Scatter Diagrams Correlation Classifications
• Scatter diagrams are used to demonstrate correlation between two • Correlation can be 19 quantitative variables. classified into three basic Regression Plot 18 categories 19 • Often, this correlation is 17 18 • Linear 17 linear . 16 16
15 15 Weight • Nonlinear • This means that a straight Weight line model can be 14 • No correlation 14 13 13 developed. 12 12 20 21 22 23 24 Length 20 21 22 23 24 Length
Chapter 5 # 1 Chapter 5 # 2
Correlation Classifications Correlation Classifications
• Two variables may be
correlated but not through Curvilinear Correlation
a linear model. 500 50 40 400 • Two quantitative • This type of model is 30 300 variables may not be 20
result 10 called non-linear 200
correlated at all C4 0 • The model might be one 100 -10 -20 0 -30 of a curve. 0 5 10 15 -40 sample 0 5 10 15 20 25 Animalno
Chapter 5 # 3 Chapter 5 # 4 Linear Correlation Linear Correlation
• Variables that are correlated through a Regression Plot 19 linear relationship can • Negatively correlated Regression Plot 18 variables vary as display either positive 17 opposites 4 or negative correlation 16
15 • Positively correlated Weight • As the value of one 14 variable increases the variables vary directly. 3 13 other decreases 12 Student GPA 20 21 22 23 24 Length
2
0 10 20 30 40 Hours Worked
Chapter 5 # 5 Chapter 5 # 6
Strength of Correlation Strength of Correlation
• Correlation may be strong, • When the data is moderate, or weak. Regression Plot distributed quite close Regression Plot to the line the 95 • You can estimate the 90 4 strength be observing the correlation is said to 85 be strong 80 variation of the points 75
70
around the line 3 • The correlation type is 65 independent of the 60 Final Exam Score • Large variation is weak Student GPA strength. 55 correlation 50
2 55 65 75 85 95 0 10 20 30 40 Midterm Stats Grade Hours Worked
Chapter 5 # 7 Chapter 5 # 8 The Correlation Coefficient Interpreting r
• The sign of the correlation coefficient tells us the • The strength of a linear relationship is measured direction of the linear relationship by the correlation coefficient If r is negative (<0) the correlation is negative. The • The sample correlation coefficient is given the line slopes down symbol “r” If r is positive (> 0) the correlation is positive. The • The population correlation coefficient has the line slopes up symbol “ρρρ”.
Chapter 5 # 9 Chapter 5 # 10
Interpreting r Cautions
• The size (magnitude) of the correlation • The correlation coefficient only gives us an coefficient tells us the strength of a linear indication about the strength of a linear relationship relationship . If | r | > 0.90 implies a strong linear association • Two variables may have a strong curvilinear For 0.65 < | r | < 0.90 implies a moderate linear relationship, but they could have a “weak” value association for r For | r | < 0.65 this is a weak linear association
Chapter 5 # 11 Chapter 5 # 12 Fundamental Rule of Correlation Setting
• Correlation DOES NOT imply causation • A chemical engineer would like to determine if a – Just because two variables are highly correlated does relationship exists between the extrusion not mean that the explanatory variable “causes ” the temperature and the strength of a certain response formulation of plastic. She oversees the production of 15 batches of plastic at various • Recall the discussion about the correlation temperatures and records the strength results. between sexual assaults and ice cream cone sales
Chapter 5 # 13 Chapter 5 # 14
The Study Variables The Experimental Data
• The two variables of interest in this study are the strength of the plastic and the extrusion temperature. • The independent variable is extrusion temp. This is the Temp 120 125 130 135 140 variable over which the experimenter has control. She Str 18 22 28 31 36 can set this at whatever level she sees as appropriate. • The response variable is strength. The value of “strength” Temp 145 150 155 160 165 is thought to be “dependent on” temperature. Str 40 47 50 52 58
Chapter 5 # 15 Chapter 5 # 16 The Scatter Plot Conclusions by Inspection
• The scatter diagram for • Does there appear to be a relationship between the the temperature versus Scatter diagram of Strength vs Temperature 60 study variables? strength data allows us to deduce the nature of the 50 • Classify the relationship as: Linear, curvilinear, no relationship between these 40 relationship Strength (psi) two variables 30 • Classify the correlation as positive, negative, or no
20 correlation
120 130 140 150 160 170 Temperature (F) • Classify the strength of the correlation as strong, moderate, weak, or none
What can we conclude simply from the scatter diagram?
Chapter 5 # 17 Chapter 5 # 18
Computing r Computing r
1 x − x y − y 1 r = Σ r = Σ[]()Z ()Z − − x y n 1 sx sy n 1
z-scores df z-scores for y data for x data
Chapter 5 # 19 Chapter 5 # 20 Computing r - Example Classifying the strength of linear correlation
•The strength of a linear correlation between the response See example handout for the plastic strength versus and the explanatory variable can be assigned based on r extrusion temperature setting