Appendix: Calculating a correlation coefficient
Correlation coefficients can be calculated using ordinary four-species calculating machines. In fact, Karl Pearson himself was the proud user of a Brunsviga four-species calculator.
This appendix illustrates a short-cut method for calculating correlation coefficients.
Suppose we have a simple pinwheel calculator with 12 digits in the input- and revolution register, and 20 in the result register:
Take xi = 12 and yi = 13. Put xi and yi in the input register (separated by zeroes!)
Shift the input register over half its width:
Turn the crank 2 times (units of xi)
Shift the input register over 1 position and turn the crank 1 time (tens of xi)
We now have xi in the revolution register, and xi² and xi yi in the result register.
Shift the input register back completely.
Turn the crank 3 times (units of yi)
Shift the input register and turn the crank 1 time (tens of yi)
We now have xi and yi in the revolution register, and xi² and 2xi yi and yi² in the result register.
Shift the input register back completely, and repeat the procedure for x(i +1) and y(i +1) without clearing the revolution- and result registers.
Finally the revolution register will contain Σxi and Σyi, and the result register Σxi² and 2Σxi yi and Σyi²
We see that each value of xi and yi has to be entered twice: once in the input register and once when cranking. But, because the revolution register is not cleared, it is difficult, after the first pair of values, to check if the correct value has been “cranked in”. Electrically driven machines that allow entering multiplicators via a separate keyboard or pin setting would be a great help in this case.
It is also clear that, depending on the number of data-points and the range of the data, the registers should be rather large: for 100 data points with a range of 0…100 (integer numbers!) Σxi² can grow to 106, needing 7 digits. The result register will have to accommodate 3 sums of this size, so should have at least 21 digits.
Karl Pearson with a Brunsviga calculator
Walther EM electrical pinwheel calculator
Archimedes LKvM, with separate keyboard
Monroe CAA-10-3-S electromechanical calculator, specially marketed for statistical calculations (note the attached Veeder counter)
