1. © 2006 Hewlett-Packard Development Company, L.P.
The information contained herein is subject to change without notice
Oleg Janke
25-June-2012
Tip of the hat to Kevin Kacmarynski
Scatterplots and
Cautions of
Correlation

2. Topics for Today’s Discussion
• Scatterplots:
− Why bother?
− Creating & Analyzing
− Good usage
• Correlation and Association
• Potential Missteps
• Summary
• Real Personal Application

3. Presumption
• Basic graphing skills
• Familiar with charting (for example, as Excel)
• YES NO

4. Scatterplots: Why bother?
• Scatterplot (Scatter diagram)
− Converts two columns of numbers
(ordered pairs) into picture
− Explores relationship between two
quantitative variables
• What value does it have?
− Determine possible cause and
effect links (control)
− Predict results of variable that is
difficult to measure if it is strongly
related to another variable that is
easier to measure (proxy)

5. Creating a scatterplot
What is the relationship between height & weight?

6. Creating a scatterplot
Plot each characteristic of interest on a standard XY plot.
Katie James

7. Analyzing a scatterplot
Is there a relationship? OR Is it just randomness? (N=40)

8. Analyzing a scatterplot
Is there a relationship? OR Is it just randomness? (N=40)
How confident are you
that there is a linear
relationship between
height and weight in
this data set? (Choose
one)
• 100%
• 99-100%
• 95-99%
• 90-95%
• 80-90%
• insufficient data to say
• no relationship

9. Analyzing a scatterplot
Is there a relationship or is it just randomness? (N=40)
• Add Median lines and count quadrant totals+
Median X
14
6
6
14
Median Y
+
Olmstead-Tukey 1947

10. Analyzing a scatterplot
Is there a relationship or is it just randomness? (N=40)
• Add Median lines and count quadrant totals+
Median X
14
6
6
14
Median Y
+
Olmstead-Tukey 1947
NO relationship
• shotgun effect
• appx equal number in
each quadrant
IS a relationship
• one diagonal will dominate

11. Analyzing a scatterplot
Is there a relationship or is it just randomness? (N=40)
• Add Median lines and count quadrant totals
Median X
14
6
6
14
Median Y
•Less than 5% chance data could
align this way simply from
randomness
• Therefore fairly confident X& Y
are related
SIGN TEST TABLE *
N 1% 5%
10 0 1
20 3 5
30 7 9
40 11 13
50 15 17
60 19 21
* Ishikawa “Guide to Quality Control”, 1976

12. Good usage of Scatterplots
• In this plot, we observe
a clear relationship
between height and
weight
• As height of individuals
increase, their weight
tends to increase as
well
• In the ideal case this
relationship is called
Body Mass Index (BMI)

13. Good usage of scatterplots
Scenario 1:
We are building
parts on one line in
one location.
What does the plot tell us about part
length and part diameter?

14. Good usage of scatterplots
Scenario 2:
We build the
same part on two
different Lines.
Now what does plot tell us about part length & part diameter?

15. Good usage of scatterplots
• IS a Line effect here
• Relationship between
Diam & Length differs by
Line
• Diam1 twice Daim2
− Tighter process control?
• Length1 < Length2
Now what does plot tell us about part length & part diameter?
• Always be alert to possible strata in the data
• Plotting your data is crucial for discovery

16. Correlation Coefficient
• Correlation is defined as measure of strength of linear
relationship between two quantitative variables
− Correlation coefficient is a mathematically calculated value:
− Correlation values are always between -1 and +1
• 0 indicates no correlation (perfect shotgun pattern)
• -1 and +1 indicates perfect correlation (all points fall on line)
• Sign indicates direction
− Positive: up and to right
− Negative: down and to left

17. Correlation and Association
• Re-visiting our first example,
we saw strong, positive
relationship between height
and weight
• Supported by correlation
coefficient value of 0.709
• Relationship exists
• Does NOT prove causality
Correlation = 0.709
Calculation provided by JMP statistical software

18. Correlation and Association
Medical Trial
• Dosage 490-510 mg
• Recorded therapeutic response
from 20 to 100
Is there a correlation between
Dosage and Response?
• Yes
• No
• Insufficient data
• Don’t know

19. Correlation and Association
• Since calculated
correlation value is zero,
there is no association
between dosage and
desired response! Right?
− No LINEAR relationship
• Correlation coefficient, by
itself, does not tell the
entire story
• Always look at your graphs
to see what the data say
Correlation coefficient = 0
Calculation provided by JMP statistical software
2 2
2 2

20. Questions?
How to create/analyze Scatterplots
Correlation and Associations

21. Missteps with scatterplots & correlation
1. Bimodal distributions
2. Stratified data
3. Lurking variables
4. Extrapolation
5. Too narrow range of X (independent variable)
6. Weak/sloppy measurement
7. Chicken and Egg Syndrome

22. Misstep #1with scatterplots &
correlation
• What is my house worth?
− Sale price data & house size
were collected on 21 houses
in the same town
− Another house (mine) in
same town is 2300 square
feet in size, so it should be
worth a little over $200K
− Correlation coefficient =
0.943 (very high)
What is the problemin this analysis and the
resulting conclusion?

23. Misstep #1with scatterplots &
correlation
What is the problem?
• Relationship/correlation dependent solely on one data point
• Why might this one point not be appropriate?
• Location (school district, suburban)
• Features (pool, lot, barn, view)
• Timing (peak of housing bubble)
• Example of Bimodal data
Need both appropriate data & proper analysis techniques
4
7
6
4
Is there a linear relationship?
• What do median lines say about the
relationship?
Sign Test Table
N 1% 5%
10 0 1
20 3 5
30 7 9
40 11 13
50 15 17

24. Misstep #2 with scatterplots & correlation
• Based on this data set,
with high Correlation
Coefficient (0.780) what’s
the relationship between
shoe size and knowledge?
• What’s missing?
Correlation = 0.780
Calculation provided by JMP statistical software

25. Misstep #2 with scatterplots & correlation
• Does this help solve mystery?
• Be sure to look for hidden
variables that might have an
impact on relationship
• Stratified Data– sub population
with different relationships – can
give erroneous conclusions
For example: CSAT data
• Those who respond to survey
• Those who do NOT respond to survey
Do both groups have similar opinions?

26. Misstep #3 with scatterplots &
correlation
Beware of Lurking Variables
• Related thru common 3rd
variable
− Ice cream sales correlates with water usage (temperature)
− Height–weight example (age)
− Call vol at hp Call Center A correlated w/call vol at hp CC--B (business)
• Related thru independent growth (decay) rates
− Population in Indonesia correlates with price of tea in NYC (growth)
− My car’s value correlates to grams of Cobalt-60 isotope (decay)
• Both have half-lives of about 5-6 years
• Related through measuring same characteristic differently
− Weight in pounds is correlated to weight in kilos
− Attendance at an event is correlated to empty seats at same event
− Area of a US state is correlated with population of that state
• Some notable exceptions (AK, MT)

27. Does School Spending Educate
Students?
States spending more per student have lower SAT* scores!
Expediture/pupil by State vs SAT
900
950
1000
1050
1100
1150
1200
$4,000 $5,000 $6,000 $7,000 $8,000 $9,000 $10,000 $11,000 $12,000
Expenditure per pupil for public school K-12 (2002-03)
SATscoresfor1998
Obvious
Negative
Correlation
* SAT test is a standardized test used by many colleges across
US to determine level of student preparedness for college

28. Does School Spending Educate
Students?
States spending more per student
have lower SAT scores!
Expediture/pupil by State vs SAT
900
950
1000
1050
1100
1150
1200
$4,000 $5,000 $6,000 $7,000 $8,000 $9,000 $10,000 $11,000 $12,000
Expenditure per pupil for public school K-12 (2002-03)
SATscoresfor1998
Negative Correlation
Do we have a measurement issue here?
What does SAT scores actually measure?
• Test performance – at a minimum
• Education – Not always correlated with Knowledge
• Knowledge – Our belief that this leads to Life Success
• Life Success -- This is what we would like to be the case
Be careful of proxies that stand in for other measures

29. Does School Spending Educate
Students?
States spending more per student
have lower SAT scores!
Expediture/pupil by State vs SAT
900
950
1000
1050
1100
1150
1200
$4,000 $5,000 $6,000 $7,000 $8,000 $9,000 $10,000 $11,000 $12,000
Expenditure per pupil for public school K-12 (2002-03)
SATscoresfor1998
Negative Correlation
Do we have a measurement issue here?
No! SAT scores are predictive of Life Success-- financial
• Life Success – college grad, good job
• Not with certainty, but on average
• Should we move to states with lower student spending?

30. Does School Spending Educate
Students?
Does Percent of students taking SAT impact SAT scores?
Correlation Coefficient = .92
Beware the Lurking Variable!
1998 SAT by State
y = 1278x-0.0575
R2
= 0.8461
900
1000
1100
1200
0 10 20 30 40 50 60 70 80 90
% Taking SAT
CompositeSAT
OR
SC
WV
DC
NH
GA
WA
AK
CO
IL
M N
TX
MS
OH
KS
WI
USA
UT
M A
NY
NJ
CT
INHI NC
NV
M T
VT
VT PA
RI
MSMSMS

31. Misstep #4 with scatterplots & correlation
Back to Height & Weight data
• How much would an 100 inch (~2.5meters) person weigh?
80 90 100
300
280
260
240
220
200
•From scatterplot he
would weight ~300#
(136 kg)
•Can we make this
prediction?

32. Misstep #4 with
scatterplots & correlation
• “Predicted” 300 pounds!
•Robert Wadlow was 8 ft 11 (2.72 m)
and weighed 439# (199 kg)
• Interpolating within range of
independent variable set --
acceptable
•Extrapolating beyond range of
independent variable is dangerous
• Relationship may not be stable

33. Misstep #5 with scatterplots & correlation
Back to Height & Weight data
• What would conclusion be if height ranged from 1600.0 mm to
1700.0 mm (64-66 inches)?
• Easily conclude no
relationship between
height and weight
• Make sure range of
independent variable (X)
sufficiently large relative
to dependent variable (Y)

34. Misstep #6 with scatterplots & correlation
Poor Measurement System
• Inappropriate tool or gage to measure
− Pixel width with standard yard/meter stick
− Monitor response time with second hand on watch
• Weak tech repeatability
− Tech visual determination of damage of NB set in for repair
− Typo-graphical errors on a written page
Ensure Measurement System Analysis performed before data are collected

35. Misstep #7 with scatterplots & correlation
Chicken and Egg Syndrome
• Which came first?
• What is the cause and what is the effect?
− Do children from poor families do poorly academically because they
are poor OR are they poor because of poor academic performance?
− Do consumers buy good product out of loyalty OR are consumers
loyal because of good product?
• Vicious/Virtuous Cycles – hard to break through
• Relationship is there; Causality is not easily determined

36. Misstep Summary
Watch out for ….
• Bimodal distribution House Size and Price
• Stratified data  Shoe Size and Knowledge
• Lurking variable  SAT Scores and Participation Rate
− Underlying third variable
− Common but unrelated growth/decay curves
− Same variable measured differently
• Extrapolation  Height and Weight for tallest man
− Generalizing from a sampled subset to a broader, larger population
• Narrow range of X  Height and Weight
• Sloppy measurement
− Can hide a real relationship
− May create one when none exists
• Chicken and Egg Syndrome
− Variables are related, but which is cause & which is effect?

37. Questions?
Seven Missteps

38. Summary
• Scatterplots
− Simple, but powerful tool to explore relationships
between two quantitative variables
• Be sure data are representative of question
− “What are we trying to accomplish?”
• Plot data to lookforanomalies orassociations
• Correlation has special meaning
− Correlation does not imply causation
− Nor does lack of correlation deny causation
• Recall Missteps that may impact scatterplot/correlation
analysis including lurking variables

39. Personal
Example

40. Memory Loss Boosts Risk of Death

41. x
Memory Loss Boosts Risk of Death
Cognitive
Impairment
Qty Lifespan
median month
Mortality
None 3157 138 57%
Mild 533 106 68%
Moderate
267 63 79%
Severe
Were there any
missteps in the
analysis?
X
Key Points in Article
• About 4000 men & women
• Aged 60 to 102
• Indianapolis, Indiana. USA
• Started in early 1990’s; ended 2006
• Lower socio-economic background
• 10 questions to assess mental status
• Primary care Dr appt
• No intervening follow-up of mental assessment

42. Memory Loss Boosts Risk of Death
Potential missteps in analysis
1) Applies equally for men and women?
 Stratified data?
1) Indy only? OR for all USA? OR World Wide?
2) Only applies to those that go to Doctor?
4) Socio-economic Background– What role does it play?
 Three possible Extrapolations
5) How repeatable was 10 question assessment?
 Measurement system
6) Why combine Moderate & Severe?
7) Depends on fewer points at extreme
 Bimodal?
6) And the Big Misstep
 Lurking variable!!!
Key Points in Article
• About 4000 men & women
• Aged 60 to 102
• Indianapolis, Indiana. USA
• Started in early 1990’s; ended 2006
• Lower socio-economic background
• 10 questions to assess mental status
• Primary care Dr appt
• No intervening follow-up of mental
assessment
Cognitive
Impa irment
Q ty Lifespa n
median month
Morta lity
None 3157 138 57%
Mild 533 106 68%
Moderate
267 63 79%
Severe
Cognitive
Impa irment
Q ty Lifespa n
median month
Morta lity
None 3157 138 57%
Mild 533 106 68%
Moderate
267 63 79%
Severe

43. Memory Loss Boosts Risk of Death
Potential missteps in analysis
1) Applies equally for men and women?
 Stratified data?
1) Indy only? OR for all USA? OR WW?
2) Only applies to those that go to Doctor?
4) Socio-economic Background– What role does it play?
 Three possible Extrapolations
5) How repeatable was 10 question assessment?
 Measurement system
6) Why combine Moderate & Severe?
7) Depends on fewer points at extreme
 Bimodal?
6) And the Biggie ….
 A Lurking variable!!!
Key Points in Article
• About 4000 men & women
• Aged 60 to 102
• Indianapolis, Indiana. USA
• Started in early 1990’s; ended 2006
• Lower socio-economic background
• 10 questions to assess mental status
• Primary care Dr appt
• No intervening follow-up of mental assessment
Cognitive
Im pa irm ent
Q ty Lifespa n
median month
Morta lity
None 3157 138 57%
Mild 533 106 68%
Moderate
267 63 79%
Severe
Cognitive
Im pa irm ent
Q ty Lifespa n
median month
Morta lity
None 3157 138 57%
Mild 533 106 68%
Moderate
267 63 79%
Severe
Does Cognitive Impairment hasten death?
OR
Does Age Boost the Risk of Death?

44. Questions?