1. Describing Relationships: Scatterplots and Correlation
Statistical Thinking
Chapter 14
Describing Relationships: Scatterplots and Correlation
Chapter 14
Chapter 13

2. Statistical Thinking
Correlation
Objective: Analyze a collection of paired data (sometimes called bivariate data).
A correlation exists between two variables when there is a relationship (or an association) between them.
We will consider only linear relationships.
- when graphed, the points approximate a
straight-line pattern.
Chapter 13
Chapter 13

3. Statistical Thinking
Scatterplot
A scatterplot is a graph in which paired (x, y) data (usually collected on the same individuals) are plotted with one variable represented on a horizontal (x -) axis and the other variable represented on a vertical (y-) axis. Each individual pair (x, y) is plotted as a single point.
Example:
Chapter 13
Chapter 13

4. Examining a Scatterplot
Statistical Thinking
Examining a Scatterplot
You can describe the overall pattern of a scatterplot by the
Form – linear or non-linear ( quadratic, exponential, no
correlation etc.)
Direction – negative, positive.
Strength – strong, very strong, moderately strong,
weak etc.
Look for outliers and how they affect the correlation.
Chapter 13
Chapter 13

5. Scatterplot Example: Draw a scatter plot for the data below.
What is the nature of the relationship between
X and Y. x 2 4 –2
– 4 y 6 x 1 2 3 4 5 y -4 -2
Strong, positive and linear.
Chapter 13

6. Examining a Scatterplot
Statistical Thinking
Examining a Scatterplot
Two variables are positively correlated when high values of the variables tend to occur together and low values of the variables tend to occur together.
The scatterplot slopes upwards from left to right.
Two variables are negatively correlated when high values of one of the variables tend to occur with low values of the other and vice versa.
The scatterplot slopes downwards from left to right.
Chapter 13
Chapter 13

7. Types of Correlation As x increases, y tends to decrease.
Statistical Thinking
Types of Correlation x y x y
As x increases, y tends to decrease.
As x increases, y tends to increase.
Negative Linear Correlation
Positive Linear Correlation x y x y
No Correlation
Non-linear Correlation
Chapter 13
Chapter 13 7

8. Examples of Relationships
Statistical Thinking
Examples of Relationships
Chapter 13
Chapter 13

9. Statistical Thinking
Thought Question 1
What type of association would the following pairs of variables have – positive, negative, or none?
Temperature during the summer and electricity bills
Temperature during the winter and heating costs
Number of years of education and height
Frequency of brushing and number of cavities
Number of churches and number of bars in cities
Height of husband and height of wife
Chapter 13
Chapter 13

10. Statistical Thinking
Thought Question 2
Consider the two scatterplots below. How does the outlier impact the correlation for each plot?
does the outlier increase the correlation, decrease the correlation, or have no impact?
Chapter 13
Chapter 13

11. Measuring Strength & Direction of a Linear Relationship
How closely does a non-horizontal straight line fit the points of a scatterplot?
The correlation coefficient (often referred to as just correlation): r
measure of the strength of the relationship: the stronger the relationship, the larger the magnitude of r.
measure of the direction of the relationship: positive r indicates a positive relationship, negative r indicates a negative relationship.
Chapter 13

12. Correlation Coefficient
Statistical Thinking
Correlation Coefficient
Greek Capital Letter Sigma – denotes summation or addition.
The <Plot> link on this slide is to the Correlation & Regression applet found on the VCU Stat 208 website.
The address is .
Chapter 13
Chapter 13

13. Correlation Coefficient
The range of the correlation coefficient is -1 to 1. -1 1
If r = -1 there is a perfect negative correlation
If r is close to 0 there is no linear correlation
If r = 1 there is a perfect positive correlation
Chapter 13

14. Linear Correlation Strong negative correlation
Statistical Thinking
Linear Correlation x y x y
r = 0.91
r = 0.88
Strong negative correlation
Strong positive correlation x y x y
r = 0.42
r = 0.07
Try
Weak positive correlation
Non-linear Correlation
Chapter 13
Chapter 13 14

15. Correlation Coefficient
Statistical Thinking
Correlation Coefficient
special values for r :
a perfect positive linear relationship would have r = +1
a perfect negative linear relationship would have r = -1
if there is no linear relationship, or if the scatterplot points are best fit by a horizontal line, then r = 0
Note: r must be between -1 and +1, inclusive
r > 0: as one variable changes, the other variable tends to change in the same direction
r < 0: as one variable changes, the other variable tends to change in the opposite direction
The <Plot> link on this slide is to the Correlation & Regression applet found on the VCU Stat 208 website.
The address is .
Chapter 13
Chapter 13

16. Examples of Correlations
Statistical Thinking
Examples of Correlations
Husband’s versus Wife’s ages
r = .94
Husband’s versus Wife’s heights
r = .36
Professional Golfer’s Putting Success: Distance of putt in feet versus percent success
r = -.94
The <Plot> link on this slide is to the Correlation & Regression applet found on the VCU Stat 208 website.
The address is .
Plot
Chapter 13
Chapter 13

17. Correlation Coefficient
Statistical Thinking
Correlation Coefficient
Because r uses the z-scores for the observations, it does not change when we change the units of measurements of x , y or both.
Correlation ignores the distinction between explanatory and response variables.
r measures the strength of only linear association between variables.
A large value of r does not necessarily mean that there is a strong linear relationship between the variables – the relationship might not be linear; always look at the scatterplot.
When r is close to 0, it does not mean that there is no relationship between the variables, it means there is no linear relationship.
Outliers can inflate or deflate correlations.
The <Plot> link on this slide is to the Correlation & Regression applet found on the VCU Stat 208 website.
The address is .
Try
Chapter 13
Chapter 13

18. Not all Relationships are Linear Miles per Gallon versus Speed
Curved relationship (r is misleading)
Speed chosen for each subject varies from 20 mph to 60 mph
MPG varies from trial to trial, even at the same speed
Statistical relationship
r=-0.06
Chapter 13

19. Common Errors Involving Correlation
Statistical Thinking
Common Errors Involving Correlation
1. Causation: It is wrong to conclude that correlation implies causality.
2. Averages: Averages suppress individual variation and may inflate the correlation coefficient.
3. Linearity: There may be some relationship between x and y even when there is no linear correlation.
page 525 of Elementary Statistics, 10th Edition
Chapter 13
Chapter 13

20. Correlation and Causation
The fact that two variables are strongly correlated does not in itself imply a cause-and-effect relationship between the variables.
If there is a significant correlation between two variables, you should consider the following possibilities.
Is there a direct cause-and-effect relationship between the variables?
Does x cause y?
Chapter 13

21. Correlation and Causation
Is there a reverse cause-and-effect relationship between the variables?
Does y cause x?
Is it possible that the relationship between the variables can be caused by a third variable or by a combination of several other variables?
Is it possible that the relationship between two variables may be a coincidence?
Chapter 13

22. Example A survey of the world’s nations in 2004 shows a strong
Statistical Thinking
Example
A survey of the world’s nations in 2004 shows a strong
positive correlation between percentage of countries
using cell phones and life expectancy in years at birth.
Does this mean that cell phones are good for your health?
No. It simply means that in countries where cell phone use is high, the life expectancy tends to be high as well.
What might explain the strong correlation?
The economy could be a lurking variable. Richer countries generally have more cell phone use and better health care.
The <Plot> link on this slide is to the Correlation & Regression applet found on the VCU Stat 208 website.
The address is .
Chapter 13
Chapter 13

23. Example The correlation between Age and Income as measured on 100
Statistical Thinking
Example
The correlation between Age and Income as measured on 100
people is r = Explain whether or not each of these
conclusions is justified.
When Age increases, Income increases as well.
The form of the relationship between Age and Income is linear.
There are no outliers in the scatterplot of Income vs. Age.
Whether we measure Age in years or months, the correlation will still be 0.75.
The <Plot> link on this slide is to the Correlation & Regression applet found on the VCU Stat 208 website.
The address is .
Chapter 13
Chapter 13

24. Example Explain the mistakes in the statements below:
Statistical Thinking
Example
Explain the mistakes in the statements below:
“My correlation of between GDP and Infant Mortality Rate shows that there is almost no association between GDP and Infant Mortality Rate”.
“There was a correlation of 0.44 between GDP and Continent”
“There was a very strong correlation of 1.22 between Life Expectancy and GDP”.
The <Plot> link on this slide is to the Correlation & Regression applet found on the VCU Stat 208 website.
The address is .
Chapter 13
Chapter 13

25. Warnings about Statistical Significance
“Statistical significance” does not imply the relationship is strong enough to be considered “practically important.”
Even weak relationships may be labeled statistically significant if the sample size is very large.
Even very strong relationships may not be labeled statistically significant if the sample size is very small.
Chapter 13

26. Key Concepts Strength of Linear Relationship
Direction of Linear Relationship
Correlation Coefficient
Problems with Correlations
r can only be calculated for quantitative data.
Chapter 13