Besides looking at the scatter plot and seeing that a line seems reasonable, how can you
tell if the line is a good predictor? Use the correlation coefficient as another indicator
(besides the scatterplot) of the strength of the relationship between xx and yy.
The correlation coefficient, r, developed by Karl Pearson in the early 1900s, is a numerical measure of the strength of association between the independent variable x and the dependent variable y.
The correlation coefficient is calculated as
r
=
n
⋅
Σ
x
⋅
y
-
(
Σ
x
)
⋅
(
Σ
y
)
[
n
⋅
Σ
x
2
-
(
Σ
x
)
2
]
⋅
[
n
⋅
Σ
y
2
-
(
Σ
y
)
2
]
r=
n
⋅
Σ
x
⋅
y
-
(
Σ
x
)
⋅
(
Σ
y
)
[
n
⋅
Σ
x
2
-
(
Σ
x
)
2
]
⋅
[
n
⋅
Σ
y
2
-
(
Σ
y
)
2
]
(1)
where nn = the number of data points.
If you suspect a linear relationship between xx and yy, then rr can measure how strong the linear relationship is.
- The value of rr is always between -1 and +1: -1≤r≤1-1≤r≤1.
- The closer the correlation coefficient rr is to -1 or 1 (and the further from 0), the stronger the evidence of a significant linear relationship between xx and yy; this would indicate that the observed data points fit more closely to the best fit line. Values of rr further from 0 indicate a stronger linear relationship between xx and yy. Values of rr closer to 0 indicate a weaker linear relationship between xx and yy.
- If r=0r=0
there is absolutely no linear relationship between xx and yy (no linear correlation).
- If r=1r=1, there is perfect positive correlation. If r=-1r=-1, there is perfect negative
correlation. In both these cases, all of the original data points lie on a straight line. Of course,
in the real world, this will not generally happen.
- A positive value of rr means that when xx increases, yy increases and when xx decreases, yy decreases (positive correlation).
- A negative value of rr means that when xx increases, yy decreases and when xx decreases, yy increases (negative correlation).
- The sign of rr is the same as the sign of the slope, bb,
of the best fit line.
Strong correlation does not suggest that xx causes yy or yy causes xx. We say "correlation does not imply causation." For example, every person who learned
math in the 17th century is dead. However, learning math does not necessarily cause
death!
The formula for rr looks formidable. However, computer spreadsheets, statistical software, and many calculators can quickly calculate rr. The correlation coefficient rr is the bottom item in the output screens for the LinRegTTest on the TI-83, TI-83+, or TI-84+ calculator (see previous section for instructions).
r
2
r
2
is called the coefficient of determination.
r
2
r
2
is the square of the correlation coefficient , but is usually stated as a percent, rather than in decimal form.
r
2
r
2
has an interpretation in the context of the data
-
r
2
r
2
, when expressed as a percent, represents the percent of variation in the dependent variable y that can be explained by variation in the independent variable x using the regression (best fit) line.
- 1-
r
2
r
2
, when expressed as a percent, represents the percent of variation in y that is NOT explained by variation in x using the regression line. This can be seen as the scattering of the observed data points about the regression line.
- The line of best fit is:
y
^
=
-173.51
+
4.83x
y
^
=-173.51+4.83x
- The correlation coefficient is
r
=
0.6631
r=0.6631
- The coefficient of determination is
r
2
r
2
=
0.6631
2
0.6631
2
= 0.4397
- Interpretation of
r
2
r
2
in the context of this example:
- Approximately 44% of the variation in the final exam grades can be explained by the variation in the grades on the third exam, using the best fit regression line.
- Therefore approximately 56% of the variation in the final exam grades can NOT be explained by the variation in the grades on the third exam, using the best fit regression line. (This is seen as the scattering of the points about the line.)
**With contributions from Roberta Bloom.
- Coefficient of Correlation:
A measure developed by Karl Pearson (early 1900s) that gives the strength of association between the independent variable and the dependent variable. The formula is:
r
=
n
∑
xy
−
(
∑
x
)
(
∑
y
)
[
n
∑
x
2
−
(
∑
x
)
2
]
[
n
∑
y
2
−
(
∑
y
)
2
]
,
r
=
n
∑
xy
−
(
∑
x
)
(
∑
y
)
[
n
∑
x
2
−
(
∑
x
)
2
]
[
n
∑
y
2
−
(
∑
y
)
2
]
,
size 12{r= { {n Sum { ital "xy"} - \( Sum {x \) \( Sum {y \) } } } over { sqrt { \[ n Sum {x rSup { size 8{2} } - \( Sum {x \) rSup { size 8{2} } \] \[ n Sum {y rSup { size 8{2} } - \( Sum {y \) rSup { size 8{2} } \] } } } } } } } ,} {}
(2)
where n is the number of data points.
The coefficient cannot be more then 1 and less then -1. The closer the coefficient is to
±1±1 size 12{ +- 1} {}, the stronger the evidence of a significant linear relationship between
xx size 12{x} {} and
yy size 12{y} {}.
(What does "Endorsed by" mean?)
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