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.

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.

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.

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) 

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.

 (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)

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

 (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)

Scatterplot- A Cartesian graph - Each ordered pair is

represented as a point

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.

 (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)

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.

+ if it's increasing,  if it's decreasing

r = + 1 -

r = +(1- 0.3)

r = 0.7 moderate

2.47.9 )(

5) Calculate the correlation coefficient, r.

real r = 0.87

r MeaningNear 0 Zero correlationNear ± 0.5

Weak correlation

Near ± 0.75

Moderate correlation

Near ± 0.87

Strong correlation

Near ± 1 Perfect correlation

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.

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.

predicted

actual

Recall:

Since the correlation is moderate/high, we can be confident that our predictions are good.

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...

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 predictionwhat is the equation for the line of best fit?