Editing 2295: Garbage Math

This comic illustrates the "{{w|garbage in, garbage out}}" concept using mathematical expressions. It shows how, if you have garbage as inputs to your calculations, then you will likely get garbage as a result, except when you multiply by zero, which eliminates all uncertainty of the result.  

The propagation of errors in {{w|arithmetic}}, other {{w|mathematical operations}}, and {{w|statistics}} is described in colloquial terms. Numbers with low precision are termed garbage, while numbers with high precision are called precise. The table below quantifies the change in precision from the operands to their result in terms of their {{w|variance}}, represented by σ, the Greek lowercase letter sigma, equal to the {{w|standard deviation}}, or the square root of the variance. Variance or standard deviation are common specifications of uncertainty (as an alternative to, for example, a {{w|tolerance interval}}.)

The {{w|accuracy and precision}} of mathematical operations correspond to the rules of {{w|Propagation_of_uncertainty#Example_formulae|propagation of uncertainty}}, where a "garbage" number would correspond to an estimate with a high degree of uncertainty, and a precise number has low uncertainty. The uncertainty of the result of such operations will usually correspond to the term with the highest uncertainty. The rule about N pieces of independent garbage used to calculate an {{w|arithmetic mean}} reflects how the {{w|central limit theorem}} predicts that the uncertainty (or {{w|standard error}}) of an estimate will be reduced when independent estimates are averaged.

!Formula as shown

!Resulting uncertainty

|<math>\mathop\sigma(X+Y)=\sqrt{\mathop\sigma(X)^2+\mathop\sigma(Y)^2}</math>

|{{Nowrap|If we know absolute error bars, then adding two precise numbers will}} at worst add the sizes of the two error bars. For example, if our precise numbers are 1 (±10^-6) and 1 (±10^-6), then our sum is 2 (±2·10^-6). It is possible to lose a lot of relative precision, if the resultant sum is close to zero as a result of adding a number to its approximate negation, a phenomenon known as {{w|catastrophic cancellation}}. Therefore, both of the numbers must be positive for the stated assertion to be true.

|<math>\mathop\sigma(X\times Y)\cong</math><br><br><math>\sqrt{\mathop\sigma(X)\times Y^2+\mathop\sigma(Y)\times X^2}</math>

|<math>\mathop\sigma(X+Y)=\sqrt{\mathop\sigma(X)^2+\mathop\sigma(Y)^2}</math>

|<math>\mathop\sigma(X\times Y)\cong</math><br><br><math>\sqrt{\mathop\sigma(X)\times Y^2+\mathop\sigma(Y)\times X^2}</math>

|√<span style="border-top:1px solid; padding:0 0.1em;">Garbage</span> = Less bad garbage

|<math>\mathop\sigma(\sqrt X)\cong\frac{\mathop\sigma(X)}{2\times\sqrt X} </math>

|<math>\mathop\sigma(X^2)\cong2\times X\times\mathop\sigma(X)</math>

|<math>\frac{1}{N}\sum(</math>N pieces of statistically independent garbage<math>)</math> = Better garbage

|<math>{\sigma}_\bar{X}\ = \frac{\sigma_X}{\sqrt{N}}</math>

|By aggregating many pieces of statistically independent observations (for instance, surveying many individuals), it is possible to reduce relative error to the {{w|Standard_error#Standard_error_of_the_mean|standard error of the mean}}. This is the basis of statistical sampling and the {{w|central limit theorem}}.

|<math>\mathop\sigma(b^X)\cong|b^X|\times\mathop{\mathrm{ln}}b\times\sigma(X)</math>

|<math>\mathop\sigma(X-Y)=\sqrt{\mathop\sigma(X)^2+\mathop\sigma(Y)^2}</math>

|<math>\mathop\sigma\left(\frac{a}{X-Y}\right)\cong</math><br><br><math>\frac {|a|}{(X-Y)^2}\times\sqrt{\mathop\sigma(X)^2+\mathop\sigma(Y)^2}</math>{{fact}}

The title text refers to the computer science maxim of "garbage in, garbage out," which states that when it comes to computer code, supplying incorrect initial data will produce incorrect results, even if the code itself accurately does what it is supposed to do. As we can see above, however, when plugging data into mathematical formulas, this can possibly magnify the error of our input data, though there are ways to reduce this error (such as aggregating data). Therefore, the quantity of garbage is not necessarily {{w|Conservation law|conserved}}, in contrast to other scientific quantities like energy and momentum that are always conserved. Alternatively, this could be take as a pun on environmental conservation efforts, which can often involve recycling one's trash. However, the computer science maxim of "garbage in, garbage out," has nothing to do with actual garbage.