1. Part II: Two - Variable Statistics  
Statistical studies often involve more than one 
variable. We are interested in knowing if there is a 
relationship between the two characteristics for the same 
subject.
 Example:  A person's age and the time spent using a mobile phone.
When the data is quantitative (numbers), the variables can be written as an ordered pair (x, y).
Correlation is the study and description of the 
 relationship (if any) that exists between the variables.

2. A) Qualitative Interpretation of Correlation
Data can be organised and displayed in a scatterplot (Cartesian plane) or a contingency table.
By inspection, we will describe the type, the direction, and the intensity (or strength) of the relation between the variables.

3. Type: Refers to the function that best fits the relation   between the variables. We will be using linear   correlation.
Direction: If both variables move in the same direction    (increase together or decrease together), then the direction is positive.
If both variables move in opposite directions, 
   then direction is negative.
Intensity: Strength may be categorised as...
   Zero, weak, moderate, strong or perfect.

4. Example:. In a gym class students were required to do
Example: In a gym class students were required to do 
    push-ups and sit-ups. Each student's 
       achievements were recorded as an ordered      pair. The first number refers to push-ups and 
    the second to sit-ups.
    (27, 30), (26, 28), (38, 45), (52, 55), (35, 36),
    (40, 54), (40, 50), (52, 46), (42, 55), (61, 62),
    (35, 38), (45, 53), (38, 42), (63, 55), (55, 54),
    (46, 46), (34, 36), (45, 45), (30, 34), (68, 62)  

5. Contingency Table   
 - A table of values (often involves classes)
 - Listed down the left side is one variable; listed across 
 the top is the other variable.
 - Each row and column is added and the totals are 
 displayed; the bottom right cell shows the total 
 frequency of the distribution.

6.  (27, 30), (26, 28), (38, 45), (52, 55), (35, 36),
 (40, 54), (40, 50), (52, 46), (42, 55), (61, 62),
 (35, 38), (45, 53), (38, 42), (63, 55), (55, 54),
 (46, 46), (34, 36), (45, 45), (30, 34), (68, 62)

7. When the majority of the values fall into a diagonal of 
a contingency table, and the corners contain mostly 
zeros, then the correlation is said to be linear and 
strong.
  Direction is positive if the diagonal is  
  Direction is negative if the diagonal is

8. - Each ordered pair is represented as a point
Scatterplot
- A Cartesian graph
- Each ordered pair is represented as a point
 (27, 30), (26, 28), (38, 45), (52, 55), (35, 36),
 (40, 54), (40, 50), (52, 46), (42, 55), (61, 62),
 (35, 38), (45, 53), (38, 42), (63, 55), (55, 54),
 (46, 46), (34, 36), (45, 45), (30, 34), (68, 62)

9. B) Quantitative Interpretation of Correlation
 The correlation will be represented by a number, 
 called the correlation coefficient.  
 This coefficient will range from to
 Its symbol is r.

10. 2) Draw a line that "best fits"
1) Draw a scatterplot
2) Draw a line that "best fits"
  the points. This line passes 
 through the middle of the 
 scatterplot.
3) Around the points, draw 
  the smallest rectangle 
  possible. Two of the sides should be parallel to the line.
4) Measure the dimensions
  of the rectangle.
 (27, 30), (26, 28), (38, 45), (52, 55), (35, 36),
 (40, 54), (40, 50), (52, 46), (42, 55), (61, 62),
 (35, 38), (45, 53), (38, 42), (63, 55), (55, 54),
 (46, 46), (34, 36), (45, 45), (30, 34), (68, 62)

11. 5) Calculate the correlation coefficient, r.
+ if it's increasing,
 if it's decreasing ( )
r =
2.4
7.9
r = +(1- 0.3)
r = 0.7
moderate
real r = 0.87

12. r Near 0 Near ± 0.5 Near ± 0.75 Near ± 0.87 Near ± 1 Meaning
Zero correlation
Near ± 0.5
Weak correlation
Near ± 0.75
Moderate correlation
Near ± 0.87
Strong correlation
Near ± 1
Perfect correlation

13. The Regression Line
Also called a line of best fit, a regression line is one that best represents (or passes through) the points of a scatterplot. It passes through as many points as possible, going through the middle of the scatterplot.
There are several different methods of drawing a regression line.
The regression line may be used to predict values that 
do not appear in the distribution.

14. Using the regression line, we can predict the value of 
one variable, given the value of the other. The 
reliability of the prediction depends on the strength of 
the correlation.
Examples: 
a) Predict the number of    b) Predict the number of 
  sit-ups a student can      push-ups a student can 
  do if he can do 49     do if she can do 70 
 push-ups.     sit-ups.
Determine the equation of the line.

15. Recall:
Since the correlation is moderate/high, we can be confident that our predictions are good.
predicted
actual

16. Interpreting a Correlation
 A strong correlation indicates that there is a 
 statistical relationship between two variables.
 It does not, however, explain the reason for the 
 relationship or its nature.  
 There are other things to consider...

18. The outlier
any point that stands alone away from the group is left outide the rectangle.
DON'T INCLUDE THIS POINT IN THE RECTANGLE
OUTLIER
Is there a strong correlation?
What is the correlation coefficient?
What would be the value of y when x = 2? Make a prediction
what is the equation for the line of best fit?