Data from two variables
(x, y) can be plotted on
x-y graphs as scatter
plots.
 There are several
possible patterns that
can show qualitatively if
there is a correlation
between the two
variables.

A
measure of the correlation is the correlation
coefficient (r).
 To find the correlation coefficient, place the
x-values in L1 and the y-values in L2.
 Then STAT TESTS LinRegTTest, being sure that
the Lists for x an y are correct.
 Scroll down to find “r”.
 The values of “r” are 1  r  1 . A negative
value indicates a negative correlation. (as x
increases y increases) A positive value
indicates a positive correlation (as x increases
y increases)
 To
find if there is a Correlation to some
significance (probability of being wrong), we
need to compare the Test Statistic “r” to a
Critical Value.
 To find the Critical Value, need to use “n”
(the number of ordered pairs (x, y)) and a
table for the given significance. Table II is
such a table for   0.05 . There are other
tables for other significance, but we will only
use   0.05 and only Table II.
 If |r|≥ Critical Value, then there is
correlation.
 If
there is a correlation, then there is a
line that represents the relationship. The
line can be approximated from the
sample of (x, y) pairs.
 A linear relationship has the form
y=mx+b.
 This line is only an approximation of the
regression line and will be given by y  ax  b
where a is the approximate slope and b is
the approximate y-intercept.
 To
find the regression line, use the same
function as for finding “r” and find the “a” &
“b”.
 If “r” is positive then “a” will be positive. If
“r” is negative then “a” will be negative.
 The equation y  ax  b will give an
approximate value of y for any given x, but
because it is approximate it will have an
associated error.
 The
error can be found for any value of x that is
in the (x, y) data.
 The
 As
error is called the residual and is y  y .
an example if one of the (x, y) pairs is (2, 5)
and y  2 x  3 then for x = 2 y  7 . Then the
residual is 5 – 7 = -2.
 Two
Cautions:
 1.
Correlation does NOT imply Causality:
Just because one variable goes up as another
goes up, does not mean that one causes the
other to go up.
 2.
The scope of the Regression Equation is
only the interval of the data from the
ordered pairs (i.e. the domain is from x-min
to x-max. Outside of this interval we can not
determine what the relationship is.
 Do
first problem of Lab 10.