1. Chapter 3 ~ Descriptive Analysis & Presentation of Bivariate Data5 4 3 2 1 6
Weight
Height
Regression Plot
Y = X
r = 0.559

2. Chapter Goals
To be able to present bivariate data in tabular and graphic form
To become familiar with the ideas of descriptive presentation
To gain an understanding of the distinction between the basic purposes of correlation analysis and regression analysis

3. 3.1 ~ Bivariate Data
Bivariate Data: Consists of the values of two different response variables that are obtained from the same population of interest
Three combinations of variable types:
1. Both variables are qualitative (attribute)
2. One variable is qualitative (attribute) and the other is quantitative (numerical)
3. Both variables are quantitative (both numerical)

4. Two Qualitative Variables
When bivariate data results from two qualitative (attribute or categorical) variables, the data is often arranged on a cross-tabulation or contingency table
Example: A survey was conducted to investigate the relationship between preferences for television, radio, or newspaper for national news, and gender. The results are given in the table below:

5. Marginal Totals
This table may be extended to display the marginal totals (or marginals). The total of the marginal totals is the grand total:
Row Totals
760
560
Col. Totals
395
450
475
1320 TV
Radio NP
Male
280
175
305
Female
115
275
170
Note: Contingency tables often show percentages (relative frequencies). These percentages are based on the entire sample or on the subsample (row or column) classifications.

6. Percentages Based on the Grand Total (Entire Sample)
The previous contingency table may be converted to percentages of the grand total by dividing each frequency by the grand total and multiplying by 100
For example, 175 becomes 13.3% TV
Radio NP
Row Totals
Male
21.2
13.3
23.1
57.6
Female
8.7
20.8
12.9
42.4
Col. Totals
29.9
34.1
36.0
100.0
175
1320
100 13 3 = æ è ç ö ø ÷ .

7. Percentages Based on Grand Total
Illustration
These same statistics (numerical values describing sample results) can be shown in a (side-by-side) bar graph: 5 10 15 20 25 TV
Radio NP
Male
Female
Percentages Based on Grand Total
Percent
Media

8. Percentages Based on Row (Column) Totals
The entries in a contingency table may also be expressed as percentages of the row (column) totals by dividing each row (column) entry by that row’s (column’s) total and multiplying by The entries in the contingency table below are expressed as percentages of the column totals:
Note: These statistics may also be displayed in a side-by-side bar graph

9. One Qualitative & One Quantitative Variable
1. When bivariate data results from one qualitative and one quantitative variable, the quantitative values are viewed as separate samples
2. Each set is identified by levels of the qualitative variable
3. Each sample is described using summary statistics, and the results are displayed for side-by-side comparison
4. Statistics for comparison: measures of central tendency, measures of variation, 5-number summary
5. Graphs for comparison: dotplot, boxplot

10. Example
Example: A random sample of households from three different parts of the country was obtained and their electric bill for June was recorded. The data is given in the table below:
The part of the country is a qualitative variable with three levels of response. The electric bill is a quantitative variable. The electric bills may be compared with numerical and graphical techniques.

11. Comparison Using Dotplots:
Northeast .
:..:. ..
Midwest :
West
The electric bills in the Northeast tend to be more spread out than those in the Midwest. The bills in the West tend to be higher than both those in the Northeast and Midwest.

12. Comparison Using Box-and-Whisker Plots2 3 4 5 6 7
Electric
Bill
The Monthly Electric Bill

13. Two Quantitative Variables
1. Expressed as ordered pairs: (x, y)
2. x: input variable, independent variable y: output variable, dependent variable
Scatter Diagram: A plot of all the ordered pairs of bivariate data on a coordinate axis system. The input variable x is plotted on the horizontal axis, and the output variable y is plotted on the vertical axis.
Note: Use scales so that the range of the y-values is equal to or slightly less than the range of the x-values. This creates a window that is approximately square.

14. Example
Example: In a study involving children’s fear related to being hospitalized, the age and the score each child made on the Child Medical Fear Scale (CMFS) are given in the table below:
Construct a scatter diagram for this data

15. Child Medical Fear Scale
Solution
age = input variable, CMFS = output variable
Child Medical Fear Scale 1 5 4 3 2 9 8 7 6
CMFS
Age

16. 3.2 ~ Linear Correlation
Measures the strength of a linear relationship between two variables
As x increases, no definite shift in y: no correlation
As x increases, a definite shift in y: correlation
Positive correlation: x increases, y increases
Negative correlation: x increases, y decreases
If the ordered pairs follow a straight-line path: linear correlation

17. Example: No Correlation
As x increases, there is no definite shift in y: 3 2 1 5 4
Output
Input

18. Example: Positive Correlation
As x increases, y also increases: 5 4 3 2 1 6
Output
Input

19. Example: Negative Correlation
As x increases, y decreases:
Output
Input 5 4 3 2 1 9 8 7 6

20. Please Note
Perfect positive correlation: all the points lie along a line with positive slope
Perfect negative correlation: all the points lie along a line with negative slope
If the points lie along a horizontal or vertical line: no correlation
If the points exhibit some other nonlinear pattern: no linear relationship, no correlation
Need some way to measure correlation

21. 3.1 ~ Bivariate Data
Coefficient of Linear Correlation: r, measures the strength of the linear relationship between two variables
Pearson’s Product Moment Formula:
Notes:
r = +1: perfect positive correlation
r = -1 : perfect negative correlation

22. Alternate Formula for rSS
“sum of squ
ares for ( ) x x” = n - å 2 SS
“sum of squ
ares for ( ) y y” = n - å 2 SS
“sum of squ
ares for ( ) xy
xy” = x y n - å

23. Example
Example: The table below presents the weight (in thousands of pounds) x and the gasoline mileage (miles per gallon) y for ten different automobiles. Find the linear correlation coefficient:

24. Completing the Calculation for rxy x y = - SS ( ) . )( 0. 42 79 7
449
1116 9 47

25. Please Note r is usually rounded to the nearest hundredth
r close to 0: little or no linear correlation
As the magnitude of r increases, towards -1 or +1, there is an increasingly stronger linear correlation between the two variables
Method of estimating r based on the scatter diagram. Window should be approximately square. Useful for checking calculations.

26. 3.3 ~ Linear Regression
Regression analysis finds the equation of the line that best describes the relationship between two variables
One use of this equation: to make predictions

27. Models or Prediction Equations
Some examples of various possible relationships: y ^ b x = + 1 a bx cx 2 ( )
log
Linear:
Quadratic:
Exponential:
Logarithmic:
Note: What would a scatter diagram look like to suggest each relationship?

28. Method of Least Squaresb x = + 1 y ^
Equation of the best-fitting line: y ^
Predicted value: ( ) )) y b x - = + å 2 1 ^
Least squares criterion:
Find the constants b0 and b1 such that the sum is as small as possible

29. Illustration Observed and predicted values of y: y y b x = + ) ( , x y1 y ^ ) ( , x y y - ^ y ^ ( , ) x

30. The Line of Best Fit Equation
The equation is determined by:
b0: y-intercept
b1: slope
Values that satisfy the least squares criterion:

31. Example
Example: A recent article measured the job satisfaction of subjects with a 14-question survey. The data below represents the job satisfaction scores, y, and the salaries, x, for a sample of similar individuals:
1) Draw a scatter diagram for this data
2) Find the equation of the line of best fit

32. Finding b1 & b0
Preliminary calculations needed to find b1 and b0:

33. Line of Best Fit ( ) å y ^ b xy x 118 75 229 5 5174 = SS ( ) . 0. b y1
133
5174
234 8
4902 = - × å
(0. )( .
Equation o
f the line
of best f
it: . 0. x = + 1 49
517 y ^
Solution 1)

34. Job Satisfaction Survey
Scatter Diagram 21 23 25 27 29 31 33 35 37 12 13 14 15 16 17 18 19 20 22
Job
Satisfaction
Salary
Job Satisfaction Survey
Solution 2)

35. Please Note
Keep at least three extra decimal places while doing the calculations to ensure an accurate answer
When rounding off the calculated values of b0 and b1, always keep at least two significant digits in the final answer
The slope b1 represents the predicted change in y per unit increase in x
The y-intercept is the value of y where the line of best fit intersects the y-axis
The line of best fit will always pass through the point

36. Making Predictions
1. One of the main purposes for obtaining a regression equation is for making predictions y ^
2. For a given value of x, we can predict a value of
3. The regression equation should be used to make predictions only about the population from which the sample was drawn
4. The regression equation should be used only to cover the sample domain on the input variable. You can estimate values outside the domain interval, but use caution and use values close to the domain interval.
5. Use current data. A sample taken in 1987 should not be used to make predictions in 1999.