https://www.explainxkcd.com/wiki/api.php?action=feedcontributions&user=172.68.51.124&feedformat=atomexplain xkcd - User contributions [en]2024-03-29T06:06:20ZUser contributionsMediaWiki 1.30.0https://www.explainxkcd.com/wiki/index.php?title=2295:_Garbage_Math&diff=1908822295: Garbage Math2020-04-18T05:51:08Z<p>172.68.51.124: </p>
<hr />
<div>{{comic<br />
| number = 2295<br />
| date = April 17, 2020<br />
| title = Garbage Math<br />
| image = garbage_math.png<br />
| titletext = 'Garbage In, Garbage Out' should not be taken to imply any sort of conservation law limiting the amount of garbage produced.<br />
}}<br />
<br />
==Explanation==<br />
{{incomplete|Created by a ZILOG Z80. Please mention here why this explanation isn't complete. Do NOT delete this tag too soon.}}<br />
This comic explains the "{{w|garbage in, garbage out}}" concept using arithmetical expressions. Just like the comic says, if you get garbage in any part of your workflow, you get garbage as a result.<br />
<br />
Some of these rules correspond to the rules of {{w|floating point arithmetic}}, while others may be inspired by 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 the uncertainty of the result of arithmetic operations will tend to be dominated by the term with the highest uncertainty. The rule about N pieces of independent garbage reflects the {{w|central limit theorem}} and how it predicts that the uncertainty (or {{w|standard error}}) of an estimate will be reduced when independent estimates are averaged. The comic oddly omits raising garbage to the 0th power, which transforms even NaN, the platonic ideal of garbage, to exactly 1.<br />
<br />
This comic is not related to the {{w|2019–20 coronavirus outbreak|2020 pandemic}} of the {{w|coronavirus}} {{w|SARS-CoV-2}}, which causes {{w|COVID-19}}, breaking the streak of comics preceding this on [[:Category:COVID-19|topics relating to COVID-19]], after (rather appropriately) 19 comics (not counting the [[2288: Collector's Edition|April Fools' comic]]).<br />
<br />
This comic is about the propagation of errors in numerical analysis and statistics, but described in much more colloquial terms. Numbers with low precision are termed "garbage" and numbers with high precision are labeled "precise".<br />
<br />
{| class="wikitable"<br />
!Formula<br />
!Statistical Expression<br />
!Explanation<br />
|-<br />
|Precise number + Precise number = Slightly less precise number<br />
|<math>\mathop\sigma(X+Y)=\sqrt{(\mathop\sigma(X))^2+(\mathop\sigma(Y))^2}</math><br />
|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<sup>-6</sup>) and 1 (±10<sup>-6</sup>), then our sum is 2 (±2·10<sup>-6</sup>). 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 and then close to its inverse. This phenomenon is known as catastrophic cancellation. Therefore, it is likely that all numbers referred here are positive numbers, which does not exhibit this phenomenon.<br />
|-<br />
|Precise number × Precise number = Slightly less precise number<br />
|<math>\mathop\sigma(X+Y)=\sqrt{(\mathop\sigma(X)\times Y)^2+(\mathop\sigma(Y)\times X)^2}</math><br />
|Here, instead of absolute error, relative error will be added. For example, if our precise numbers are 1 (±10<sup>-6</sup>) and 1 (±10<sup>-6</sup>), then our product is 1 (±2·10<sup>-6</sup>).<br />
|-<br />
|Precise number + Garbage = Garbage<br />
|<br />
|If one of the numbers has a high absolute error, and the numbers being added are of comparable size, then this error will be propagated to the sum. <br />
|-<br />
|Precise number × Garbage = Garbage<br />
|<br />
|Likewise, if one of the numbers has a high relative error, then this error will be propagated to the product. Here, this is independent of the sizes of the numbers.<br />
|-<br />
|<math>\sqrt{\text{Garbage}} = \text{Less bad garbage}</math><br />
|<math>\mathop\sigma(\sqrt X)=\frac{\mathop\sigma(X)}{2\times\sqrt X} </math><br />
| When the square root of a number is computed, its relative error will be halved. Depending on the application, this might not be all that much ''better'', but it's at least ''less bad''.<br />
|-<br />
|Garbage<sup>2</sup> = Worse garbage<br />
|<math>\mathop\sigma(X^2)=2\times X\times\mathop\sigma(X)</math><br />
|Likewise, when a number is squared, its relative error will be doubled. This is a corollary to multiplication adding relative errors.<br />
|-<br />
|<math>\frac{1}{N}\sum(\text{N pieces of statistically independent garbage}) = \text{Better garbage}</math><br />
|<br />
|By aggregating many pieces of statistically independent observations (for instance, surveying many individuals), it is possible to reduce relative error. This is the basis of statistical sampling.<br />
|-<br />
|Precise number<sup>Garbage</sup> = Much worse garbage<br />
|<math>\mathop\sigma(b^X)=b^{2\times X}\times\mathop{\mathrm{ln}}b\times\sigma(X)</math><br />
|The exponent is very sensitive to changes, which may also magnify the effect based on the magnitude of the precise number.<br />
|-<br />
|Garbage – Garbage = Much worse garbage<br />
|<br />
|This line involves catastrophic cancellation. If both pieces of garbage are about the same (e.g. if their error bars overlap), then it is possible that the answer is positive, zero, or negative.<br />
|-<br />
|<math>\frac{\text{Precise number}}{\text{Garbage}-\text{Garbage}}</math> = Much worse garbage, possible division by zero<br />
|<br />
|Indeed, as with above, if error bars overlap then we might end up dividing by zero.<br />
|-<br />
|Garbage × 0 = Precise number<br />
|<br />
|Multiplying anything by 0 results in 0, an extremely precise number in the sense that it has no error whatsoever since we supply the 0 ourselves. This is equivalent to discarding garbage data from a statistical analysis.<br />
|}<br />
<br />
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 conserved.<br />
<br />
==Transcript==<br />
{{incomplete transcript|Do NOT delete this tag too soon.}}<br />
<br />
[A series of mathematical equations are written from top to bottom]<br />
<br />
Precise number + Precise number = Slightly less precise number<br />
<br />
Precise number × Precise number = Slightly less precise number<br />
<br />
Precise number + Garbage = Garbage<br />
<br />
Precise number × Garbage = Garbage<br />
<br />
√<span style="border-top:1px solid; padding:0 0.1em;">Garbage</span> = Less bad garbage<br />
<br />
1/N Σ (N pieces of statistically independent garbage) = Better garbage<br />
<br />
(Precise number)<sup>Garbage</sup> = Much worse garbage<br />
<br />
Garbage – Garbage = Much worse garbage<br />
<br />
Precise number / ( Garbage – Garbage ) = Much worse garbage, possible division by zero<br />
<br />
Garbage × 0 = Precise number<br />
<br />
{{comic discussion}}<br />
[[Category:Math]]</div>172.68.51.124https://www.explainxkcd.com/wiki/index.php?title=2295:_Garbage_Math&diff=1908812295: Garbage Math2020-04-18T05:45:28Z<p>172.68.51.124: </p>
<hr />
<div>{{comic<br />
| number = 2295<br />
| date = April 17, 2020<br />
| title = Garbage Math<br />
| image = garbage_math.png<br />
| titletext = 'Garbage In, Garbage Out' should not be taken to imply any sort of conservation law limiting the amount of garbage produced.<br />
}}<br />
<br />
==Explanation==<br />
{{incomplete|Created by a ZILOG Z80. Please mention here why this explanation isn't complete. Do NOT delete this tag too soon.}}<br />
This comic explains the "{{w|garbage in, garbage out}}" concept using arithmetical expressions. Just like the comic says, if you get garbage in any part of your workflow, you get garbage as a result.<br />
<br />
Some of these rules correspond to the rules of {{w|floating point arithmetic}}, while others may be inspired by 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 the uncertainty of the result of arithmetic operations will tend to be dominated by the term with the highest uncertainty. The rule about N pieces of independent garbage reflects the {{w|central limit theorem}} and how it predicts that the uncertainty (or {{w|standard error}}) of an estimate will be reduced when independent estimates are averaged. The comic oddly omits raising garbage to the 0th power, which transforms even NaN, the platonic ideal of garbage, to exactly 1.<br />
<br />
This comic is not related to the {{w|2019–20 coronavirus outbreak|2020 pandemic}} of the {{w|coronavirus}} {{w|SARS-CoV-2}}, which causes {{w|COVID-19}}, breaking the streak of comics preceding this on [[:Category:COVID-19|topics relating to COVID-19]], after (rather appropriately) 19 comics (not counting the [[2288: Collector's Edition|April Fools' comic]]).<br />
<br />
This comic is about the propagation of errors in numerical analysis and statistics, but described in much more colloquial terms. Numbers with low precision are termed "garbage" and numbers with high precision are labeled "precise".<br />
<br />
{| class="wikitable"<br />
!Formula<br />
!Statistical Expression<br />
!Explanation<br />
|-<br />
|Precise number + Precise number = Slightly less precise number<br />
|<math>\mathop\sigma(X+Y)=\sqrt{(\mathop\sigma(X))^2+(\mathop\sigma(Y))^2}</math><br />
|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<sup>-6</sup>) and 1 (±10<sup>-6</sup>), then our sum is 2 (±2·10<sup>-6</sup>). 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 and then close to its inverse. This phenomenon is known as catastrophic cancellation. Therefore, it is likely that all numbers referred here are positive numbers, which does not exhibit this phenomenon.<br />
|-<br />
|Precise number × Precise number = Slightly less precise number<br />
|<math>\mathop\sigma(X+Y)=\sqrt{(\mathop\sigma(X)\times Y)^2+(\mathop\sigma(Y)\times X)^2}</math><br />
|Here, instead of absolute error, relative error will be added. For example, if our precise numbers are 1 (±10<sup>-6</sup>) and 1 (±10<sup>-6</sup>), then our product is 1 (±2·10<sup>-6</sup>).<br />
|-<br />
|Precise number + Garbage = Garbage<br />
|<br />
|If one of the numbers has a high absolute error, and the numbers being added are of comparable size, then this error will be propagated to the sum. <br />
|-<br />
|Precise number × Garbage = Garbage<br />
|<br />
|Likewise, if one of the numbers has a high relative error, then this error will be propagated to the product. Here, this is independent of the sizes of the numbers.<br />
|-<br />
|<math>\sqrt{\text{Garbage}} = \text{Less bad garbage}</math><br />
|<math>\mathop\sigma(\sqrt X)=\frac{\mathop\sigma(X)}{2\times\sqrt X} </math><br />
| When the square root of a number is computed, its relative error will be halved. Depending on the application, this might not be all that much ''better'', but it's at least ''less bad''.<br />
|-<br />
|Garbage<sup>2</sup> = Worse garbage<br />
|<math>\mathop\sigma(X^2)=2\times\sqrt X\times\mathop\sigma(X)</math><br />
|Likewise, when a number is squared, its relative error will be doubled. This is a corollary to multiplication adding relative errors.<br />
|-<br />
|<math>\frac{1}{N}\sum(\text{N pieces of statistically independent garbage}) = \text{Better garbage}</math><br />
|<br />
|By aggregating many pieces of statistically independent observations (for instance, surveying many individuals), it is possible to reduce relative error. This is the basis of statistical sampling.<br />
|-<br />
|Precise number<sup>Garbage</sup> = Much worse garbage<br />
|<math>\mathop\sigma(b^X)=b^{2\times X}\times\mathop{\mathrm{ln}}b\times\sigma(X)</math><br />
|The exponent is very sensitive to changes, which may also magnify the effect based on the magnitude of the precise number.<br />
|-<br />
|Garbage – Garbage = Much worse garbage<br />
|<br />
|This line involves catastrophic cancellation. If both pieces of garbage are about the same (e.g. if their error bars overlap), then it is possible that the answer is positive, zero, or negative.<br />
|-<br />
|<math>\frac{\text{Precise number}}{\text{Garbage}-\text{Garbage}}</math> = Much worse garbage, possible division by zero<br />
|<br />
|Indeed, as with above, if error bars overlap then we might end up dividing by zero.<br />
|-<br />
|Garbage × 0 = Precise number<br />
|<br />
|Multiplying anything by 0 results in 0, an extremely precise number in the sense that it has no error whatsoever since we supply the 0 ourselves. This is equivalent to discarding garbage data from a statistical analysis.<br />
|}<br />
<br />
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 conserved.<br />
<br />
==Transcript==<br />
{{incomplete transcript|Do NOT delete this tag too soon.}}<br />
<br />
[A series of mathematical equations are written from top to bottom]<br />
<br />
Precise number + Precise number = Slightly less precise number<br />
<br />
Precise number × Precise number = Slightly less precise number<br />
<br />
Precise number + Garbage = Garbage<br />
<br />
Precise number × Garbage = Garbage<br />
<br />
√<span style="border-top:1px solid; padding:0 0.1em;">Garbage</span> = Less bad garbage<br />
<br />
1/N Σ (N pieces of statistically independent garbage) = Better garbage<br />
<br />
(Precise number)<sup>Garbage</sup> = Much worse garbage<br />
<br />
Garbage – Garbage = Much worse garbage<br />
<br />
Precise number / ( Garbage – Garbage ) = Much worse garbage, possible division by zero<br />
<br />
Garbage × 0 = Precise number<br />
<br />
{{comic discussion}}<br />
[[Category:Math]]</div>172.68.51.124https://www.explainxkcd.com/wiki/index.php?title=2295:_Garbage_Math&diff=1908802295: Garbage Math2020-04-18T05:40:07Z<p>172.68.51.124: </p>
<hr />
<div>{{comic<br />
| number = 2295<br />
| date = April 17, 2020<br />
| title = Garbage Math<br />
| image = garbage_math.png<br />
| titletext = 'Garbage In, Garbage Out' should not be taken to imply any sort of conservation law limiting the amount of garbage produced.<br />
}}<br />
<br />
==Explanation==<br />
{{incomplete|Created by a ZILOG Z80. Please mention here why this explanation isn't complete. Do NOT delete this tag too soon.}}<br />
This comic explains the "{{w|garbage in, garbage out}}" concept using arithmetical expressions. Just like the comic says, if you get garbage in any part of your workflow, you get garbage as a result.<br />
<br />
Some of these rules correspond to the rules of {{w|floating point arithmetic}}, while others may be inspired by 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 the uncertainty of the result of arithmetic operations will tend to be dominated by the term with the highest uncertainty. The rule about N pieces of independent garbage reflects the {{w|central limit theorem}} and how it predicts that the uncertainty (or {{w|standard error}}) of an estimate will be reduced when independent estimates are averaged. The comic oddly omits raising garbage to the 0th power, which transforms even NaN, the platonic ideal of garbage, to exactly 1.<br />
<br />
This comic is not related to the {{w|2019–20 coronavirus outbreak|2020 pandemic}} of the {{w|coronavirus}} {{w|SARS-CoV-2}}, which causes {{w|COVID-19}}, breaking the streak of comics preceding this on [[:Category:COVID-19|topics relating to COVID-19]], after (rather appropriately) 19 comics (not counting the [[2288: Collector's Edition|April Fools' comic]]).<br />
<br />
This comic is about the propagation of errors in numerical analysis and statistics, but described in much more colloquial terms. Numbers with low precision are termed "garbage" and numbers with high precision are labeled "precise".<br />
<br />
{| class="wikitable"<br />
!Formula<br />
!Statistical Expression<br />
!Explanation<br />
|-<br />
|Precise number + Precise number = Slightly less precise number<br />
|<math>\mathop\sigma(X+Y)=\sqrt{(\mathop\sigma(X))^2+(\mathop\sigma(Y))^2}</math><br />
|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<sup>-6</sup>) and 1 (±10<sup>-6</sup>), then our sum is 2 (±2·10<sup>-6</sup>). 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 and then close to its inverse. This phenomenon is known as catastrophic cancellation. Therefore, it is likely that all numbers referred here are positive numbers, which does not exhibit this phenomenon.<br />
|-<br />
|Precise number × Precise number = Slightly less precise number<br />
|<math>\mathop\sigma(X+Y)=\sqrt{(\mathop\sigma(X)\times Y)^2+(\mathop\sigma(Y)\times X)^2}</math><br />
|Here, instead of absolute error, relative error will be added. For example, if our precise numbers are 1 (±10<sup>-6</sup>) and 1 (±10<sup>-6</sup>), then our product is 1 (±2·10<sup>-6</sup>).<br />
|-<br />
|Precise number + Garbage = Garbage<br />
|<br />
|If one of the numbers has a high absolute error, and the numbers being added are of comparable size, then this error will be propagated to the sum. <br />
|-<br />
|Precise number × Garbage = Garbage<br />
|<br />
|Likewise, if one of the numbers has a high relative error, then this error will be propagated to the product. Here, this is independent of the sizes of the numbers.<br />
|-<br />
|<math>\sqrt{\text{Garbage}} = \text{Less bad garbage}</math><br />
|<math>\mathop\sigma(\sqrt X)=\frac{\mathop\sigma(X)}{2\times\sqrt X} </math><br />
| When the square root of a number is computed, its relative error will be halved. Depending on the application, this might not be all that much ''better'', but it's at least ''less bad''.<br />
|-<br />
|Garbage<sup>2</sup> = Worse garbage<br />
|<math>\mathop\sigma(\sqrt X)=2\times\sqrt X\times\mathop\sigma(X)</math><br />
|Likewise, when a number is squared, its relative error will be doubled. This is a corollary to multiplication adding relative errors.<br />
|-<br />
|<math>\frac{1}{N}\sum(\text{N pieces of statistically independent garbage}) = \text{Better garbage}</math><br />
|<br />
|By aggregating many pieces of statistically independent observations (for instance, surveying many individuals), it is possible to reduce relative error. This is the basis of statistical sampling.<br />
|-<br />
|Precise number<sup>Garbage</sup> = Much worse garbage<br />
|<br />
|The exponent is very sensitive to changes, which may also magnify the effect based on the magnitude of the precise number.<br />
|-<br />
|Garbage – Garbage = Much worse garbage<br />
|<br />
|This line involves catastrophic cancellation. If both pieces of garbage are about the same (e.g. if their error bars overlap), then it is possible that the answer is positive, zero, or negative.<br />
|-<br />
|<math>\frac{\text{Precise number}}{\text{Garbage}-\text{Garbage}}</math> = Much worse garbage, possible division by zero<br />
|<br />
|Indeed, as with above, if error bars overlap then we might end up dividing by zero.<br />
|-<br />
|Garbage × 0 = Precise number<br />
|<br />
|Multiplying anything by 0 results in 0, an extremely precise number in the sense that it has no error whatsoever since we supply the 0 ourselves. This is equivalent to discarding garbage data from a statistical analysis.<br />
|}<br />
<br />
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 conserved.<br />
<br />
==Transcript==<br />
{{incomplete transcript|Do NOT delete this tag too soon.}}<br />
<br />
[A series of mathematical equations are written from top to bottom]<br />
<br />
Precise number + Precise number = Slightly less precise number<br />
<br />
Precise number × Precise number = Slightly less precise number<br />
<br />
Precise number + Garbage = Garbage<br />
<br />
Precise number × Garbage = Garbage<br />
<br />
√<span style="border-top:1px solid; padding:0 0.1em;">Garbage</span> = Less bad garbage<br />
<br />
1/N Σ (N pieces of statistically independent garbage) = Better garbage<br />
<br />
(Precise number)<sup>Garbage</sup> = Much worse garbage<br />
<br />
Garbage – Garbage = Much worse garbage<br />
<br />
Precise number / ( Garbage – Garbage ) = Much worse garbage, possible division by zero<br />
<br />
Garbage × 0 = Precise number<br />
<br />
{{comic discussion}}<br />
[[Category:Math]]</div>172.68.51.124https://www.explainxkcd.com/wiki/index.php?title=2295:_Garbage_Math&diff=1908792295: Garbage Math2020-04-18T05:34:41Z<p>172.68.51.124: </p>
<hr />
<div>{{comic<br />
| number = 2295<br />
| date = April 17, 2020<br />
| title = Garbage Math<br />
| image = garbage_math.png<br />
| titletext = 'Garbage In, Garbage Out' should not be taken to imply any sort of conservation law limiting the amount of garbage produced.<br />
}}<br />
<br />
==Explanation==<br />
{{incomplete|Created by a ZILOG Z80. Please mention here why this explanation isn't complete. Do NOT delete this tag too soon.}}<br />
This comic explains the "{{w|garbage in, garbage out}}" concept using arithmetical expressions. Just like the comic says, if you get garbage in any part of your workflow, you get garbage as a result.<br />
<br />
Some of these rules correspond to the rules of {{w|floating point arithmetic}}, while others may be inspired by 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 the uncertainty of the result of arithmetic operations will tend to be dominated by the term with the highest uncertainty. The rule about N pieces of independent garbage reflects the {{w|central limit theorem}} and how it predicts that the uncertainty (or {{w|standard error}}) of an estimate will be reduced when independent estimates are averaged. The comic oddly omits raising garbage to the 0th power, which transforms even NaN, the platonic ideal of garbage, to exactly 1.<br />
<br />
This comic is not related to the {{w|2019–20 coronavirus outbreak|2020 pandemic}} of the {{w|coronavirus}} {{w|SARS-CoV-2}}, which causes {{w|COVID-19}}, breaking the streak of comics preceding this on [[:Category:COVID-19|topics relating to COVID-19]], after (rather appropriately) 19 comics (not counting the [[2288: Collector's Edition|April Fools' comic]]).<br />
<br />
This comic is about the propagation of errors in numerical analysis and statistics, but described in much more colloquial terms. Numbers with low precision are termed "garbage" and numbers with high precision are labeled "precise".<br />
<br />
{| class="wikitable"<br />
!Formula<br />
!Statistical Expression<br />
!Explanation<br />
|-<br />
|Precise number + Precise number = Slightly less precise number<br />
|<math>\mathop\sigma(X+Y)=\sqrt{(\mathop\sigma(X))^2+(\mathop\sigma(Y))^2}</math><br />
|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<sup>-6</sup>) and 1 (±10<sup>-6</sup>), then our sum is 2 (±2·10<sup>-6</sup>). 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 and then close to its inverse. This phenomenon is known as catastrophic cancellation. Therefore, it is likely that all numbers referred here are positive numbers, which does not exhibit this phenomenon.<br />
|-<br />
|Precise number × Precise number = Slightly less precise number<br />
|<math>\mathop\sigma(X+Y)=\sqrt{(\mathop\sigma(X)\times Y)^2+(\mathop\sigma(Y)\times X)^2}</math><br />
|Here, instead of absolute error, relative error will be added. For example, if our precise numbers are 1 (±10<sup>-6</sup>) and 1 (±10<sup>-6</sup>), then our product is 1 (±2·10<sup>-6</sup>).<br />
|-<br />
|Precise number + Garbage = Garbage<br />
|<br />
|If one of the numbers has a high absolute error, and the numbers being added are of comparable size, then this error will be propagated to the sum. <br />
|-<br />
|Precise number × Garbage = Garbage<br />
|<br />
|Likewise, if one of the numbers has a high relative error, then this error will be propagated to the product. Here, this is independent of the sizes of the numbers.<br />
|-<br />
|<math>\sqrt{\text{Garbage}} = \text{Less bad garbage}</math><br />
|<math>\mathop\sigma(\sqrt X)=\frac{\mathop\sigma(X)}{2\times\sqrt X} </math><br />
| When the square root of a number is computed, its relative error will be halved. Depending on the application, this might not be all that much ''better'', but it's at least ''less bad''.<br />
|-<br />
|Garbage<sup>2</sup> = Worse garbage<br />
|<br />
|Likewise, when a number is squared, its relative error will be doubled. This is a corollary to multiplication adding relative errors.<br />
|-<br />
|<math>\frac{1}{N}\sum(\text{N pieces of statistically independent garbage}) = \text{Better garbage}</math><br />
|<br />
|By aggregating many pieces of statistically independent observations (for instance, surveying many individuals), it is possible to reduce relative error. This is the basis of statistical sampling.<br />
|-<br />
|Precise number<sup>Garbage</sup> = Much worse garbage<br />
|<br />
|The exponent is very sensitive to changes, which may also magnify the effect based on the magnitude of the precise number.<br />
|-<br />
|Garbage – Garbage = Much worse garbage<br />
|<br />
|This line involves catastrophic cancellation. If both pieces of garbage are about the same (e.g. if their error bars overlap), then it is possible that the answer is positive, zero, or negative.<br />
|-<br />
|<math>\frac{\text{Precise number}}{\text{Garbage}-\text{Garbage}}</math> = Much worse garbage, possible division by zero<br />
|<br />
|Indeed, as with above, if error bars overlap then we might end up dividing by zero.<br />
|-<br />
|Garbage × 0 = Precise number<br />
|<br />
|Multiplying anything by 0 results in 0, an extremely precise number in the sense that it has no error whatsoever since we supply the 0 ourselves. This is equivalent to discarding garbage data from a statistical analysis.<br />
|}<br />
<br />
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 conserved.<br />
<br />
==Transcript==<br />
{{incomplete transcript|Do NOT delete this tag too soon.}}<br />
<br />
[A series of mathematical equations are written from top to bottom]<br />
<br />
Precise number + Precise number = Slightly less precise number<br />
<br />
Precise number × Precise number = Slightly less precise number<br />
<br />
Precise number + Garbage = Garbage<br />
<br />
Precise number × Garbage = Garbage<br />
<br />
√<span style="border-top:1px solid; padding:0 0.1em;">Garbage</span> = Less bad garbage<br />
<br />
1/N Σ (N pieces of statistically independent garbage) = Better garbage<br />
<br />
(Precise number)<sup>Garbage</sup> = Much worse garbage<br />
<br />
Garbage – Garbage = Much worse garbage<br />
<br />
Precise number / ( Garbage – Garbage ) = Much worse garbage, possible division by zero<br />
<br />
Garbage × 0 = Precise number<br />
<br />
{{comic discussion}}<br />
[[Category:Math]]</div>172.68.51.124https://www.explainxkcd.com/wiki/index.php?title=2295:_Garbage_Math&diff=1908782295: Garbage Math2020-04-18T05:33:27Z<p>172.68.51.124: </p>
<hr />
<div>{{comic<br />
| number = 2295<br />
| date = April 17, 2020<br />
| title = Garbage Math<br />
| image = garbage_math.png<br />
| titletext = 'Garbage In, Garbage Out' should not be taken to imply any sort of conservation law limiting the amount of garbage produced.<br />
}}<br />
<br />
==Explanation==<br />
{{incomplete|Created by a ZILOG Z80. Please mention here why this explanation isn't complete. Do NOT delete this tag too soon.}}<br />
This comic explains the "{{w|garbage in, garbage out}}" concept using arithmetical expressions. Just like the comic says, if you get garbage in any part of your workflow, you get garbage as a result.<br />
<br />
Some of these rules correspond to the rules of {{w|floating point arithmetic}}, while others may be inspired by 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 the uncertainty of the result of arithmetic operations will tend to be dominated by the term with the highest uncertainty. The rule about N pieces of independent garbage reflects the {{w|central limit theorem}} and how it predicts that the uncertainty (or {{w|standard error}}) of an estimate will be reduced when independent estimates are averaged. The comic oddly omits raising garbage to the 0th power, which transforms even NaN, the platonic ideal of garbage, to exactly 1.<br />
<br />
This comic is not related to the {{w|2019–20 coronavirus outbreak|2020 pandemic}} of the {{w|coronavirus}} {{w|SARS-CoV-2}}, which causes {{w|COVID-19}}, breaking the streak of comics preceding this on [[:Category:COVID-19|topics relating to COVID-19]], after (rather appropriately) 19 comics (not counting the [[2288: Collector's Edition|April Fools' comic]]).<br />
<br />
This comic is about the propagation of errors in numerical analysis and statistics, but described in much more colloquial terms. Numbers with low precision are termed "garbage" and numbers with high precision are labeled "precise".<br />
<br />
{| class="wikitable"<br />
!Formula<br />
!Statistical Expression<br />
!Explanation<br />
|-<br />
|Precise number + Precise number = Slightly less precise number<br />
|<math>\mathop\sigma(X+Y)=\sqrt{(\mathop\sigma(X))^2+(\mathop\sigma(Y))^2}</math><br />
|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<sup>-6</sup>) and 1 (±10<sup>-6</sup>), then our sum is 2 (±2·10<sup>-6</sup>). 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 and then close to its inverse. This phenomenon is known as catastrophic cancellation. Therefore, it is likely that all numbers referred here are positive numbers, which does not exhibit this phenomenon.<br />
|-<br />
|Precise number × Precise number = Slightly less precise number<br />
|<math>\mathop\sigma(X+Y)=\sqrt{(\mathop\sigma(X)\times Y)^2+(\mathop\sigma(Y)\times X)^2}</math><br />
|Here, instead of absolute error, relative error will be added. For example, if our precise numbers are 1 (±10<sup>-6</sup>) and 1 (±10<sup>-6</sup>), then our product is 1 (±2·10<sup>-6</sup>).<br />
|-<br />
|Precise number + Garbage = Garbage<br />
|<br />
|If one of the numbers has a high absolute error, and the numbers being added are of comparable size, then this error will be propagated to the sum. <br />
|-<br />
|Precise number × Garbage = Garbage<br />
|<br />
|Likewise, if one of the numbers has a high relative error, then this error will be propagated to the product. Here, this is independent of the sizes of the numbers.<br />
|-<br />
|<math>\sqrt{\text{Garbage}} = \text{Less bad garbage}</math><br />
|<math>\mathop\sigma(\sqrt X)=\frac{\mathop\sigma X}{2\times\sqrt X} </math><br />
| When the square root of a number is computed, its relative error will be halved. Depending on the application, this might not be all that much ''better'', but it's at least ''less bad''.<br />
|-<br />
|Garbage<sup>2</sup> = Worse garbage<br />
|<br />
|Likewise, when a number is squared, its relative error will be doubled. This is a corollary to multiplication adding relative errors.<br />
|-<br />
|<math>\frac{1}{N}\sum(\text{N pieces of statistically independent garbage}) = \text{Better garbage}</math><br />
|<br />
|By aggregating many pieces of statistically independent observations (for instance, surveying many individuals), it is possible to reduce relative error. This is the basis of statistical sampling.<br />
|-<br />
|Precise number<sup>Garbage</sup> = Much worse garbage<br />
|<br />
|The exponent is very sensitive to changes, which may also magnify the effect based on the magnitude of the precise number.<br />
|-<br />
|Garbage – Garbage = Much worse garbage<br />
|<br />
|This line involves catastrophic cancellation. If both pieces of garbage are about the same (e.g. if their error bars overlap), then it is possible that the answer is positive, zero, or negative.<br />
|-<br />
|<math>\frac{\text{Precise number}}{\text{Garbage}-\text{Garbage}}</math> = Much worse garbage, possible division by zero<br />
|<br />
|Indeed, as with above, if error bars overlap then we might end up dividing by zero.<br />
|-<br />
|Garbage × 0 = Precise number<br />
|<br />
|Multiplying anything by 0 results in 0, an extremely precise number in the sense that it has no error whatsoever since we supply the 0 ourselves. This is equivalent to discarding garbage data from a statistical analysis.<br />
|}<br />
<br />
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 conserved.<br />
<br />
==Transcript==<br />
{{incomplete transcript|Do NOT delete this tag too soon.}}<br />
<br />
[A series of mathematical equations are written from top to bottom]<br />
<br />
Precise number + Precise number = Slightly less precise number<br />
<br />
Precise number × Precise number = Slightly less precise number<br />
<br />
Precise number + Garbage = Garbage<br />
<br />
Precise number × Garbage = Garbage<br />
<br />
√<span style="border-top:1px solid; padding:0 0.1em;">Garbage</span> = Less bad garbage<br />
<br />
1/N Σ (N pieces of statistically independent garbage) = Better garbage<br />
<br />
(Precise number)<sup>Garbage</sup> = Much worse garbage<br />
<br />
Garbage – Garbage = Much worse garbage<br />
<br />
Precise number / ( Garbage – Garbage ) = Much worse garbage, possible division by zero<br />
<br />
Garbage × 0 = Precise number<br />
<br />
{{comic discussion}}<br />
[[Category:Math]]</div>172.68.51.124https://www.explainxkcd.com/wiki/index.php?title=2295:_Garbage_Math&diff=1908772295: Garbage Math2020-04-18T05:22:57Z<p>172.68.51.124: </p>
<hr />
<div>{{comic<br />
| number = 2295<br />
| date = April 17, 2020<br />
| title = Garbage Math<br />
| image = garbage_math.png<br />
| titletext = 'Garbage In, Garbage Out' should not be taken to imply any sort of conservation law limiting the amount of garbage produced.<br />
}}<br />
<br />
==Explanation==<br />
{{incomplete|Created by a ZILOG Z80. Please mention here why this explanation isn't complete. Do NOT delete this tag too soon.}}<br />
This comic explains the "{{w|garbage in, garbage out}}" concept using arithmetical expressions. Just like the comic says, if you get garbage in any part of your workflow, you get garbage as a result.<br />
<br />
Some of these rules correspond to the rules of {{w|floating point arithmetic}}, while others may be inspired by 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 the uncertainty of the result of arithmetic operations will tend to be dominated by the term with the highest uncertainty. The rule about N pieces of independent garbage reflects the {{w|central limit theorem}} and how it predicts that the uncertainty (or {{w|standard error}}) of an estimate will be reduced when independent estimates are averaged. The comic oddly omits raising garbage to the 0th power, which transforms even NaN, the platonic ideal of garbage, to exactly 1.<br />
<br />
This comic is not related to the {{w|2019–20 coronavirus outbreak|2020 pandemic}} of the {{w|coronavirus}} {{w|SARS-CoV-2}}, which causes {{w|COVID-19}}, breaking the streak of comics preceding this on [[:Category:COVID-19|topics relating to COVID-19]], after (rather appropriately) 19 comics (not counting the [[2288: Collector's Edition|April Fools' comic]]).<br />
<br />
This comic is about the propagation of errors in numerical analysis and statistics, but described in much more colloquial terms. Numbers with low precision are termed "garbage" and numbers with high precision are labeled "precise".<br />
<br />
{| class="wikitable"<br />
!Formula<br />
!Statistical Expression<br />
!Explanation<br />
|-<br />
|Precise number + Precise number = Slightly less precise number<br />
|<math>\mathop\sigma(X+Y)=\sqrt{(\mathop\sigma(X))^2+(\mathop\sigma(Y))^2}</math><br />
|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<sup>-6</sup>) and 1 (±10<sup>-6</sup>), then our sum is 2 (±2·10<sup>-6</sup>). 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 and then close to its inverse. This phenomenon is known as catastrophic cancellation. Therefore, it is likely that all numbers referred here are positive numbers, which does not exhibit this phenomenon.<br />
|-<br />
|Precise number × Precise number = Slightly less precise number<br />
|<math>\mathop\sigma(X+Y)=\sqrt{(\mathop\sigma(X)\times Y)^2+(\mathop\sigma(Y)\times X)^2}</math><br />
|Here, instead of absolute error, relative error will be added. For example, if our precise numbers are 1 (±10<sup>-6</sup>) and 1 (±10<sup>-6</sup>), then our product is 1 (±2·10<sup>-6</sup>).<br />
|-<br />
|Precise number + Garbage = Garbage<br />
|<br />
|If one of the numbers has a high absolute error, and the numbers being added are of comparable size, then this error will be propagated to the sum. <br />
|-<br />
|Precise number × Garbage = Garbage<br />
|<br />
|Likewise, if one of the numbers has a high relative error, then this error will be propagated to the product. Here, this is independent of the sizes of the numbers.<br />
|-<br />
|<math>\sqrt{\text{Garbage}} = \text{Less bad garbage}</math><br />
|<br />
| When the square root of a number is computed, its relative error will be halved. Depending on the application, this might not be all that much ''better'', but it's at least ''less bad''.<br />
|-<br />
|Garbage<sup>2</sup> = Worse garbage<br />
|<br />
|Likewise, when a number is squared, its relative error will be doubled. This is a corollary to multiplication adding relative errors.<br />
|-<br />
|<math>\frac{1}{N}\sum(\text{N pieces of statistically independent garbage}) = \text{Better garbage}</math><br />
|<br />
|By aggregating many pieces of statistically independent observations (for instance, surveying many individuals), it is possible to reduce relative error. This is the basis of statistical sampling.<br />
|-<br />
|Precise number<sup>Garbage</sup> = Much worse garbage<br />
|<br />
|The exponent is very sensitive to changes, which may also magnify the effect based on the magnitude of the precise number.<br />
|-<br />
|Garbage – Garbage = Much worse garbage<br />
|<br />
|This line involves catastrophic cancellation. If both pieces of garbage are about the same (e.g. if their error bars overlap), then it is possible that the answer is positive, zero, or negative.<br />
|-<br />
|<math>\frac{\text{Precise number}}{\text{Garbage}-\text{Garbage}}</math> = Much worse garbage, possible division by zero<br />
|<br />
|Indeed, as with above, if error bars overlap then we might end up dividing by zero.<br />
|-<br />
|Garbage × 0 = Precise number<br />
|<br />
|Multiplying anything by 0 results in 0, an extremely precise number in the sense that it has no error whatsoever since we supply the 0 ourselves. This is equivalent to discarding garbage data from a statistical analysis.<br />
|}<br />
<br />
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 conserved.<br />
<br />
==Transcript==<br />
{{incomplete transcript|Do NOT delete this tag too soon.}}<br />
<br />
[A series of mathematical equations are written from top to bottom]<br />
<br />
Precise number + Precise number = Slightly less precise number<br />
<br />
Precise number × Precise number = Slightly less precise number<br />
<br />
Precise number + Garbage = Garbage<br />
<br />
Precise number × Garbage = Garbage<br />
<br />
√<span style="border-top:1px solid; padding:0 0.1em;">Garbage</span> = Less bad garbage<br />
<br />
1/N Σ (N pieces of statistically independent garbage) = Better garbage<br />
<br />
(Precise number)<sup>Garbage</sup> = Much worse garbage<br />
<br />
Garbage – Garbage = Much worse garbage<br />
<br />
Precise number / ( Garbage – Garbage ) = Much worse garbage, possible division by zero<br />
<br />
Garbage × 0 = Precise number<br />
<br />
{{comic discussion}}<br />
[[Category:Math]]</div>172.68.51.124https://www.explainxkcd.com/wiki/index.php?title=2295:_Garbage_Math&diff=1908762295: Garbage Math2020-04-18T05:15:54Z<p>172.68.51.124: </p>
<hr />
<div>{{comic<br />
| number = 2295<br />
| date = April 17, 2020<br />
| title = Garbage Math<br />
| image = garbage_math.png<br />
| titletext = 'Garbage In, Garbage Out' should not be taken to imply any sort of conservation law limiting the amount of garbage produced.<br />
}}<br />
<br />
==Explanation==<br />
{{incomplete|Created by a ZILOG Z80. Please mention here why this explanation isn't complete. Do NOT delete this tag too soon.}}<br />
This comic explains the "{{w|garbage in, garbage out}}" concept using arithmetical expressions. Just like the comic says, if you get garbage in any part of your workflow, you get garbage as a result.<br />
<br />
Some of these rules correspond to the rules of {{w|floating point arithmetic}}, while others may be inspired by 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 the uncertainty of the result of arithmetic operations will tend to be dominated by the term with the highest uncertainty. The rule about N pieces of independent garbage reflects the {{w|central limit theorem}} and how it predicts that the uncertainty (or {{w|standard error}}) of an estimate will be reduced when independent estimates are averaged. The comic oddly omits raising garbage to the 0th power, which transforms even NaN, the platonic ideal of garbage, to exactly 1.<br />
<br />
This comic is not related to the {{w|2019–20 coronavirus outbreak|2020 pandemic}} of the {{w|coronavirus}} {{w|SARS-CoV-2}}, which causes {{w|COVID-19}}, breaking the streak of comics preceding this on [[:Category:COVID-19|topics relating to COVID-19]], after (rather appropriately) 19 comics (not counting the [[2288: Collector's Edition|April Fools' comic]]).<br />
<br />
This comic is about the propagation of errors in numerical analysis and statistics, but described in much more colloquial terms. Numbers with low precision are termed "garbage" and numbers with high precision are labeled "precise".<br />
<br />
{| class="wikitable"<br />
!Formula<br />
!Statistical Expression<br />
!Explanation<br />
|-<br />
|Precise number + Precise number = Slightly less precise number<br />
|<math>\mathop\sigma(X+Y)=\sqrt{(\mathop\sigma(X))^2+(\mathop\sigma(Y))^2}</math><br />
|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<sup>-6</sup>) and 1 (±10<sup>-6</sup>), then our sum is 2 (±2·10<sup>-6</sup>). 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 and then close to its inverse. This phenomenon is known as catastrophic cancellation. Therefore, it is likely that all numbers referred here are positive numbers, which does not exhibit this phenomenon.<br />
|-<br />
|Precise number × Precise number = Slightly less precise number<br />
|<br />
|Here, instead of absolute error, relative error will be added. For example, if our precise numbers are 1 (±10<sup>-6</sup>) and 1 (±10<sup>-6</sup>), then our product is 1 (±2·10<sup>-6</sup>).<br />
|-<br />
|Precise number + Garbage = Garbage<br />
|<br />
|If one of the numbers has a high absolute error, and the numbers being added are of comparable size, then this error will be propagated to the sum. <br />
|-<br />
|Precise number × Garbage = Garbage<br />
|<br />
|Likewise, if one of the numbers has a high relative error, then this error will be propagated to the product. Here, this is independent of the sizes of the numbers.<br />
|-<br />
|<math>\sqrt{\text{Garbage}} = \text{Less bad garbage}</math><br />
|<br />
| When the square root of a number is computed, its relative error will be halved. Depending on the application, this might not be all that much ''better'', but it's at least ''less bad''.<br />
|-<br />
|Garbage<sup>2</sup> = Worse garbage<br />
|<br />
|Likewise, when a number is squared, its relative error will be doubled. This is a corollary to multiplication adding relative errors.<br />
|-<br />
|<math>\frac{1}{N}\sum(\text{N pieces of statistically independent garbage}) = \text{Better garbage}</math><br />
|<br />
|By aggregating many pieces of statistically independent observations (for instance, surveying many individuals), it is possible to reduce relative error. This is the basis of statistical sampling.<br />
|-<br />
|Precise number<sup>Garbage</sup> = Much worse garbage<br />
|<br />
|The exponent is very sensitive to changes, which may also magnify the effect based on the magnitude of the precise number.<br />
|-<br />
|Garbage – Garbage = Much worse garbage<br />
|<br />
|This line involves catastrophic cancellation. If both pieces of garbage are about the same (e.g. if their error bars overlap), then it is possible that the answer is positive, zero, or negative.<br />
|-<br />
|<math>\frac{\text{Precise number}}{\text{Garbage}-\text{Garbage}}</math> = Much worse garbage, possible division by zero<br />
|<br />
|Indeed, as with above, if error bars overlap then we might end up dividing by zero.<br />
|-<br />
|Garbage × 0 = Precise number<br />
|<br />
|Multiplying anything by 0 results in 0, an extremely precise number in the sense that it has no error whatsoever since we supply the 0 ourselves. This is equivalent to discarding garbage data from a statistical analysis.<br />
|}<br />
<br />
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 conserved.<br />
<br />
==Transcript==<br />
{{incomplete transcript|Do NOT delete this tag too soon.}}<br />
<br />
[A series of mathematical equations are written from top to bottom]<br />
<br />
Precise number + Precise number = Slightly less precise number<br />
<br />
Precise number × Precise number = Slightly less precise number<br />
<br />
Precise number + Garbage = Garbage<br />
<br />
Precise number × Garbage = Garbage<br />
<br />
√<span style="border-top:1px solid; padding:0 0.1em;">Garbage</span> = Less bad garbage<br />
<br />
1/N Σ (N pieces of statistically independent garbage) = Better garbage<br />
<br />
(Precise number)<sup>Garbage</sup> = Much worse garbage<br />
<br />
Garbage – Garbage = Much worse garbage<br />
<br />
Precise number / ( Garbage – Garbage ) = Much worse garbage, possible division by zero<br />
<br />
Garbage × 0 = Precise number<br />
<br />
{{comic discussion}}<br />
[[Category:Math]]</div>172.68.51.124https://www.explainxkcd.com/wiki/index.php?title=2295:_Garbage_Math&diff=1908752295: Garbage Math2020-04-18T05:15:16Z<p>172.68.51.124: </p>
<hr />
<div>{{comic<br />
| number = 2295<br />
| date = April 17, 2020<br />
| title = Garbage Math<br />
| image = garbage_math.png<br />
| titletext = 'Garbage In, Garbage Out' should not be taken to imply any sort of conservation law limiting the amount of garbage produced.<br />
}}<br />
<br />
==Explanation==<br />
{{incomplete|Created by a ZILOG Z80. Please mention here why this explanation isn't complete. Do NOT delete this tag too soon.}}<br />
This comic explains the "{{w|garbage in, garbage out}}" concept using arithmetical expressions. Just like the comic says, if you get garbage in any part of your workflow, you get garbage as a result.<br />
<br />
Some of these rules correspond to the rules of {{w|floating point arithmetic}}, while others may be inspired by 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 the uncertainty of the result of arithmetic operations will tend to be dominated by the term with the highest uncertainty. The rule about N pieces of independent garbage reflects the {{w|central limit theorem}} and how it predicts that the uncertainty (or {{w|standard error}}) of an estimate will be reduced when independent estimates are averaged. The comic oddly omits raising garbage to the 0th power, which transforms even NaN, the platonic ideal of garbage, to exactly 1.<br />
<br />
This comic is not related to the {{w|2019–20 coronavirus outbreak|2020 pandemic}} of the {{w|coronavirus}} {{w|SARS-CoV-2}}, which causes {{w|COVID-19}}, breaking the streak of comics preceding this on [[:Category:COVID-19|topics relating to COVID-19]], after (rather appropriately) 19 comics (not counting the [[2288: Collector's Edition|April Fools' comic]]).<br />
<br />
This comic is about the propagation of errors in numerical analysis and statistics, but described in much more colloquial terms. Numbers with low precision are termed "garbage" and numbers with high precision are labeled "precise".<br />
<br />
{| class="wikitable"<br />
!Formula<br />
!Statistical Expression<br />
!Explanation<br />
|-<br />
|Precise number + Precise number = Slightly less precise number<br />
|<math>\mathop\sigma(X+Y)=\sqrt{(\mathop\sigma(X))^2(\mathop\sigma(Y))^2}</math><br />
|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<sup>-6</sup>) and 1 (±10<sup>-6</sup>), then our sum is 2 (±2·10<sup>-6</sup>). 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 and then close to its inverse. This phenomenon is known as catastrophic cancellation. Therefore, it is likely that all numbers referred here are positive numbers, which does not exhibit this phenomenon.<br />
|-<br />
|Precise number × Precise number = Slightly less precise number<br />
|<br />
|Here, instead of absolute error, relative error will be added. For example, if our precise numbers are 1 (±10<sup>-6</sup>) and 1 (±10<sup>-6</sup>), then our product is 1 (±2·10<sup>-6</sup>).<br />
|-<br />
|Precise number + Garbage = Garbage<br />
|<br />
|If one of the numbers has a high absolute error, and the numbers being added are of comparable size, then this error will be propagated to the sum. <br />
|-<br />
|Precise number × Garbage = Garbage<br />
|<br />
|Likewise, if one of the numbers has a high relative error, then this error will be propagated to the product. Here, this is independent of the sizes of the numbers.<br />
|-<br />
|<math>\sqrt{\text{Garbage}} = \text{Less bad garbage}</math><br />
|<br />
| When the square root of a number is computed, its relative error will be halved. Depending on the application, this might not be all that much ''better'', but it's at least ''less bad''.<br />
|-<br />
|Garbage<sup>2</sup> = Worse garbage<br />
|<br />
|Likewise, when a number is squared, its relative error will be doubled. This is a corollary to multiplication adding relative errors.<br />
|-<br />
|<math>\frac{1}{N}\sum(\text{N pieces of statistically independent garbage}) = \text{Better garbage}</math><br />
|<br />
|By aggregating many pieces of statistically independent observations (for instance, surveying many individuals), it is possible to reduce relative error. This is the basis of statistical sampling.<br />
|-<br />
|Precise number<sup>Garbage</sup> = Much worse garbage<br />
|<br />
|The exponent is very sensitive to changes, which may also magnify the effect based on the magnitude of the precise number.<br />
|-<br />
|Garbage – Garbage = Much worse garbage<br />
|<br />
|This line involves catastrophic cancellation. If both pieces of garbage are about the same (e.g. if their error bars overlap), then it is possible that the answer is positive, zero, or negative.<br />
|-<br />
|<math>\frac{\text{Precise number}}{\text{Garbage}-\text{Garbage}}</math> = Much worse garbage, possible division by zero<br />
|<br />
|Indeed, as with above, if error bars overlap then we might end up dividing by zero.<br />
|-<br />
|Garbage × 0 = Precise number<br />
|<br />
|Multiplying anything by 0 results in 0, an extremely precise number in the sense that it has no error whatsoever since we supply the 0 ourselves. This is equivalent to discarding garbage data from a statistical analysis.<br />
|}<br />
<br />
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 conserved.<br />
<br />
==Transcript==<br />
{{incomplete transcript|Do NOT delete this tag too soon.}}<br />
<br />
[A series of mathematical equations are written from top to bottom]<br />
<br />
Precise number + Precise number = Slightly less precise number<br />
<br />
Precise number × Precise number = Slightly less precise number<br />
<br />
Precise number + Garbage = Garbage<br />
<br />
Precise number × Garbage = Garbage<br />
<br />
√<span style="border-top:1px solid; padding:0 0.1em;">Garbage</span> = Less bad garbage<br />
<br />
1/N Σ (N pieces of statistically independent garbage) = Better garbage<br />
<br />
(Precise number)<sup>Garbage</sup> = Much worse garbage<br />
<br />
Garbage – Garbage = Much worse garbage<br />
<br />
Precise number / ( Garbage – Garbage ) = Much worse garbage, possible division by zero<br />
<br />
Garbage × 0 = Precise number<br />
<br />
{{comic discussion}}<br />
[[Category:Math]]</div>172.68.51.124https://www.explainxkcd.com/wiki/index.php?title=2295:_Garbage_Math&diff=1908742295: Garbage Math2020-04-18T05:13:38Z<p>172.68.51.124: </p>
<hr />
<div>{{comic<br />
| number = 2295<br />
| date = April 17, 2020<br />
| title = Garbage Math<br />
| image = garbage_math.png<br />
| titletext = 'Garbage In, Garbage Out' should not be taken to imply any sort of conservation law limiting the amount of garbage produced.<br />
}}<br />
<br />
==Explanation==<br />
{{incomplete|Created by a ZILOG Z80. Please mention here why this explanation isn't complete. Do NOT delete this tag too soon.}}<br />
This comic explains the "{{w|garbage in, garbage out}}" concept using arithmetical expressions. Just like the comic says, if you get garbage in any part of your workflow, you get garbage as a result.<br />
<br />
Some of these rules correspond to the rules of {{w|floating point arithmetic}}, while others may be inspired by 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 the uncertainty of the result of arithmetic operations will tend to be dominated by the term with the highest uncertainty. The rule about N pieces of independent garbage reflects the {{w|central limit theorem}} and how it predicts that the uncertainty (or {{w|standard error}}) of an estimate will be reduced when independent estimates are averaged. The comic oddly omits raising garbage to the 0th power, which transforms even NaN, the platonic ideal of garbage, to exactly 1.<br />
<br />
This comic is not related to the {{w|2019–20 coronavirus outbreak|2020 pandemic}} of the {{w|coronavirus}} {{w|SARS-CoV-2}}, which causes {{w|COVID-19}}, breaking the streak of comics preceding this on [[:Category:COVID-19|topics relating to COVID-19]], after (rather appropriately) 19 comics (not counting the [[2288: Collector's Edition|April Fools' comic]]).<br />
<br />
This comic is about the propagation of errors in numerical analysis and statistics, but described in much more colloquial terms. Numbers with low precision are termed "garbage" and numbers with high precision are labeled "precise".<br />
<br />
{| class="wikitable"<br />
!Formula<br />
!Statistical Expression<br />
!Explanation<br />
|-<br />
|Precise number + Precise number = Slightly less precise number<br />
|<math>\mathop\sigma(X+Y)=\sqrt{(\mathop\sigma(X))²+(\mathop\sigma(Y))²}</math><br />
|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<sup>-6</sup>) and 1 (±10<sup>-6</sup>), then our sum is 2 (±2·10<sup>-6</sup>). 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 and then close to its inverse. This phenomenon is known as catastrophic cancellation. Therefore, it is likely that all numbers referred here are positive numbers, which does not exhibit this phenomenon.<br />
|-<br />
|Precise number × Precise number = Slightly less precise number<br />
|<br />
|Here, instead of absolute error, relative error will be added. For example, if our precise numbers are 1 (±10<sup>-6</sup>) and 1 (±10<sup>-6</sup>), then our product is 1 (±2·10<sup>-6</sup>).<br />
|-<br />
|Precise number + Garbage = Garbage<br />
|<br />
|If one of the numbers has a high absolute error, and the numbers being added are of comparable size, then this error will be propagated to the sum. <br />
|-<br />
|Precise number × Garbage = Garbage<br />
|<br />
|Likewise, if one of the numbers has a high relative error, then this error will be propagated to the product. Here, this is independent of the sizes of the numbers.<br />
|-<br />
|<math>\sqrt{\text{Garbage}} = \text{Less bad garbage}</math><br />
|<br />
| When the square root of a number is computed, its relative error will be halved. Depending on the application, this might not be all that much ''better'', but it's at least ''less bad''.<br />
|-<br />
|Garbage<sup>2</sup> = Worse garbage<br />
|<br />
|Likewise, when a number is squared, its relative error will be doubled. This is a corollary to multiplication adding relative errors.<br />
|-<br />
|<math>\frac{1}{N}\sum(\text{N pieces of statistically independent garbage}) = \text{Better garbage}</math><br />
|<br />
|By aggregating many pieces of statistically independent observations (for instance, surveying many individuals), it is possible to reduce relative error. This is the basis of statistical sampling.<br />
|-<br />
|Precise number<sup>Garbage</sup> = Much worse garbage<br />
|<br />
|The exponent is very sensitive to changes, which may also magnify the effect based on the magnitude of the precise number.<br />
|-<br />
|Garbage – Garbage = Much worse garbage<br />
|<br />
|This line involves catastrophic cancellation. If both pieces of garbage are about the same (e.g. if their error bars overlap), then it is possible that the answer is positive, zero, or negative.<br />
|-<br />
|<math>\frac{\text{Precise number}}{\text{Garbage}-\text{Garbage}}</math> = Much worse garbage, possible division by zero<br />
|<br />
|Indeed, as with above, if error bars overlap then we might end up dividing by zero.<br />
|-<br />
|Garbage × 0 = Precise number<br />
|<br />
|Multiplying anything by 0 results in 0, an extremely precise number in the sense that it has no error whatsoever since we supply the 0 ourselves. This is equivalent to discarding garbage data from a statistical analysis.<br />
|}<br />
<br />
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 conserved.<br />
<br />
==Transcript==<br />
{{incomplete transcript|Do NOT delete this tag too soon.}}<br />
<br />
[A series of mathematical equations are written from top to bottom]<br />
<br />
Precise number + Precise number = Slightly less precise number<br />
<br />
Precise number × Precise number = Slightly less precise number<br />
<br />
Precise number + Garbage = Garbage<br />
<br />
Precise number × Garbage = Garbage<br />
<br />
√<span style="border-top:1px solid; padding:0 0.1em;">Garbage</span> = Less bad garbage<br />
<br />
1/N Σ (N pieces of statistically independent garbage) = Better garbage<br />
<br />
(Precise number)<sup>Garbage</sup> = Much worse garbage<br />
<br />
Garbage – Garbage = Much worse garbage<br />
<br />
Precise number / ( Garbage – Garbage ) = Much worse garbage, possible division by zero<br />
<br />
Garbage × 0 = Precise number<br />
<br />
{{comic discussion}}<br />
[[Category:Math]]</div>172.68.51.124https://www.explainxkcd.com/wiki/index.php?title=2295:_Garbage_Math&diff=1908732295: Garbage Math2020-04-18T05:12:06Z<p>172.68.51.124: </p>
<hr />
<div>{{comic<br />
| number = 2295<br />
| date = April 17, 2020<br />
| title = Garbage Math<br />
| image = garbage_math.png<br />
| titletext = 'Garbage In, Garbage Out' should not be taken to imply any sort of conservation law limiting the amount of garbage produced.<br />
}}<br />
<br />
==Explanation==<br />
{{incomplete|Created by a ZILOG Z80. Please mention here why this explanation isn't complete. Do NOT delete this tag too soon.}}<br />
This comic explains the "{{w|garbage in, garbage out}}" concept using arithmetical expressions. Just like the comic says, if you get garbage in any part of your workflow, you get garbage as a result.<br />
<br />
Some of these rules correspond to the rules of {{w|floating point arithmetic}}, while others may be inspired by 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 the uncertainty of the result of arithmetic operations will tend to be dominated by the term with the highest uncertainty. The rule about N pieces of independent garbage reflects the {{w|central limit theorem}} and how it predicts that the uncertainty (or {{w|standard error}}) of an estimate will be reduced when independent estimates are averaged. The comic oddly omits raising garbage to the 0th power, which transforms even NaN, the platonic ideal of garbage, to exactly 1.<br />
<br />
This comic is not related to the {{w|2019–20 coronavirus outbreak|2020 pandemic}} of the {{w|coronavirus}} {{w|SARS-CoV-2}}, which causes {{w|COVID-19}}, breaking the streak of comics preceding this on [[:Category:COVID-19|topics relating to COVID-19]], after (rather appropriately) 19 comics (not counting the [[2288: Collector's Edition|April Fools' comic]]).<br />
<br />
This comic is about the propagation of errors in numerical analysis and statistics, but described in much more colloquial terms. Numbers with low precision are termed "garbage" and numbers with high precision are labeled "precise".<br />
<br />
{| class="wikitable"<br />
!Formula<br />
!Statistical Expression<br />
!Explanation<br />
|-<br />
|Precise number + Precise number = Slightly less precise number<br />
|<br />
|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<sup>-6</sup>) and 1 (±10<sup>-6</sup>), then our sum is 2 (±2·10<sup>-6</sup>). 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 and then close to its inverse. This phenomenon is known as catastrophic cancellation. Therefore, it is likely that all numbers referred here are positive numbers, which does not exhibit this phenomenon.<br />
|-<br />
|Precise number × Precise number = Slightly less precise number<br />
|<math>\mathop\sigma(X+Y)=\sqrt{(\mathop\sigma(X))²+(\mathop\sigma(Y))²}<br />
|Here, instead of absolute error, relative error will be added. For example, if our precise numbers are 1 (±10<sup>-6</sup>) and 1 (±10<sup>-6</sup>), then our product is 1 (±2·10<sup>-6</sup>).<br />
|-<br />
|Precise number + Garbage = Garbage<br />
|<br />
|If one of the numbers has a high absolute error, and the numbers being added are of comparable size, then this error will be propagated to the sum. <br />
|-<br />
|Precise number × Garbage = Garbage<br />
|<br />
|Likewise, if one of the numbers has a high relative error, then this error will be propagated to the product. Here, this is independent of the sizes of the numbers.<br />
|-<br />
|<math>\sqrt{\text{Garbage}} = \text{Less bad garbage}</math><br />
|<br />
| When the square root of a number is computed, its relative error will be halved. Depending on the application, this might not be all that much ''better'', but it's at least ''less bad''.<br />
|-<br />
|Garbage<sup>2</sup> = Worse garbage<br />
|<br />
|Likewise, when a number is squared, its relative error will be doubled. This is a corollary to multiplication adding relative errors.<br />
|-<br />
|<math>\frac{1}{N}\sum(\text{N pieces of statistically independent garbage}) = \text{Better garbage}</math><br />
|<br />
|By aggregating many pieces of statistically independent observations (for instance, surveying many individuals), it is possible to reduce relative error. This is the basis of statistical sampling.<br />
|-<br />
|Precise number<sup>Garbage</sup> = Much worse garbage<br />
|<br />
|The exponent is very sensitive to changes, which may also magnify the effect based on the magnitude of the precise number.<br />
|-<br />
|Garbage – Garbage = Much worse garbage<br />
|<br />
|This line involves catastrophic cancellation. If both pieces of garbage are about the same (e.g. if their error bars overlap), then it is possible that the answer is positive, zero, or negative.<br />
|-<br />
|<math>\frac{\text{Precise number}}{\text{Garbage}-\text{Garbage}}</math> = Much worse garbage, possible division by zero<br />
|<br />
|Indeed, as with above, if error bars overlap then we might end up dividing by zero.<br />
|-<br />
|Garbage × 0 = Precise number<br />
|<br />
|Multiplying anything by 0 results in 0, an extremely precise number in the sense that it has no error whatsoever since we supply the 0 ourselves. This is equivalent to discarding garbage data from a statistical analysis.<br />
|}<br />
<br />
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 conserved.<br />
<br />
==Transcript==<br />
{{incomplete transcript|Do NOT delete this tag too soon.}}<br />
<br />
[A series of mathematical equations are written from top to bottom]<br />
<br />
Precise number + Precise number = Slightly less precise number<br />
<br />
Precise number × Precise number = Slightly less precise number<br />
<br />
Precise number + Garbage = Garbage<br />
<br />
Precise number × Garbage = Garbage<br />
<br />
√<span style="border-top:1px solid; padding:0 0.1em;">Garbage</span> = Less bad garbage<br />
<br />
1/N Σ (N pieces of statistically independent garbage) = Better garbage<br />
<br />
(Precise number)<sup>Garbage</sup> = Much worse garbage<br />
<br />
Garbage – Garbage = Much worse garbage<br />
<br />
Precise number / ( Garbage – Garbage ) = Much worse garbage, possible division by zero<br />
<br />
Garbage × 0 = Precise number<br />
<br />
{{comic discussion}}<br />
[[Category:Math]]</div>172.68.51.124https://www.explainxkcd.com/wiki/index.php?title=2295:_Garbage_Math&diff=1908722295: Garbage Math2020-04-18T05:04:00Z<p>172.68.51.124: </p>
<hr />
<div>{{comic<br />
| number = 2295<br />
| date = April 17, 2020<br />
| title = Garbage Math<br />
| image = garbage_math.png<br />
| titletext = 'Garbage In, Garbage Out' should not be taken to imply any sort of conservation law limiting the amount of garbage produced.<br />
}}<br />
<br />
==Explanation==<br />
{{incomplete|Created by a ZILOG Z80. Please mention here why this explanation isn't complete. Do NOT delete this tag too soon.}}<br />
This comic explains the "{{w|garbage in, garbage out}}" concept using arithmetical expressions. Just like the comic says, if you get garbage in any part of your workflow, you get garbage as a result.<br />
<br />
Some of these rules correspond to the rules of {{w|floating point arithmetic}}, while others may be inspired by 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 the uncertainty of the result of arithmetic operations will tend to be dominated by the term with the highest uncertainty. The rule about N pieces of independent garbage reflects the {{w|central limit theorem}} and how it predicts that the uncertainty (or {{w|standard error}}) of an estimate will be reduced when independent estimates are averaged. The comic oddly omits raising garbage to the 0th power, which transforms even NaN, the platonic ideal of garbage, to exactly 1.<br />
<br />
This comic is not related to the {{w|2019–20 coronavirus outbreak|2020 pandemic}} of the {{w|coronavirus}} {{w|SARS-CoV-2}}, which causes {{w|COVID-19}}, breaking the streak of comics preceding this on [[:Category:COVID-19|topics relating to COVID-19]], after (rather appropriately) 19 comics (not counting the [[2288: Collector's Edition|April Fools' comic]]).<br />
<br />
This comic is about the propagation of errors in numerical analysis and statistics, but described in much more colloquial terms. Numbers with low precision are termed "garbage" and numbers with high precision are labeled "precise".<br />
<br />
{| class="wikitable"<br />
!Formula<br />
!Statistical Expression<br />
!Explanation<br />
|-<br />
|Precise number + Precise number = Slightly less precise number<br />
|<br />
|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<sup>-6</sup>) and 1 (±10<sup>-6</sup>), then our sum is 2 (±2·10<sup>-6</sup>). 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 and then close to its inverse. This phenomenon is known as catastrophic cancellation. Therefore, it is likely that all numbers referred here are positive numbers, which does not exhibit this phenomenon.<br />
|-<br />
|Precise number × Precise number = Slightly less precise number<br />
|<br />
|Here, instead of absolute error, relative error will be added. For example, if our precise numbers are 1 (±10<sup>-6</sup>) and 1 (±10<sup>-6</sup>), then our product is 1 (±2·10<sup>-6</sup>).<br />
|-<br />
|Precise number + Garbage = Garbage<br />
|<br />
|If one of the numbers has a high absolute error, and the numbers being added are of comparable size, then this error will be propagated to the sum. <br />
|-<br />
|Precise number × Garbage = Garbage<br />
|<br />
|Likewise, if one of the numbers has a high relative error, then this error will be propagated to the product. Here, this is independent of the sizes of the numbers.<br />
|-<br />
|<math>\sqrt{\text{Garbage}} = \text{Less bad garbage}</math><br />
|<br />
| When the square root of a number is computed, its relative error will be halved. Depending on the application, this might not be all that much ''better'', but it's at least ''less bad''.<br />
|-<br />
|Garbage<sup>2</sup> = Worse garbage<br />
|<br />
|Likewise, when a number is squared, its relative error will be doubled. This is a corollary to multiplication adding relative errors.<br />
|-<br />
|<math>\frac{1}{N}\sum(\text{N pieces of statistically independent garbage}) = \text{Better garbage}</math><br />
|<br />
|By aggregating many pieces of statistically independent observations (for instance, surveying many individuals), it is possible to reduce relative error. This is the basis of statistical sampling.<br />
|-<br />
|Precise number<sup>Garbage</sup> = Much worse garbage<br />
|<br />
|The exponent is very sensitive to changes, which may also magnify the effect based on the magnitude of the precise number.<br />
|-<br />
|Garbage – Garbage = Much worse garbage<br />
|<br />
|This line involves catastrophic cancellation. If both pieces of garbage are about the same (e.g. if their error bars overlap), then it is possible that the answer is positive, zero, or negative.<br />
|-<br />
|<math>\frac{\text{Precise number}}{\text{Garbage}-\text{Garbage}}</math> = Much worse garbage, possible division by zero<br />
|<br />
|Indeed, as with above, if error bars overlap then we might end up dividing by zero.<br />
|-<br />
|Garbage × 0 = Precise number<br />
|<br />
|Multiplying anything by 0 results in 0, an extremely precise number in the sense that it has no error whatsoever since we supply the 0 ourselves. This is equivalent to discarding garbage data from a statistical analysis.<br />
|}<br />
<br />
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 conserved.<br />
<br />
==Transcript==<br />
{{incomplete transcript|Do NOT delete this tag too soon.}}<br />
<br />
[A series of mathematical equations are written from top to bottom]<br />
<br />
Precise number + Precise number = Slightly less precise number<br />
<br />
Precise number × Precise number = Slightly less precise number<br />
<br />
Precise number + Garbage = Garbage<br />
<br />
Precise number × Garbage = Garbage<br />
<br />
√<span style="border-top:1px solid; padding:0 0.1em;">Garbage</span> = Less bad garbage<br />
<br />
1/N Σ (N pieces of statistically independent garbage) = Better garbage<br />
<br />
(Precise number)<sup>Garbage</sup> = Much worse garbage<br />
<br />
Garbage – Garbage = Much worse garbage<br />
<br />
Precise number / ( Garbage – Garbage ) = Much worse garbage, possible division by zero<br />
<br />
Garbage × 0 = Precise number<br />
<br />
{{comic discussion}}<br />
[[Category:Math]]</div>172.68.51.124https://www.explainxkcd.com/wiki/index.php?title=Talk:2208:_Drone_Fishing&diff=180639Talk:2208: Drone Fishing2019-09-30T10:08:32Z<p>172.68.51.124: </p>
<hr />
<div><!--Please sign your posts with ~~~~ and don't delete this text. New comments should be added at the bottom.--><br />
<br />
So kite fishing is a thing for recreational deep sea fishing. I think some people are experimenting with using drones instead of kites. I think I've also read about using a drone to allow long "casts" when shore fishing. This seems to be Randall just mixing all that up in a fun (?) way.<br />
[[Special:Contributions/172.69.63.105|172.69.63.105]] 16:04, 27 September 2019 (UTC)<br />
<br />
What kind of bait do you use to attract a drone, anyway? Or would you use some kind of electronic lure? [[User:Barmar|Barmar]] ([[User talk:Barmar|talk]]) 20:01, 27 September 2019 (UTC)<br />
:Perhaps no bait is required and you either wait for the drone to fly into the streaming lines and foul itself; or fly the kite in such a manner as to 'snag' the drone similar to the way one snags salmon during the mating runs. [[User:RAGBRAIvet|RAGBRAIvet]] ([[User talk:RAGBRAIvet|talk]]) 17:43, 29 September 2019 (UTC)<br />
<br />
Isn't this comic a reference to How To? There was a comic in that book about fishing while suspended from drones. [[Special:Contributions/172.69.63.73|172.69.63.73]] 21:47, 27 September 2019 (UTC)<br />
<br />
Could this comic possibly be a reference to [[2148: Cubesat Launch]]?--[[User:XRENEGADEx|XRENEGADEx]] ([[User talk:XRENEGADEx|talk]]) 23:11, 27 September 2019 (UTC)<br />
:And/or the space junk removal experiments? [[Special:Contributions/172.68.143.18|172.68.143.18]] 03:23, 28 September 2019 (UTC)<br />
::Those experiments are awesome. And Randall actually predicted one (kinda) by making a line go from solid to dotted at the right time, see trivia of [[1402: Harpoons]] --[[User:Lupo|Lupo]] ([[User talk:Lupo|talk]]) 07:21, 30 September 2019 (UTC)<br />
<br />
Nobody notices drone fishing is an actual thing? [[Special:Contributions/172.69.55.22|172.69.55.22]] 01:44, 28 September 2019 (UTC)<br />
<br />
It isn't clear to me that this would be illegal, at least the "fishing" part, although caught drones would have to be returned. Unless the airspace has been reserved, kites and drones have equal access to the airspace. Perhaps the extra dangling strings could be seen as a deliberate attempt to trap drones, but any justification ("testing kite tail designs", "testing ion content in the air" etc.) could be sufficient to make these OK. Likewise, the kite owner could complain about the drones being "armed" with unjustifiably sharp propellers and such "designed" to damage the kite. Umm, are drone fights a thing (yet)?[[Special:Contributions/108.162.241.154|108.162.241.154]] 12:16, 28 September 2019 (UTC)<br />
:Mini-drone racing has been a thing for years, but IRL fights are considered gauche in the extreme. [[Special:Contributions/172.69.22.134|172.69.22.134]] 16:51, 28 September 2019 (UTC)<br />
: [http://unleashthebot.com/best-battle-drones/ Battle Drones] are a real thing. [[User:These Are Not The Comments You Are Looking For|These Are Not The Comments You Are Looking For]] ([[User talk:These Are Not The Comments You Are Looking For|talk]]) 04:13, 29 September 2019 (UTC)<br />
:: What could possibly go wrong? Thank goodness income inequality is stabilizing globally. [[Special:Contributions/172.68.189.211|172.68.189.211]] 20:54, 29 September 2019 (UTC)<br />
<br />
Alright ladies and gentlemen, I've come to [https://www.theverge.com/2017/12/12/16767000/police-netherlands-eagles-rogue-drones train eagles] and write content, and I'm [https://www.reddit.com/r/facepalm/comments/db5f2l/dutch_police_arrest_bird_for_participating_in/ all out of eagles.] [[Special:Contributions/172.69.22.248|172.69.22.248]] 06:46, 30 September 2019 (UTC)<br />
<br />
I believe 1523: Microdrones should be mentioned, as it also mentions stealing drones. [[User:Magic9mushroom|Magic9mushroom]] ([[User talk:Magic9mushroom|talk]]) 08:09, 30 September 2019 (UTC)<br />
<br />
When I was much younger I remember seeing a documantary film where people use kites and hooks to "fish" for bats or megabats or fruit bats. However I don't remember where that scene has taken place. [[Special:Contributions/172.68.51.124|172.68.51.124]] 10:08, 30 September 2019 (UTC)</div>172.68.51.124https://www.explainxkcd.com/wiki/index.php?title=Talk:2192:_Review&diff=178374Talk:2192: Review2019-08-21T13:42:19Z<p>172.68.51.124: new</p>
<hr />
<div><!--Please sign your posts with ~~~~ and don't delete this text. New comments should be added at the bottom.--><br />
Earth : Terrible storyline, feel depressed afterward. Controls buggy.<br />
<br />
Mostly harmless - [[User:GreenWyvern|GreenWyvern]] ([[User talk:GreenWyvern|talk]]) 13:39, 21 August 2019 (UTC)<br />
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New servers opening soon! [[Special:Contributions/172.68.51.124|172.68.51.124]] 13:42, 21 August 2019 (UTC)</div>172.68.51.124https://www.explainxkcd.com/wiki/index.php?title=Talk:1367:_Installing&diff=177278Talk:1367: Installing2019-07-31T10:20:33Z<p>172.68.51.124: </p>
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<div>explainxkcd has it pretty easy with this one since the comic explains a lot of itself. Maybe explain what a smartphone is and how apps work? [[Special:Contributions/108.162.237.218|108.162.237.218]] 05:28, 12 May 2014 (UTC)<br />
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For some reason, this reminded me of the old Snaptu app. (https://en.wikipedia.org/wiki/Snaptu) [[Special:Contributions/108.162.225.147|108.162.225.147]] 07:02, 12 May 2014 (UTC)<br />
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Maybe it should be mentioned that sometimes you DON'T want to auto-install every application and give it access to all your phone resources. Because, you know, malware. -- [[User:Hkmaly|Hkmaly]] ([[User talk:Hkmaly|talk]]) 09:47, 12 May 2014 (UTC)<br />
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I wonder why he chose <code>cookies</code> over <code>localStorage</code>... seems like <code>localStorage</code> does a better job of storing configs. {{User:Grep/signature|12:06, 12 May 2014}}<br />
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: <code>document.cookies</code> was invented before <code>localStorage</code>. --[[Special:Contributions/108.162.246.4|108.162.246.4]] 22:19, 12 May 2014 (UTC)<br />
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Firefox OS can technically do this, and technically does this. {{User:Grep/signature|12:07, 12 May 2014}}<br />
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Web pages and native apps still has a few essential differences that prevent us to interchange them practically, at least for now. The latter can be compiled and optimized into binaries that executes performantly on the specific device/platform. Current web standards don't make pages/sites/apps this way, the web browser needs to load the text codes then interpret and run them on the fly, which is much slower. [[Special:Contributions/199.27.128.79|199.27.128.79]] 08:33, 16 May 2014 (UTC)<br />
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:Native apps on PCs? Sure. But on phones? Apps on phones rarely contains any native code and in fact often ARE written in web-compatible languages. I mean in java or javascript. Also, in many situations, combination of extremely optimized Java virtual machine and poorly optimized native code results in interpreted code running FASTER that compiled one. Not speaking about fact that not many applications NEEDS to run so fast - they spend most time waiting for disk, net, user input or screen refresh anyway. -- [[User:Hkmaly|Hkmaly]] ([[User talk:Hkmaly|talk]]) 10:22, 16 May 2014 (UTC)<br />
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::On windows/Mac most programs are distributed as binaries - already compiled to support a wide variety of platforms, very rarely do they contain control-paths for specific hardware implementations. Java on the other hand: It gets compiled at runtime and can get hardware specific optimisations (and if the JRE detects critical sections it will dedicate more time and resources on optimizing that part making it even faster). Javascript also can get compiled, depending on the webpage this can also be done on the server-side as to make it harder to manipulate the js. 10:20, 31 July 2019 (UTC)<br />
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The phrase "[...] a phone that has every app "installed" [...]" from Cueball's dialogue seems to conflict with the explanation. I understood it as the phone would have all the apps installed, but with only the "header" data. In the Android context, I suppose that would be the AndroidManifest.xml. In the Windows context, I suppose that would be the registry entries. [[Special:Contributions/188.114.99.189|188.114.99.189]] 00:30, 11 November 2015 (UTC)<br />
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== progressive web apps ==<br />
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seem similar to this</div>172.68.51.124