CAS MA 115 Lecture Notes - Lecture 4: Dependent And Independent Variables, Regional Policy Of The European Union, Robust Statistics
1 May 2018
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CHAPTER 4 – DESCRIBING THE RELATION BETWEEN TWO
VARIABLES
Section 4.1 – Scatter Diagrams and Correlation
Objective 1: Draw and Interpret Scatter Diagrams
• Response Variable – the variable whose value can be explained by the value of the
explanatory variable (ex: the time to drill is the response variable while the depth is the
explanatory variable)
• Scatter Diagram – a graph that shows the relationship between two quantitative variables
measured on the same individual
o Each individual is represented by a point and the explanatory variable is on the
horizontal axis while the response variable is plotted on the vertical axis
o Types of Scatter Diagram Relationships:
o
• Positive v. Negative Association
o Positive Association – two linearly related variables when above-average/below-
average values of one are associated with above-average/below-average of
another
▪ Essentially, two variables are positively associated if they are directly
related (ex: above x-bar values are associated with above y-bar values
while below x-bar values are associated with below y-bar values)
▪ Most of the data will fall within the 1st and 3rd quadrants
o Negative Association – two linearly related variables when above-average values
of one are associated with below-average of another and vice versa
▪ Essentially, two variables are negatively associated if they are inversely
related (ex: above x-bar values are associated with below y-bar values
while below x-bar values are associated with above y-bar values)
▪ Most of the data will fall within the 2nd and 4th quadrants
Objective 2: Describe the Properties of the Linear Correlation Coefficient
• Linear Correlation Coefficient – a measure of the strength and direction of the linear
relation between two quantitative variables
find more resources at oneclass.com
find more resources at oneclass.com
o ex: positive = increasing linear relationship & negative = decreasing linear
relationship
o Population Linear Correlation Coefficient (ρ)
o Sample Linear Correlation Coefficient (r)
▪ Formula:
• Properties of the Linear Correlation Coefficient
o 1. The linear correlation coefficient is always between −1 and 1.
▪ −1 ≤ r ≤ 1.
o 2. If r = + 1, then a perfect positive linear relation exists between the two
variables.
o 3. If r = −1, then a perfect negative linear relation exists between the two
variables.
o 4. The closer r is to +1, the stronger the evidence is of a positive association
between the two variables.
o 5. The closer r is to −1, the stronger the evidence is of a negative association
between the two variables.
o 6. If r is close to 0, then little or no evidence exists of a linear relation between the
two variables. But this does not imply no relation at all, just no linear relation.
o 7. The linear correlation coefficient is a unit less measure.
o 8. The correlation coefficient is not a resistant statistic.
o 9. If an observation does not follow the overall pattern of the data, it could affect
the value of the linear correlation coefficient.
o 10. Having more data points will give a stronger correlation and a higher r value
whereas having less data points will give a weaker correlation and a lower r value.
• Types of Linear Correlation Coefficient Relationships:
find more resources at oneclass.com
find more resources at oneclass.com
Document Summary
Chapter 4 describing the relation between two. The linear correlation coefficient is always between 1 and 1. If r = + 1, then a perfect positive linear relation exists between the two variables: 3. If r = 1, then a perfect negative linear relation exists between the two variables: 4. The closer r is to +1, the stronger the evidence is of a positive association between the two variables: 5. The closer r is to 1, the stronger the evidence is of a negative association between the two variables: 6. If r is close to 0, then little or no evidence exists of a linear relation between the two variables. But this does not imply no relation at all, just no linear relation: 7. The linear correlation coefficient is a unit less measure: 8. The correlation coefficient is not a resistant statistic: 9.
