## WAPP+: Formula Entry of Errors

Often x and y errors are known as a formula. Examples:
• Poisson statistical error in a count, where the uncertainty in the count is given by the square root of the count
• measurement errors using a ruler where the uncertainty is constant and taken as half the finest division on the ruler
• measurement device specifications, which often list likely calibration errors as a formula (see below)
Alternatively, error estimates may be individualized and not expressible as a simple formula. Examples:
• judgments about when a condition applies (problem of definition errors) like: judging the range of lens locations that results in a focused image or judging if the star is exactly in the crosshairs now
• errors found by the deviations observed on repeated re-measurement.
In the former case (where the error can be reduced to a simple equation), WAPP+ provides a way to type in the formula rather than calculating the result for each datapoint. Much like a spreadsheet you can type in your error formula using the usual fortran syntax:

#### Example: DMM Specifications

What follows is a simplified version of the "resolution and accuracy" of the DM-441B Digital Multimeter used in introductory physics.

FUNCTIONACCURACY
DC Voltage ±(0.1% + 4 digits)
AC Voltage ±(0.5% + 20 digits)
DC Current ±(0.5% + 1 digit)
AC Current ±(1.0% + 10 digits)
Resistance ±(0.2% + 2 digits)

A "digit" above refers to the rightmost (least significant) digit displayed. For example the "+1 digit" in a DC current reading "1.4869 A" would be 0.0001 A. You could report the error in DC currents on the 2 A scale as: The "+4 digits" in a DC voltage reading "166.51 mV" would be 0.04 mV so you could report the error in DC voltage on the 200 mV scale as:. However since different scales have different sized "digits", the above formulas do not apply if the scale was changed. The SCALE function helps with this problem if (as usual, but not universal) scale changes happen at "2". Viz:

```Reading X     SCALE(X)
2.566         10.
1.4869         1.
0.8473         1.
0.3213         1.
0.19865         .1
0.03213         .1
0.019865        .01
```
Thus any DC voltage reading error could be written as: as long as all y values use the same unit (say mV). Note: I've assumed above that y is positive, otherwise I would need to write, for example:

#### Functions

The functions available for your error formula include the usual fortran functions: ABS, ACOS, ASIN, ATAN, COS, COSH, ERF, ERFC, EXP, INT, LOG, LOG10, NINT, RAN, SIGN, SIN, SINH, SQRT, TAN, TANH plus GAMMA (the gamma function), K (complex elliptic integral), NORM (the area under the normal curve, from minus infinity to x), and INORM (the inverse of the previous function). The value of X, XE, Y, YE for the current datapoint are available using the appropriate symbol (however note that if both xe and ye are calculated from formulas, ye will still be zero when xe is calculated). PI is a handy constant. For example, if x is in degrees, would give you what your calculator might record as "cos(x)". (That is fortran trigonometric functions are designed for radian measure.)