E1C04 09/14/2010
14:7:44 Page 144
Linear Polynomials
For linear polynomials a correlation coefficient r can be found by
r ¼ r xy ¼
N
P N
i¼1
x i y i À
P N
i¼1
x i
P N
i¼1
y i
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
N
P N
i¼1
x 2
i À
P N
i¼1
x i
2
s
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
N
P N
i¼1
y 2
i À
P N
i¼1
y i
2
s
ð4:41Þ
The correlation coefficient provides a measure of the association between x and y as predicted by the
form of the curve fit equation. It is bounded by Æ1, which represents perfect correlation; the sign
indicates that y increases or decreases with x. For Æ0:9 < r Æ1, a linear regression can be
considered a reliable relation between y and x. Alternatively, the value r
2 is often reported, which is
indicative of how well the variance in y is accounted for by the fit. However, the correlation
coefficient and the r
2 value are only indicators of the hypothesis that y and x are associated;
correlation does not imply cause and effect. The r and r
2 values are not effective estimators of the
random error in y c ; instead the s yx value is used for that purpose.
The precision estimate in the slope of the fit can be estimated by
s a 1 ¼ s yx
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1
P N
i¼1 ðx i À xÞ
2
s
ð4:42Þ
The precision estimate of the zero intercept can be estimated by
s a 0 ¼ s yx
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P N
i¼1 x 2
i
N
P N
i¼1 ðx i À xÞ
2
s
ð4:43Þ
An error in a 0 would offset a calibration curve from its y intercept. The derivation and further
discussion on Equations 4.41 to 4.43 can be found elsewhere (1, 3, 4, 6).
Example 4.9
Compute the correlation coefficient and the standard error of the fit for the data in Example 4.8. Estimate
the random uncertainty associated with the fit. State the correlation with its 95% confidence interval.
KNOWN y c ¼ 0:02 þ 1:04x V
ASSUMPTIONS Errors are normally distributed. No systematic errors.
FIND r and s yx
SOLUTION Direct substitution of the data set into Equation 4.41 yields the correlation
coefficient of r ¼ 0.996. An equivalent estimator is r
2 . Here r
2
¼ 0.99, which indicates that
99% of the variance in y is accounted for by the fit, whereas only 1% is unaccountable. These values
suggest that a linear fit is a reliable relation between x and y.
144 Chapter 4 Probability and Statistics
14:7:44 Page 144
Linear Polynomials
For linear polynomials a correlation coefficient r can be found by
r ¼ r xy ¼
N
P N
i¼1
x i y i À
P N
i¼1
x i
P N
i¼1
y i
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
N
P N
i¼1
x 2
i À
P N
i¼1
x i
2
s
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
N
P N
i¼1
y 2
i À
P N
i¼1
y i
2
s
ð4:41Þ
The correlation coefficient provides a measure of the association between x and y as predicted by the
form of the curve fit equation. It is bounded by Æ1, which represents perfect correlation; the sign
indicates that y increases or decreases with x. For Æ0:9 < r Æ1, a linear regression can be
considered a reliable relation between y and x. Alternatively, the value r
2 is often reported, which is
indicative of how well the variance in y is accounted for by the fit. However, the correlation
coefficient and the r
2 value are only indicators of the hypothesis that y and x are associated;
correlation does not imply cause and effect. The r and r
2 values are not effective estimators of the
random error in y c ; instead the s yx value is used for that purpose.
The precision estimate in the slope of the fit can be estimated by
s a 1 ¼ s yx
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1
P N
i¼1 ðx i À xÞ
2
s
ð4:42Þ
The precision estimate of the zero intercept can be estimated by
s a 0 ¼ s yx
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P N
i¼1 x 2
i
N
P N
i¼1 ðx i À xÞ
2
s
ð4:43Þ
An error in a 0 would offset a calibration curve from its y intercept. The derivation and further
discussion on Equations 4.41 to 4.43 can be found elsewhere (1, 3, 4, 6).
Example 4.9
Compute the correlation coefficient and the standard error of the fit for the data in Example 4.8. Estimate
the random uncertainty associated with the fit. State the correlation with its 95% confidence interval.
KNOWN y c ¼ 0:02 þ 1:04x V
ASSUMPTIONS Errors are normally distributed. No systematic errors.
FIND r and s yx
SOLUTION Direct substitution of the data set into Equation 4.41 yields the correlation
coefficient of r ¼ 0.996. An equivalent estimator is r
2 . Here r
2
¼ 0.99, which indicates that
99% of the variance in y is accounted for by the fit, whereas only 1% is unaccountable. These values
suggest that a linear fit is a reliable relation between x and y.
144 Chapter 4 Probability and Statistics