1. Bivariate Data analysis

2. Bivariate Data
In this PowerPoint we look at sets of data which contain two variables.
Scatter plots
Correlation
Outliers
Causation

3. Quantitative (Numerical) Qualitative (categorical)
We are only going to consider
quantitative variables in this AS
Variables
Quantitative
(Numerical)
(measurements and counts)
Qualitative
(categorical)
(define groups)
Continuous
Discrete
Categorical
(no idea of order)
Ordinal
(fall in natural order)

4. Quantitative Discrete Many repeated values Age groups Marks Continuous
Few repeated values
Height
Length
Weight

5. Qualitative Categorical Gender Religious denomination Blood types
Sport’s numbers (e.g. He wears the number ‘8’ jersey)
Ordinal
Grades
Places in a race (e.g. 1st, 2nd, 3rd)

6. We often want to know if there is a relationship between two numerical variables.
A scatter plot, which gives a visual display of the relationship between two variables, provides a good starting point.

7. In a relationship involving two variables, if the values of one variable ‘depend’ on the values of another variable, then the former variable is referred to as the dependent (or response) variable and the latter variable is referred to as the independent (or explanatory) variable.
y - axis dependent (response) variable
x - axis independent (explanatory) variable

8. Consider data on ‘hours of study’ vs ‘ test score’18 59 14 54 17 16 67 72 76 22 74 63 27 90 19 29 89 15 62 20 58 30 93 28 10 47 96 71 85 23 82 60 25 75 26 35 84 78 98 61

9. We may want to see if we could predict the test score (response variable) based on the hours of study (explanatory variable).
y - axis: Test score
x - axis: Hours of study

10. This is called correlation
We look for a
pattern in the way
the points lie
Certain patterns
tell us about
the relationship
This point
is an outlier
This is called correlation

11. We could describe
the rest of the
data as having
a linear form.

12. Scatter plots Use hollow circles for points
Label axes correctly with units
What you want to predict goes on the y-axis (response variable)
Title of graph
No background; No gridlines
Unless you need to show categories- no legend
Show different categories on a single graph in different colours rather than on separate graphs.
Adjust scale and size of font (14pt for pasting)

13. What to look for in your plot?
Direction of the relationship - positive or negative
Form of the graph - linear or curved
The strength - whether it is strong, moderate or weak
Scatter - constant scatter, a fan effect…
Outliers
Groupings

14. Page 22

19. What do you see in this scatter plot?
There appears to be a linear trend.
There appears to be moderate constant scatter.
Negative Association.
No outliers or groupings visible. 45 40 35 20 19 18 17 16 15 14
Latitude (°S)   
Mean January Air Temperatures for 30 New Zealand Locations
Temperature (°C)

20. What do you see in this scatter plot?
There appears to be a non-linear trend.
There appears to be non-constant scatter about the trend line.
Positive Association.
One possible outlier (Large GDP, low % Internet Users).
% of population who are Internet Users vs GDP per capita for 202 Countries 10 20 30 40
GDP per capita (thousands of dollars) 50 60 70 80
Internet Users (%)

21. What do you see in this scatter plot?
Two non-linear trends (Male and Female).
Very little scatter about the trend lines
Negative association until about 1970, then a positive association.
Gap in the data collection (Second World War).
Year
1990
1980
1970
1960
1950
1940
1930 30 28 26 24 22 20
Age
Average Age New Zealanders are First Married

22. Rank these relationships from weakest (1) to strongest (4):2 1 3

23. Describe these relationships
Perfect,
positive,
linear
relationship
Perfect,
negative,
linear
relationship No
relationship
Moderate,
negative
linear
relationship
Weak,
positive
linear
relationship

24. Describe this relationship.

25. As the hours of study increase, the test score . . . .? . . .

26. Pearson’s product-moment correlation coefficient, r
Points fall exactly on a straight line
Correlation measures the strength of the linear association between two quantitative variables.
No linear relationship
(uncorrelated)
Points fall exactly on a straight line
r = -1
The correlation coefficient may take any value between -1.0 and +1.0
r = -0.7
r = -0.4
r = 0.3
r = 0
r = 0.8
r = 1

27. r - what does it tell you? How close the points in the
scatter plot come to lying on the line.
r = 0.99 x y *
r = 0.57
r = 0.99
r = 0.57

28. Interpreting r 0.75-1 Strong positive linear association
Moderate positive linear association
Weak positive linear association
No association or weak linear association
Weak negative linear association
Moderate negative linear association
Strong negative linear association

29. Useful websites http://istics.net/stat/Correlations/ Guessing
Regression by eye
Guessing
effect of outliers

30. Assumptions linear relationship between x and y
continuous random variables
The residuals must be normally distributed
x and y must be independent of each other
all individuals must be selected at random from the population
all individuals must have equal chance of being selected

31. What is correlation?
A measure of the strength of a LINEAR association between two quantitative variables.

32. Sure you can calculate a correlation coefficient for any pair of variables but correlation measures the strength only of the linear association and will be misleading if the relationship is not linear.

33. Do you know that:
Correlation applies only to quantitative variables. Check you know the units and what they measure.
Outliers can distort the correlation dramatically.

34. Some facts about the correlation coefficient
The sign gives the direction of the association.
Correlation is always between -1 and 1.
Correlation treats x and y symmetrically. The correlation of x and y is the same as the correlation of y with x.
Correlation has no units and is generally given as a decimal.
r is a multiple of the slope
Note: variables can have a strong association but still have a small correlation if the association isn’t linear.
Correlation is sensitive to outliers. A single outlying value can make a small correlation large or make a large one small.

35. The sign gives the direction of the association.
Positive
Negative

36. Correlation treats x and y symmetrically
Correlation treats x and y symmetrically. The correlation of x and y is the same as the correlation of y with x.

37. r is a multiple of the slope

38. Variables can have a strong association but still have a small correlation if the association isn’t linear.
Always plot the data before looking at the correlation!

39. Would it be OK to use a correlation coefficient to describe the strength of the relationship?9 8 7 6 5 4 3 2 1
4000
3000
2000
1000
Position Number
Distance (million miles)
Distances of Planets from the Sun
Reaction Times (seconds) for 30 Year 10 Students
0.2
0.4
0.6
0.8 1
Non-dominant Hand
Dominant Hand
Female ($)
Average Weekly Income for Employed New Zealanders in 2001
Male ($)
200
400
600
800
1000
1200 45 40 35 20 19 18 17 16 15 14
Latitude (°S)   
Mean January Air Temperatures for 30 New Zealand Locations
Temperature (°C) X X √ √

40. Correlation is sensitive to outliers
Correlation is sensitive to outliers. A single outlying value can make a small correlation large or make a large one small.

41. You should be cautious in interpreting the correlation - these graphs all have the same correlation coefficient (0.817)

42. Data set 1

43. Data set 2

44. Data set 3

45. Data set 4

46. Outliers can distort the correlation dramatically
Outliers can distort the correlation dramatically. An outlier can make an otherwise small correlation look big or hide a large correlation. It can even give an otherwise positive association a negative correlation coefficient (and vice versa).

47. What do you see in this scatterplot?22 23 24 25 26 27 28 29
150
160
170
180
190
200
Foot size (cm)
Height (cm)
Height and Foot Size for 30 Year 10 Students
Appears to be a linear trend, with a possible outlier (tall person with a small foot size.)
Appears to be constant scatter.
Positive association.

48. Height and Foot Size for 30 Year 10 Students
What will happen to the correlation coefficient if the tallest Year 10 student is removed? 22 23 24 25 26 27 28 29
150
160
170
180
190
200
Foot size (cm)
Height (cm)
Height and Foot Size for 30 Year 10 Students
It will get smaller
It will get bigger
It will stay the same

49. What do you see in this scatter plot?
Appears to be a strong linear trend.
Outlier in X (the elephant).
Appears to be constant scatter.
Positive association.
600
500
400
300
200
100 40 30 20 10
Gestation (Days)
Life Expectancy (Years)
Life Expectancies and Gestation Period for a sample of non-human Mammals
Elephant

50. What will happen to the correlation coefficient if the elephant is removed?
600
500
400
300
200
100 40 30 20 10
Gestation (Days)
Life Expectancy (Years)
Life Expectancies and Gestation Period for a sample of non-human Mammals
Elephant
It will get smaller
It will get bigger
It will stay the same

51. How does the outlier affect the r - value?

52. How does the outlier affect the r - value?

53. How does the outlier affect the r - value?

54. How does the outlier affect the r - value?

55. How does the outlier affect the r - value?

56. How does the outlier affect the r - value?

57. When you see an outlier, it’s often a good idea to report the correlations with and without the point.

58. Don’t confuse Correlation with causation
Don’t confuse Correlation with causation. Scatterplots and correlation never prove causation.

59. Using the information in the plot, can you suggest what needs to be done in a country to increase the life expectancy? Explain.
Life Expectancy and Availability of Doctors for a Sample of 40 Countries
40000
30000
20000
10000 80 70 60 50
People per Doctor
Life Expectancy
Perhaps if you have less people per Doctor (i.e. more Doctors per person), then the life expectancy will increase.

60. Life Expectancy and Availability of Televisions for a
Using the information in this plot, can you make another suggestion as to what needs to be done in a country to increase life expectancy?
Life Expectancy and Availability of Televisions for a
Sample of 40 Countries
It looks like if you decrease the number of people per television (i.e. have more TVs per person), then the life expectancy will increase!
600
500
400
300
200
100 80 70 60 50
People per Television
Life Expectancy

61. Can you suggest another variable that is linked to life expectancy and the availability of doctors (and televisions) which explains the association between the life expectancy and the availability of doctors (and televisions)?
Some measure of wealth of a country.
Eg Average income per person or GDP.

62. Damaged for life by too much TV

63. Damaged for life by too much TV
Watching too much television as a child causes serious health problems years later, and raises the risk of heart disease, a New Zealand study of 1000 children has found….
It links the amount of time spent in front of the box as a child with obesity, high cholesterol, poor fitness and smoking….

64. Damaged for life by too much TV
Health Score
TV watching

65. Causal relationships
Two general types of studies: experiments and observational studies
In an experiment, the experimenter determines which experimental units receive which treatments.
In an observational study, we simply compare units that happen to have received each of the treatments.

66. Causal relationships
Only properly designed and carefully executed experiments can reliably demonstrate causation.
An observational study is often useful for identifying possible causes of effects, but it cannot reliably establish causation

67. Causal relationships
In observational studies, strong relationships are not necessarily causal relationships.
Correlation does not imply causation.
Be aware of the possibility of lurking variables.

68. Watch out for lurking variables
Watch out for lurking variables. Damage ($) vs number of firemen would show a strong correlation, but damage doesn’t cause firemen and firemen do seem to cause damage (spraying water and chopping holes). The underlying variable is the size of the blaze.

69. Although there was plenty of evidence that increased smoking was associated with increased levels of lung cancer, it took years to provide evidence that smoking actually causes lung cancer.

70. It would be a good idea to read the two pages of notes you have that discusses correlation and causation!

71. So now you want to know how to calculate the correlation coefficient, r. Here is one version of the formula!

73. Luckily the computer will calculate R2 and you can square root this to get r. Remember only when the association is linear.

74. r measures the strength of the relationship NOT R2
r measures the strength of the relationship NOT R2!!!! r measures the strength of the relationship NOT R2!!!! r measures the strength of the relationship NOT R2!!!!

75. The words you use
There is a strong, positive, linear relationship between ‘x’ and ‘y’ and when the x- values increase, the y-values increase also. This is indicated by the value of the correlation coefficient i.e. r = 0.85 which is close to 1.
(Note: Do not use ‘x’ and ‘y’ use what they represent.)