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5.11 Error Forms

An error form is a number with an associated standard deviation, as in ‘2.3 +/- 0.12’. The notation `x +/- sigma' stands for an uncertain value which follows a normal or Gaussian distribution of mean ‘x’ and standard deviation or “error” sigma’. Both the mean and the error can be either numbers or formulas. Generally these are real numbers but the mean may also be complex. If the error is negative or complex, it is changed to its absolute value. An error form with zero error is converted to a regular number by the Calculator.

All arithmetic and transcendental functions accept error forms as input. Operations on the mean-value part work just like operations on regular numbers. The error part for any function ‘f(x)’ (such as sin(x)’) is defined by the error of ‘x’ times the derivative of ‘f’ evaluated at the mean value of ‘x’. For a two-argument function ‘f(x,y)’ (such as addition) the error is the square root of the sum of the squares of the errors due to ‘x’ and ‘y’. Note that this definition assumes the errors in ‘x’ and ‘y’ are uncorrelated. A side effect of this definition is that ‘(2 +/- 1) * (2 +/- 1)’ is not the same as ‘(2 +/- 1)^2’; the former represents the product of two independent values which happen to have the same probability distributions, and the latter is the product of one random value with itself. The former will produce an answer with less error, since on the average the two independent errors can be expected to cancel out.

Consult a good text on error analysis for a discussion of the proper use of standard deviations. Actual errors often are neither Gaussian-distributed nor uncorrelated, and the above formulas are valid only when errors are small. As an example, the error arising from `sin(x +/- sigma)' is `sigma abs(cos(x))'. When ‘x’ is close to zero, cos(x)’ is close to one so the error in the sine is close to sigma’; this makes sense, since sin(x)’ is approximately ‘x’ near zero, so a given error in ‘x’ will produce about the same error in the sine. Likewise, near 90 degrees cos(x)’ is nearly zero and so the computed error is small: The sine curve is nearly flat in that region, so an error in ‘x’ has relatively little effect on the value of sin(x)’. However, consider ‘sin(90 +/- 1000)’. The cosine of 90 is zero, so Calc will report zero error! We get an obviously wrong result because we have violated the small-error approximation underlying the error analysis. If the error in ‘x’ had been small, the error in sin(x)’ would indeed have been negligible.

To enter an error form during regular numeric entry, use the p (“plus-or-minus”) key to type the ‘+/-’ symbol. (If you try actually typing ‘+/-’ the + key will be interpreted as the Calculator's + command!) Within an algebraic formula, you can press M-+ to type the ‘+/-’ symbol, or type it out by hand.

Error forms and complex numbers can be mixed; the formulas shown above are used for complex numbers, too; note that if the error part evaluates to a complex number its absolute value (or the square root of the sum of the squares of the absolute values of the two error contributions) is used. Mathematically, this corresponds to a radially symmetric Gaussian distribution of numbers on the complex plane. However, note that Calc considers an error form with real components to represent a real number, not a complex distribution around a real mean.

Error forms may also be composed of HMS forms. For best results, both the mean and the error should be HMS forms if either one is.

The algebraic function ‘sdev(a, b)’ builds the error form ‘a +/- b’.