Transcript Document 7664521

Chapter 10
Relationships between
variables
Definition
 A Scatter Plot is a picture of
bivariate numerical data in
which each observation ( ie
each pair of values (x,y)) is
represented by a point located
on a rectangular co-ordinate
system. The Horizontal Axis is
identified with values of x and
the vertical axis with values of
y.
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Example:
Draw a Scatter Plot to represent the
following dataset:
x: 1, 3, 2, 4, 7, 6, 5
y: 4, 2, 5, 6, 9, 8, 7
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Another Example:
Draw a Scatter Plot to represent the
following dataset:
x: 1, 3, 2, 4, 7, 6, 5
y: 4, 6, 1, 3, 2, 4, 1
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Question
Any comments on these two
datasets?
Is there anything special about
them?
Looking at a scatter plot can
sometimes allow us to
determine if a relationship
exists between two variables.
 But in general we need to go
beyond pictures and develop a
numerical measure of how
strongly the two variables x and
y are related.
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Definition
 Pearson’s Sample Correlation
Coefficient, r, is a measure of
the strength of the linear
relationship between two
variables x and y.
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r
 ( x  x )( y  y )
 (x  x)  ( y  y)
2
2
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SS xy
SS xx SS yy
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Properties of r
The correct interpretation of r
requires an appreciation of some
general properties:
The value of r does not depend on
the unit of measurement for either
variable, nor does it depend on
which variable is labelled x or y.
The value of r is between -1 and 1.
A positive value of r indicates a
positive linear relationship between
the variables. So as x increases so
does y.
A negative value of r corresponds to
a negative relationship. As x
increases y decreases.
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The value r = 1, which indicates the
strongest possible positive
relationship between x and y results
only when all points in the scatter
plot lie exactly on a straight line that
slopes upward.
The value r = -1, which indicates the
strongest possible negative
relationship between x and y results
only when all points in the scatter
plot lie exactly on a straight line that
slopes downward.
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The value of r is a measure of the
extent to which x and y are linearly
related i.e. the extent to which the
points in the scatter plot lie close to a
straight line.
A value close to zero does not rule
out any strong relationship between
x and y; there could still be a strong
relationship but one that is not linear.
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Examples For each of the following
pairs of variables, indicate whether
you would expect a positive
correlation, a negative correlation or
no correlation.
Minimum daily temperature and
heating costs
Interest rate and number of loan
applications
Incomes of husbands and wives
when both have full-time jobs
Ages of boyfriends and girlfriends
Height and IQ
Height and shoe size
Your Maths score in the Leaving
Cert and your Irish score in the
Leaving Cert
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Correlation and causation
Years of research have established
several facts:
There is a strong correlation between
the numbers of storks in a country
and the number of births in that
country. Countries with many storks
have a high number of births and
countries with low stork counts have
low numbers of births.
There is a high correlation among
primary school children between
vocabulary and numbers of tooth
fillings. Children with many fillings
have a larger vocabulary than
children with only a small number or
with no fillings.
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Correlation and causation
What should we conclude from these
facts?
That storks really are responsible for
bringing babies.
That eating Mars bars will increase
your vocabulary.
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No, these examples illustrate a very
important point.
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Correlation is not the same as
causation.
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Correlation and causation
Larger countries have larger stork
populations and usually have higher
human populations as well and so
there will be higher numbers of
babies born than in smaller
countries.
Young children have very few
fillings because they have only been
around for a few years whereas older
children have had time to eat lots of
sweets, get a lot of bad teeth and
learn a lot of new words.
So be careful before you interpret a
correlation as causation. It may be
that a third confounding variable is
causing the correlation: Size of
country, Age of child.
Least Squares
Introduction
 We have just mentioned that one
should not always conclude that
because two variables are correlated
that one variable is causing the other
to behave a certain way. However,
sometimes this is the case, eg:
interest rate and number of loan
applications.
 In this section we will deal with
datasets which are correlated and in
which one variable, x, is classed as
an independent variable and the
other variable, y, is called a
dependent variable as the value of
y depends on x.
Least Squares
 We saw that correlation implies a
linear relationship. Well a line is
described by the equation
 y = a +bx
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where b is the slope of the line and a
is the intercept i.e. where the line
cuts the y axis.
The intercept a is just the value that
y takes when x is zero.
The slope b is how much y increases
by when x increases by one unit.
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Suppose we have a dataset which is
strongly correlated and so exhibits a
linear relationship, how would we
draw a line through this data so that
it fits all points best?
We use the principle of least squares,
we draw a line through the dataset so
that the sum of the squares of the
deviations of all the points from the
line is minimised.
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Regression
Suppose we have a dataset and we
have calculated the equation of the
Least Squares Line
 y = a +bx
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Then we can use this line to predict a
value for Y if we know a value for
X.
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Note we should only predict for
values of X which are bigger than
the smallest X value in the dataset
and smaller than the largest value in
the dataset.
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Example of Regression:
A study performed in the UK
examined the relationship between
husband’s and wives’ ages.
The data were analysed and a Least
Squares Line computed:
Y = 3.6 + (0.97) X
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Where
Y is Husband’s age
X is Wife’s age
Predict the age of the husband of a
20 year old woman.
Predict the age of the husband of a
25 year old woman.
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Regression Answers:
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20Yr old Woman
Y = 3.6 + (0.97) 20
Y = 23.0
So Husband is probably 23 years old
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25Yr old Woman
Y = 3.6 + (0.97) 25
Y = 27.9
So Husband is probably 27.9 years old
 Congratulations!
 It’s
over!
 You
have survived the
dreaded course on
STATISTICS.
 Hopefully
none of you
have died of Boredom.