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:
directions,
If both variables move in the same direction
(increase together or decrease together),
then the direction is positive.
If both variables move in opposite
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
Scatterplot
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)
A Cartesian graph
Each ordered pair is
B) Quantitative Interpretation of Correlation
The correlation will be represented by a number, called the
correlation coefficient.
This coefficient will range from
Its symbol is r.
to
.
1) Draw a scatterplot
(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)
2) Draw a line that "best fits"
the points. This line passes through the
middle of the scatterplot.
Around the points, draw
the
smallest rectangle
possible. Two of the
sides should be
parallel to the line.
3)
4) Measure the dimensions
of the rectangle.
5) Calculate the correlation coefficient, r.
+ if it's increasing,
if it's decreasing
(
2.4
7.9
r=+ 1-
)
r = +(1- 0.3)
r = 0.7
moderate
real r = 0.87
r
Meaning
Near 0
Zero correlation
Near ± 0.5 Weak correlation
Near ±
0.75
Near ±
0.87
Near ± 1
Moderate correlation
Strong correlation
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.
Determine the equation of
the line.
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.
Recall:
predicted
actual
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 prediction
what is the equation for the line of best fit?