loose one indicates weak correlation.

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