With number pairs which have about the same percent error, the least-error method will give an equation which will give estimates of the dependent variable to about the same percent error because each point used in the determinations of the equation parameters is assumed equally inaccurate percentage-wise. The example given here is for determination of the slope (m) and constant (c) for a linear equation:

Y = mX + c

For each X,Y point, the difference between the measured Y and the Y calculated from the equation [Y - (mX + c)] is used to calculate the fractional error from the measured Y. In all the following equations, Y refers to the values measured by experiment:

Fractional error

For the derivation, it is unnecessary to multiply by 100 to give percent as the 100 will cancel out later. This method seeks to determine m and c through minimizing the standard deviation of the fractional error. Again the actual division by (n-1) and taking of the square-root is superfluous:

Sum of fractional errors squared

Differentiate with respect to m for one equation then with respect to c for the second. Set the two derivatives each equal to zero to minimize the sum:

1st derivatives

Solving the two simultaneous equations gives the following equations for m (the slope) and c (the constant):

Equations for slope and constant

I wrote a program in APPLESOFT BASIC to calculate the m and c pairs using an Apple IIc computer - - it was 1986! Dulong & Petit couldn't have done this in 1819 because, with Petit very sick with tuberculosis and dying the next year, they wouldn't have had time to do the tedious calculations without a modern computer. Finally, Mendeleyev's Periodic Table was not available until 1869 - - and, in 1869, this itself was a hard sell.

There is no correlation coefficient for this method yet. To tell just how well things are going, one can calculate the standard deviation of the experimental Y values with those calculated from the equation. However, this may come out, it will be known that no other linear equation will give a lower average error.

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