339:
checking, and to find faulty assumptions if the figure produced is far beyond what we might reasonably expect. By contrast, precise calculations can be extremely complex but with the expectation that the answer they produce is correct. The far larger number of factors and operations involved can obscure a very significant error, either in mathematical process or in the assumptions the equation is based on, but the result may still be assumed to be right because it has been derived from a precise formula that is expected to yield good results. Without a reasonable frame of reference to work from it is seldom clear if a result is acceptably precise or is many degrees of magnitude (tens or hundreds of times) too big or too small. The Fermi estimation gives a quick, simple way to obtain this frame of reference for what might reasonably be expected to be the answer.
343:
tuners in
Chicago. If their initial estimate told them there should be a hundred or so, but the precise answer tells them there are many thousands, then they know they need to find out why there is this divergence from the expected result. First looking for errors, then for factors the estimation did not take account of – does Chicago have a number of music schools or other places with a disproportionately high ratio of pianos to people? Whether close or very far from the observed results, the context the estimation provides gives useful information both about the process of calculation and the assumptions that have been used to look at problems.
359:
typical day, or look up an accurate number for the population of
Chicago. It also gives a rough estimate that may be good enough for some purposes: if a person wants to start a store in Chicago that sells piano tuning equipment, and calculates that they need 10,000 potential customers to stay in business, they can reasonably assume that the above estimate is far enough below 10,000 that they should consider a different business plan (and, with a little more work, they could compute a rough upper bound on the number of piano tuners by considering the most extreme
1472:
278:
43:
371:
Fermi estimates generally work because the estimations of the individual terms are often close to correct, and overestimates and underestimates help cancel each other out. That is, if there is no consistent bias, a Fermi calculation that involves the multiplication of several estimated factors (such
358:
Although Fermi calculations are often not accurate, as there may be many problems with their assumptions, this sort of analysis does inform one what to look for to get a better answer. For the above example, one might try to find a better estimate of the number of pianos tuned by a piano tuner in a
342:
As long as the initial assumptions in the estimate are reasonable quantities, the result obtained will give an answer within the same scale as the correct result, and if not gives a base for understanding why this is the case. For example, suppose a person was asked to determine the number of piano
338:
Scientists often look for Fermi estimates of the answer to a problem before turning to more sophisticated methods to calculate a precise answer. This provides a useful check on the results. While the estimate is almost certainly incorrect, it is also a simple calculation that allows for easy error
346:
Fermi estimates are also useful in approaching problems where the optimal choice of calculation method depends on the expected size of the answer. For instance, a Fermi estimate might indicate whether the internal stresses of a structure are low enough that it can be accurately described by
484:
For instance, if one makes a 9-step Fermi estimate, at each step overestimating or underestimating the correct number by a factor of 2 (or with a standard deviation 2), then after 9 steps the standard error will have grown by a logarithmic factor of
1017:
There are or have been a number of university-level courses devoted to estimation and the solution of Fermi problems. The materials for these courses are a good source for additional Fermi problem examples and material about solution strategies:
260:, which seeks to estimate the number of intelligent civilizations in the galaxy. The basic question of why, if there were a significant number of such civilizations, human civilization has never encountered any others is called the
529:, and much less than the worst case of erring by a factor of 2 = 512 (about 2.71 orders of magnitude). If one has a shorter chain or estimates more accurately, the overall estimate will be correspondingly better.
242:"If the mass of one teaspoon of water could be converted entirely into energy in the form of heat, what volume of water, initially at room temperature, could it bring to a boil? (litres)."
195:
as he was known for his ability to make good approximate calculations with little or no actual data. Fermi problems typically involve making justified guesses about quantities and their
455:
513:
479:
409:
996:
An example of a Fermi
Problem relating to total gasoline consumed by cars since the invention of cars and comparison to the output of the energy released by the sun.
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1198:
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relative to some other value, for example, if a structure will be over-engineered to withstand loads several times greater than the estimate.
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1030:
905:
890:
842:
819:
1238:
1060:
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1149:
1080:
1050:
952:
937:
867:
796:
1293:
1268:
1218:
1119:
922:
882:
834:
811:
788:
773:
758:
727:
662:
325:
126:
1329:
1006:
188:
850:
1273:
1228:
1213:
235:
Fermi questions are often extreme in nature, and cannot usually be solved using common mathematical or scientific information.
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107:
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1112:
1288:
1258:
1208:
223:, based on the distance traveled by pieces of paper he dropped from his hand during the blast. Fermi's estimate of 10
288:
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913:
307:
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53:
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619:
93:
1040:
596:
1415:
558:
75:
1506:
1070:
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416:
372:
as the number of piano tuners in
Chicago) will probably be more accurate than might be first supposed.
975:
30:
This article is about the estimation technique. For Fermi's question about extraterrestrial life, see
1430:
1334:
1003:
by Nuño
Sempere, which has a proof sketch of why Fermi-style decompositions produce better estimates.
430:
units on the log scale from the actual value, then the overall estimate will have standard deviation
200:
180:
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375:
In detail, multiplying estimates corresponds to adding their logarithms; thus one obtains a sort of
1098:
488:
1348:
1135:
526:
460:
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390:
199:
or lower and upper bounds. In some cases, order-of-magnitude estimates can also be derived using
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792:
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723:
668:
658:
384:
348:
168:
1056:
650:
631:
252:"What is the mass of all the automobiles scrapped in North America this month? (kilograms)."
851:
Street-Fighting
Mathematics: The Art of Educated Guessing and Opportunistic Problem Solving
100:
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1036:
1278:
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989:
635:
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216:
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1370:
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352:
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184:
31:
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415:). In discrete terms, the number of overestimates minus underestimates will have a
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192:
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was well within an order of magnitude of the now-accepted value of 21 kilotons.
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17:
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548:
945:
Maths on the Back of an
Envelope: Clever ways to (roughly) calculate anything
672:
1363:
1358:
804:
Guesstimation: Solving the World's
Problems on the Back of a Cocktail Napkin
766:
Consider a
Cylindrical Cow: More Adventures in Environmental Problem Solving
745:
The following books contain many examples of Fermi problems with solutions:
553:
196:
1353:
172:
1104:
976:
Fermi
Questions: A Guide for Teachers, Students, and Event Supervisors
1046:
986:
What if? Serious Scientific Answers to Absurd Hypothetical Questions
1023:
6.055J / 2.038J The Art of Approximation in Science and Engineering
898:
Guesstimation 2.0: Solving Today's Problems on the Back of a Napkin
751:
Consider a Spherical Cow: A Course in Environmental Problem Solving
1108:
981:
351:; or if the estimate already bears significant relationship in
187:
of extreme scientific calculations. Fermi problems are usually
271:
36:
1026:
720:
The Great Silence: Science and Philosophy of Fermi's Paradox
238:
Example questions given by the official Fermi Competition:
245:"How much does the Thames River heat up in going over the
1007:"How should mathematics be taught to non-mathematicians?"
1022:
419:. In continuous terms, if one makes a Fermi estimate of
363:
values that could appear in each of their assumptions).
827:
How Many Licks?: Or, How to Estimate Damn Near Anything
930:
Physics on your feet. Berkeley Graduate Exam Questions
27:
Estimation problem in physics or engineering education
491:
463:
436:
393:
191:. The estimation technique is named after physicist
1389:
1322:
1142:
622:(September 1988). "How Fermi Would Have Fixed It".
67:. Unsourced material may be challenged and removed.
507:
473:
457:, since the standard deviation of a sum scales as
449:
403:
1087:Chapter 2: Discoveries on the Back of an Envelope
1091:Frontiers of Science: Scientific Habits of Mind
256:Possibly the most famous Fermi Question is the
240:
515:, so 2 = 8. Thus one will expect to be within
1120:
914:The Art of Insight in Science and Engineering
600:(2). Los Alamos National Laboratory: 45. 2005
8:
589:"A Backward Glance: Eyewitnesses to Trinity"
306:. Unsourced material may be challenged and
1127:
1113:
1105:
875:Quantify! A Crash Course in Smart Thinking
525:to 8 times the correct value – within an
492:
490:
464:
462:
440:
435:
394:
392:
326:Learn how and when to remove this message
127:Learn how and when to remove this message
708:Fermi Questions. Richard K Curtis. 2001.
877:Johns Hopkins University Press. 2010.
802:Lawrence Weinstein & John A. Adam,
783:Johns Hopkins University Press. 2003.
580:
1031:Massachusetts Institute of Technology
928:Dmitry Budker, Alexander O. Sushkov,
657:. Dover Publications. pp. 3–12.
655:The Fermi Solution: Essays on Science
7:
1061:University of California, Santa Cruz
969:Physics Education Group maintains a
304:adding citations to reliable sources
65:adding citations to reliable sources
900:Princeton University Press. 2012.
806:Princeton University Press. 2008.
649:Von Baeyer, Hans Christian (2001).
215:'s estimate of the strength of the
1081:University of California, Berkeley
1067:Order of Magnitude Problem Solving
1051:California Institute of Technology
1037:Physics on the Back of an Envelope
718:Ćirković, Milan M. (10 May 2018).
636:10.1002/j.2326-1951.1988.tb03037.x
450:{\displaystyle \sigma {\sqrt {n}}}
25:
1001:"Introduction to Fermi estimates"
768:University Science Books. 2001.
753:University Science Books. 1988.
189:back-of-the-envelope calculations
1471:
1470:
1402:Earth's location in the Universe
1330:Back-of-the-envelope calculation
1059:taught by Patrick Chuang at the
1039:taught by Lawrence Weinstein at
822:. A textbook on Fermi problems.
276:
41:
1335:Best-selling electronic devices
1079:taught by Eugene Chiang at the
1069:taught by Linda Strubbe at the
1049:taught by Sterl Phinney at the
932:Oxford University Press. 2015.
52:needs additional citations for
1:
1057:Order of Magnitude Estimation
971:collection of Fermi problems.
508:{\displaystyle {\sqrt {9}}=3}
179:education, designed to teach
1397:Astronomical system of units
781:Back-of-the-Envelope Physics
722:. Oxford University Press.
481:in the number of summands.
474:{\displaystyle {\sqrt {n}}}
404:{\displaystyle {\sqrt {n}}}
161:order-of-magnitude estimate
1523:
1077:Order of Magnitude Physics
1047:Order of Magnitude Physics
982:"What if? Paint the Earth"
620:Von Baeyer, Hans Christian
157:order-of-magnitude problem
29:
1466:
1448:The Scale of the Universe
690:. Old Dominion University
686:Weinstein, L.B. (2012).
1041:Old Dominion University
829:. Running Press. 2009.
597:Nuclear Weapons Journal
1416:To the Moon and Beyond
1284:Specific heat capacity
967:University of Maryland
559:Order of approximation
509:
475:
451:
405:
254:
219:that detonated at the
1434:(1968 and 1977 films)
1071:University of Toronto
947:HarperCollins. 2019.
510:
476:
452:
417:binomial distribution
406:
249:? (Celsius degrees)."
207:Historical background
1502:Dimensional analysis
896:Lawrence Weinstein,
651:"The Fermi Solution"
489:
461:
434:
411:(in number of terms
391:
387:, which diffuses as
300:improve this section
268:Advantages and scope
201:dimensional analysis
181:dimensional analysis
61:improve this article
1136:Orders of magnitude
1099:Columbia University
155:), also known as a
1442:(1996 documentary)
1371:Metric (SI) prefix
854:MIT Press. 2010.
527:order of magnitude
505:
471:
447:
425:standard deviation
401:
1497:Physics education
1484:
1483:
1381:Microscopic scale
1376:Macroscopic scale
917:MIT Press. 2014.
906:978-0-691-15080-2
891:978-0-8018-9717-7
843:978-0-7624-3560-9
820:978-1-4008-2444-1
779:Clifford Swartz,
688:"Fermi Questions"
497:
469:
445:
399:
385:logarithmic scale
349:linear elasticity
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328:
137:
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129:
111:
16:(Redirected from
1514:
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1155:Angular momentum
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978:by Lloyd Abrams.
911:Sanjoy Mahajan,
873:Göran Grimvall,
848:Sanjoy Mahajan,
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76:"Fermi problem"
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1027:Sanjoy Mahajan
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1011:Timothy Gowers
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990:Randall Munroe
984:from the book
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960:External links
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953:978-0008324582
943:Rob Eastaway,
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938:978-0199681655
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868:978-0262514293
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825:Aaron Santos,
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797:978-0801872631
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72:Find sources:
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50:This article
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33:
32:Fermi paradox
19:
1507:Enrico Fermi
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1345:Powers of 10
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1197: /
1150:Acceleration
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764:John Harte,
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749:John Harte,
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692:. Retrieved
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1427:(1968 film)
1424:Cosmic Zoom
1419:(1964 film)
1411:(1957 book)
1408:Cosmic View
1294:Temperature
1269:Probability
1219:Illuminance
539:Guesstimate
381:random walk
367:Explanation
316:August 2023
217:atomic bomb
177:engineering
171:problem in
1491:Categories
1456:Cosmic Eye
1093:taught by
1025:taught by
860:026251429X
694:27 October
630:(5): 2–4.
604:27 October
575:References
549:Handwaving
361:reasonable
169:estimation
145:Fermi quiz
87:newspapers
1364:1000000th
1274:Radiation
1229:Luminance
1214:Frequency
1175:Computing
673:775985788
554:Heuristic
438:σ
287:does not
167:), is an
117:July 2015
1476:Category
1323:See also
1264:Pressure
1249:Molarity
1165:Bit rate
1143:Quantity
533:See also
231:Examples
197:variance
1390:Related
1349:decades
1309:Voltage
1254:Numbers
1204:Entropy
1190:Density
1180:Current
1029:at the
520:⁄
383:on the
308:removed
293:sources
173:physics
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108:JSTOR
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