The Method of Least Squares
77
linear dependence of one variable on another; it is called the coefficient of
determination, r'-.
r- = mm'
(6-16)
The more commonly used term is r itself, which is called the correlation coefficient:
r = (mm
1 )*
(6-17)
Calculators that have built-in least-squares programs, and programmable
calculators that perform the same function, always provide for the simultaneous calculation of r, because exactly the same sums are needed for its calculation as for the calculation of m and b. For a nonprogrammable calculator (with
at least five storage registers), there would be the same effort to calculate m' as
to find m; then Equation 6-17 would be used to calculate r.
PROBLEM:
Determine the correlation coefficient r for the best-fit equation obtained in the
problem on p 75 involving thermocouple voltage versus temperature.
SOLUTION:
Work the problem exactly as you did previously, but with the additional knowledge of how to display r with your make of calculator, or with the program you
use. If you use a nonprogrammable calculator with at least five storage registers,
first accumulate 2U,, I*?, £y ( , 2yf, and 2x,;y,. Then calculate m with Equation
6-12b, m' with Equation 6-14b, and finally r with Equation 6-17. The answer will
be r = 0.9999, a very good correlation indeed.
Reliability of Slope and the y Intercept
It is a common practice to derive some important physical or chemical characteristics from the slope or the y intercept of a graph constructed from experimental data points. These data points have, of course, some error associated
with them and, as a consequence, even the "best-fit" line must have some
uncertainty associated with it. Just how good is a value derived from the slope
and intercept of a best-fit line?
In the same way that one uses the variance and standard deviation to describe the scatter of points around their average, one can also use the variance
and standard deviation to describe the scatter, in the vertical (y) direction, of
the points about the best-fit line. Statisticians have shown that the variance is
given by
77
linear dependence of one variable on another; it is called the coefficient of
determination, r'-.
r- = mm'
(6-16)
The more commonly used term is r itself, which is called the correlation coefficient:
r = (mm
1 )*
(6-17)
Calculators that have built-in least-squares programs, and programmable
calculators that perform the same function, always provide for the simultaneous calculation of r, because exactly the same sums are needed for its calculation as for the calculation of m and b. For a nonprogrammable calculator (with
at least five storage registers), there would be the same effort to calculate m' as
to find m; then Equation 6-17 would be used to calculate r.
PROBLEM:
Determine the correlation coefficient r for the best-fit equation obtained in the
problem on p 75 involving thermocouple voltage versus temperature.
SOLUTION:
Work the problem exactly as you did previously, but with the additional knowledge of how to display r with your make of calculator, or with the program you
use. If you use a nonprogrammable calculator with at least five storage registers,
first accumulate 2U,, I*?, £y ( , 2yf, and 2x,;y,. Then calculate m with Equation
6-12b, m' with Equation 6-14b, and finally r with Equation 6-17. The answer will
be r = 0.9999, a very good correlation indeed.
Reliability of Slope and the y Intercept
It is a common practice to derive some important physical or chemical characteristics from the slope or the y intercept of a graph constructed from experimental data points. These data points have, of course, some error associated
with them and, as a consequence, even the "best-fit" line must have some
uncertainty associated with it. Just how good is a value derived from the slope
and intercept of a best-fit line?
In the same way that one uses the variance and standard deviation to describe the scatter of points around their average, one can also use the variance
and standard deviation to describe the scatter, in the vertical (y) direction, of
the points about the best-fit line. Statisticians have shown that the variance is
given by