4884:
420:
1270:
31:
4872:
Libertz invites us to engage critically not only with statistical information such as averages, but also with the language used to describe the data and its uses, saying: "If statistics rely on interpretation, rhetors should invite their audience to interpret rather than insist on an interpretation." In many cases, data and specific calculations are provided to help facilitate this audience-based interpretation.
5151:. The majority view is that the meaning of "an expense" was an expansion from "damage and damage expense", and the chronological order of the meanings in the records supports this view, and the broad meaning "an expense" was never the most commonly used meaning. On the basis of the above points, the inferential step is made that the Latinate word came or probably came from the Arabic word.
1053:
4750:
moon. Using the mean of several measured values, scientists assumed that the errors add up to a relatively small number when compared to the total of all measured values. The method of taking the mean for reducing observation errors was indeed mainly developed in astronomy. A possible precursor to the arithmetic mean is the
3256:
3003:
1718:
1421:
4642:
A type of average used in finance is the average percentage return. It is an example of a geometric mean. When the returns are annual, it is called the
Compound Annual Growth Rate (CAGR). For example, if we are considering a period of two years, and the investment return in the first year is −10% and
4866:
Due to the aforementioned colloquial nature of the term "average", the term can be used to obfuscate the true meaning of data and suggest varying answers to questions based on the averaging method (most frequently arithmetic mean, median, or mode) used. In his article "Framed for Lying: Statistics
4767:
Even older potential references exist. There are records that from about 700 BC, merchants and shippers agreed that damage to the cargo and ship (their "contribution" in case of damage by the sea) should be shared equally among themselves. This might have been calculated using the average, although
4749:
was in the sixteenth century. From the late sixteenth century onwards, it gradually became a common method to use for reducing errors of measurement in various areas. At the time, astronomers wanted to know a real value from noisy measurement, such as the position of a planet or the diameter of the
4667:
This method can be generalized to examples in which the periods are not equal. For example, consider a period of a half of a year for which the return is −23% and a period of two and a half years for which the return is +13%. The average percentage return for the combined period is the single year
446:
The most frequently occurring number in a list is called the mode. For example, the mode of the list (1, 2, 2, 3, 3, 3, 4) is 3. It may happen that there are two or more numbers which occur equally often and more often than any other number. In this case there is no agreed definition of mode. Some
464:
Thus to find the median, order the list according to its elements' magnitude and then repeatedly remove the pair consisting of the highest and lowest values until either one or two values are left. If exactly one value is left, it is the median; if two values, the median is the arithmetic mean of
4816:
still have the primary meaning of "damage". The huge transformation of the meaning in
English began with the practice in later medieval and early modern Western merchant-marine law contracts under which if the ship met a bad storm and some of the goods had to be thrown overboard to make the ship
4762:
In the first place, we must set out in a row the sequence of numbers from the monad up to nine: 1, 2, 3, 4, 5, 6, 7, 8, 9. Then we must add up the amount of all of them together, and since the row contains nine terms, we must look for the ninth part of the total to see if it is already naturally
4757:
However, there are various older vague references to the use of the arithmetic mean (which are not as clear, but might reasonably have to do with our modern definition of the mean). In a text from the 4th century, it was written that (text in square brackets is a possible missing text that might
4871:
faculty member Daniel
Libertz comments that statistical information is frequently dismissed from rhetorical arguments for this reason. However, due to their persuasive power, averages and other statistical values should not be discarded completely, but instead used and interpreted with caution.
4821:. From there the word was adopted by British insurers, creditors, and merchants for talking about their losses as being spread across their whole portfolio of assets and having a mean proportion. Today's meaning developed out of that, and started in the mid-18th century, and started in English.
4663:
that makes this equation true is 0.2, or 20%. This means that the total return over the 2-year period is the same as if there had been 20% growth each year. The order of the years makes no difference – the average percentage returns of +60% and −10% is the same result as that for −10% and +60%.
1265:{\displaystyle {\underset {{\vec {x}}\in \mathbb {R} ^{d}}{\operatorname {argmax} }}\,{\underset {{\vec {u}}\in \mathbb {R} ^{d}}{\operatorname {min} }}\,\sum _{i=1}^{n}\left({\begin{cases}1,{\text{ if }}({\vec {x}}_{i}-{\vec {x}})\cdot {\vec {u}}\geq 0\\0,{\text{ otherwise}}\end{cases}}\right)}
3717:
1932:
4010:
3483:
2551:
2835:
1005:
2662:
2390:
3094:
2841:
1544:
3816:
2169:
1286:
5328:
Eisenhart, Churchill. "The development of the concept of the best mean of a set of measurements from antiquity to the present day." Unpublished presidential address, American
Statistical Association, 131st Annual Meeting, Fort Collins, Colorado.
723:
465:
these two. This method takes the list 1, 7, 3, 13 and orders it to read 1, 3, 7, 13. Then the 1 and 13 are removed to obtain the list 3, 7. Since there are two elements in this remaining list, the median is their arithmetic mean, (3 + 7)/2 = 5.
4727:
the term "moving average" is used even when the sum of the weights is not 1.0 (so the output series is a scaled version of the averages). The reason for this is that the analyst is usually interested only in the trend or the periodic behavior.
4160:
819:
634:
2070:
1815:
1538:
50: – the sum of the numbers divided by how many numbers are in the list. For example, the mean average of the numbers 2, 3, 4, 7, and 9 (summing to 25) is 5. Depending on the context, the most representative
4832:, where the owner can claim a proportional contribution from all the parties to the marine venture. The type of calculations used in adjusting general average gave rise to the use of "average" to mean "arithmetic mean".
3500:
2280:
4714:
values, then moving forward one place by dropping the oldest value and introducing a new value at the other end of the list, and so on. This is the simplest form of moving average. More complicated forms involve using a
4842:
There is earlier (from at least the 11th century), unrelated use of the word. It appears to be an old legal term for a tenant's day labour obligation to a sheriff, probably anglicised from "avera" found in the
English
1821:
327:
4817:
lighter and safer, then all merchants whose goods were on the ship were to suffer proportionately (and not whoever's goods were thrown overboard); and more generally there was to be proportionate distribution of any
3354:
3088:
3914:
3360:
2405:
5122:
to be found in Latin, and (#4) a substantial number of Arabic words entered
Italian, Catalan and Provençal in the 12th and 13th centuries starting as terms of Mediterranean sea-commerce, and (#5) the Arabic
2675:
874:
4702:, such as daily stock market prices or yearly temperatures, people often want to create a smoother series. This helps to show underlying trends or perhaps periodic behavior. An easy way to do this is the
4338:
3908:
152:-insensitivity: all items count equally in determining their average value and their positions in the list are irrelevant; the average of (1, 2, 3, 4, 6) is the same as that of (3, 2, 6, 4, 1).
2557:
5078:. See also the definition of English "average" in English dictionaries published in the early 18th century, i.e., in the time period just before the big transformation of the meaning:
4835:
A second
English usage, documented as early as 1674 and sometimes spelled "averish", is as the residue and second growth of field crops, which were considered suited to consumption by
2286:
4997:
3251:{\displaystyle {\underset {x\in \mathbb {R} _{\geq 0}}{\operatorname {argmin} }}\,\sum _{i=1}^{n}(x^{p}-x_{i}^{p})^{2},\qquad {\text{if }}x_{i}\geq 0\,\forall \,i\in \{1,\dots ,n\}}
2998:{\displaystyle {\underset {x\in \mathbb {R} _{\geq 0}}{\operatorname {argmin} }}\,\sum _{i=1}^{n}(x^{3}-x_{i}^{3})^{2},\qquad {\text{if }}x_{i}\geq 0\,\forall \,i\in \{1,\dots ,n\}}
1713:{\displaystyle {\underset {x\in \mathbb {R} _{>0}}{\operatorname {argmin} }}\,\sum _{i=1}^{n}(\ln(x)-\ln(x_{i}))^{2},\qquad {\text{if }}x_{i}>0\,\forall \,i\in \{1,\dots ,n\}}
1416:{\displaystyle {\underset {x\in \mathbb {R} }{\operatorname {argmax} }}\,\sum _{i=1}^{n}\left({\begin{cases}1,{\text{ if }}x=x_{i}\\0,{\text{ if }}x\neq x_{i}\end{cases}}\right)}
3723:
2076:
86:. For this reason, it is recommended to avoid using the word "average" when discussing measures of central tendency and specify which type of measure of average is being used.
1043:
868:
4053:
4465:. This most general definition still captures the important property of all averages that the average of a list of identical elements is that element itself. The function
640:
78:
is often given as the median – the number below which are 50% of personal incomes and above which are 50% of personal incomes – because the
4080:
739:
518:
4792:, a state of partial damage". Within the Western languages the word's history begins in medieval sea-commerce on the Mediterranean. 12th and 13th century Genoa Latin
1945:
5135:
is a suffix in
Italian, and the Western word's earliest records are in Italian-speaking locales (writing in Latin). And most commentators agree that (#6) the Arabic
1731:
4719:. The weighting can be used to enhance or suppress various periodic behavior and there is very extensive analysis of what weightings to use in the literature on
5354:
4073:
94:
If all numbers in a list are the same number, then their average is also equal to this number. This property is shared by each of the many types of average.
4374:
However, this method for generating means is not general enough to capture all averages. A more general method for defining an average takes any function
461:
The median is the middle number of the group when they are ranked in order. (If there are an even number of numbers, the mean of the middle two is taken.)
1434:
4754:(the mean of the two extreme values), used for example in Arabian astronomy of the ninth to eleventh centuries, but also in metallurgy and navigation.
3712:{\displaystyle {\frac {\sum _{i=1}^{n}w_{i}x_{i}}{\sum _{i=1}^{n}w_{i}}}={\frac {w_{1}x_{1}+w_{2}x_{2}+\cdots +w_{n}x_{n}}{w_{1}+w_{2}+\cdots +w_{n}}}}
46:
is a single number or value that best represents a set of data. The type of average taken as most typically representative of a list of numbers is the
734:
A middle value that separates the higher half from the lower half of the data set; may not be unique if the data set contains an even number of points
4188:
2182:
1927:{\displaystyle {\underset {x\in \mathbb {R} _{\neq 0}}{\operatorname {argmin} }}\,\sum _{i=1}^{n}\left({\frac {1}{x}}-{\frac {1}{x_{i}}}\right)^{2}}
5355:"Waterfield, Robin. "The theology of arithmetic." On the Mystical, mathematical and Cosmological Symbolism of the First Ten Number (1988). page 70"
257:
4021:
Similar to the truncated mean, but, rather than deleting the extreme values, they are set equal to the largest and smallest values that remain
5143:= "damage or damage expenses". A minority of commentators have been dubious about this on the grounds that the early records of Italian-Latin
3269:
4005:{\displaystyle {\underset {x\in \mathbb {R} }{\operatorname {argmin} }}\,{\underset {i\in \{1,\dots ,n\}}{\operatorname {max} }}\,|x-x_{i}|}
3478:{\displaystyle {\underset {x\in \operatorname {dom} (f)}{\operatorname {argmin} }}\,\sum _{i=1}^{n}(f(x)-f(x_{i}))^{2},\qquad {\text{if }}f}
3016:
2546:{\displaystyle {\sqrt {{\frac {1}{n}}\sum _{i=1}^{n}x_{i}^{2}}}={\sqrt {{\frac {1}{n}}\left(x_{1}^{2}+x_{2}^{2}+\cdots +x_{n}^{2}\right)}}}
5343:
Bakker, Arthur. "The early history of average values and implications for education." Journal of
Statistics Education 11.1 (2003): 17-26.
5036:. The seaport of Genoa is the location of the earliest-known record in European languages, year 1157. A set of medieval Latin records of
4184:
2830:{\displaystyle {\sqrt{{\frac {1}{n}}\sum _{i=1}^{n}x_{i}^{3}}}={\sqrt{{\frac {1}{n}}\left(x_{1}^{3}+x_{2}^{3}+\cdots +x_{n}^{3}\right)}}}
1000:{\displaystyle {\underset {{\vec {x}}\in \mathbb {R} ^{d}}{\operatorname {argmin} }}\,\sum _{i=1}^{n}||{\vec {x}}-{\vec {x}}_{i}||_{2}}
5172:
Merigo, Jose M.; Cananovas, Montserrat (2009). "The
Generalized Hybrid Averaging Operator and its Application in Decision Making".
132:
In some types of average, the items in the list are assigned different weights before the average is determined. These include the
3827:
The arithmetic mean of data values after a certain number or proportion of the highest and lowest data values have been discarded
5062:. At the port of Marseille in the 1st half of the 13th century notarized commercial contracts have dozens of instances of Latin
4804:
had the same meaning, and it begot English "averay" (1491) and English "average" (1502) with the same meaning. Today, Italian
4208:
3861:
4763:
present among the numbers in the row; and we will find that the property of being ninth only belongs to the mean itself...
5079:
5290:
Plackett, R. L. (1958). "Studies in the History of Probability and Statistics: VII. The Principle of the Arithmetic Mean".
4796:
meant "damage, loss and non-normal expenses arising in connection with a merchant sea voyage"; and the same meaning for
4055:
of objects with minimal sum of dissimilarities to all the objects in the set, according to some dissimilarity function
2657:{\displaystyle {\underset {x\in \mathbb {R} _{\geq 0}}{\operatorname {argmin} }}\,\sum _{i=1}^{n}(x^{2}-x_{i}^{2})^{2}}
5361:
492:
223:
129:: if all numbers of a list are multiplied by the same positive number, then its average changes by the same factor.
5029:
5118:
started in the 12th century and it started as a term of Mediterranean sea-commerce, and (#3) there is no root for
2385:{\displaystyle {\underset {x\in \mathbb {R} }{\operatorname {argmin} }}\,\sum _{i=1}^{n}x_{i}^{p-1}(x-x_{i})^{2}}
5271:
5246:
5544:
4168:
5416:
5208:
3811:{\displaystyle {\underset {x\in \mathbb {R} }{\operatorname {argmin} }}\,\sum _{i=1}^{n}w_{i}(x-x_{i})^{2}}
2164:{\displaystyle {\underset {x\in \mathbb {R} }{\operatorname {argmin} }}\,\sum _{i=1}^{n}x_{i}(x-x_{i})^{2}}
5103:
5091:
5087:
5022:. Summary information about the word's early records in Italian-Latin, Italian, Catalan, and French is at
5017:
5554:
5099:
5083:
4637:
4897:
1019:
844:
718:{\displaystyle {\underset {x\in \mathbb {R} }{\operatorname {argmin} }}\,\sum _{i=1}^{n}(x-x_{i})^{2}}
5447:
5016:
was first reported by Reinhart Dozy in the 19th century. Dozy's original summary is in his 1869 book
4953:
4724:
148:, the weight of an item depends on its position in the list. Most types of average, however, satisfy
17:
4417:
is then the value that, when replacing each member of the list, results in the same function value:
4034:
1341:
1161:
4868:
4155:{\displaystyle {\underset {y\in {\mathcal {X}}}{\operatorname {argmin} }}\sum _{i=1}^{n}d(y,x_{i})}
3493:
133:
814:{\displaystyle {\underset {x\in \mathbb {R} }{\operatorname {argmin} }}\,\sum _{i=1}^{n}|x-x_{i}|}
5452:
4780:, a defect, or anything defective or damaged, including partially spoiled merchandise; and عواري
834:
629:{\displaystyle {\bar {x}}={\frac {1}{n}}\sum _{i=1}^{n}x_{i}={\frac {1}{n}}(x_{1}+\cdots +x_{n})}
436:
137:
2065:{\displaystyle {\frac {x_{1}^{2}+x_{2}^{2}+\cdots +x_{n}^{2}}{{x_{1}}+{x_{2}}+\cdots +{x_{n}}}}}
5106:. Today there is consensus that: (#1) today's English "average" descends from medieval Italian
4949:
meant "any defect, or anything defective or damaged". Some medieval Arabic dictionaries are at
831:
219:
5095:
4800:
is in Marseille in 1210, Barcelona in 1258 and Florence in the late 13th. 15th-century French
5468:
4902:
3262:
1810:{\displaystyle {\frac {n}{{\frac {1}{x_{1}}}+{\frac {1}{x_{2}}}+\cdots +{\frac {1}{x_{n}}}}}}
5342:
5327:
3841:. A special case of the inter-quantile truncated mean, which operates on quantiles (often
126:
8:
5486:
5075:
4572:(where the list elements are positive numbers) provides the geometric mean. The function
4402:
3838:
1938:
5529:
5307:
5041:
4889:
4195:
4058:
3832:
3486:
4960:, and some translation to English of what's in the medieval Arabic dictionaries is in
5508:
5267:
5242:
5181:
4883:
1276:
432:
414:
363:
203:
161:
67:
5204:"Axiomatisations of the average and a further generalisation of monotonic sequences"
1533:{\displaystyle {\sqrt{\prod _{i=1}^{n}x_{i}}}={\sqrt{x_{1}\cdot x_{2}\dotsb x_{n}}}}
5498:
5299:
5217:
5071:
4716:
4643:
the return in the second year is +60%, then the average percentage return or CAGR,
3009:
825:
55:
5033:
4957:
4917:
4912:
4829:
4742:
4016:
511:
249:
173:
141:
47:
479:
The mid-range is the arithmetic mean of the highest and lowest values of a set.
197:
5549:
5530:
Calculations and comparison between arithmetic and geometric mean of two values
5390:
5050:
in Volume 1 pages 115-116. Many more records in medieval Latin at Genoa are at
4996:
or damage – this can be seen in the searchable collection of medieval texts at
4907:
4858:
loss, damage, have been "quite disposed of" and the word has a Romance origin.
4836:
4720:
4693:
3822:
2396:
1427:
177:
145:
71:
5222:
5203:
2275:{\displaystyle {\frac {\sum _{i=1}^{n}x_{i}^{p}}{\sum _{i=1}^{n}x_{i}^{p-1}}}}
419:
5538:
5512:
5185:
4844:
1724:
181:
167:
5385:
5098:. Some complexities surrounding the English word's history are discussed in
4628:(where the list elements are positive numbers) provides the harmonic mean.
4406:
1011:
98:
4850:
The Oxford English Dictionary, however, says that derivations from German
4347:
is any invertible function. The harmonic mean is an example of this using
322:{\displaystyle \scriptstyle {\bar {x}}={\frac {1}{n}}\sum _{i=1}^{n}x_{i}}
5174:
Journal of Quantitative Methods for Economics and Business Administration
5148:
5147:
have, in some cases, a meaning of "an expense" in a more general sense –
4922:
4699:
4410:
2175:
1046:
149:
83:
5503:
5023:
3349:{\displaystyle f^{-1}\left({\frac {1}{n}}\sum _{k=1}^{n}f(x_{k})\right)}
5441:
5311:
4976:
can be naturally formed in Arabic grammar to refer to things that have
4746:
3846:
2668:
488:
51:
39:
4751:
3083:{\displaystyle {\sqrt{{\frac {1}{n}}\cdot \sum _{i=1}^{n}x_{i}^{p}}}}
474:
389:
222:. These can all be seen as minimizing variation by some measure; see
211:
59:
5303:
4710:
and creates a new series by taking the arithmetic mean of the first
346:
Middle value separating the greater and lesser halves of a data set
5412:
5074:. Some information about the English word over the centuries is at
3854:
440:
5139:= "damage | relating to damage" is semantically a good match for
4180:
1016:
Another rotation invariant extension of the median for points in
5131:, as conversion of w to v was regular in Latin and Italian, and
4961:
230:
Comparison of common averages of values { 1, 2, 2, 3, 4, 7, 9 }
30:
5051:
4828:, which is borne only by the owner of the damaged property, or
4026:
3842:
838:
729:
456:
428:
341:
207:
75:
63:
447:
authors say they are all modes and some say there is no mode.
394:
The arithmetic mean of the highest and lowest values of a set
4968:. The medieval dictionaries do not list the word-form عوارية
3849:) that are equidistant but on opposite sides of the median.
1405:
1254:
424:
215:
79:
4950:
493:
Central tendency § Solutions to variational problems
254:
Sum of values of a data set divided by number of values:
224:
Central tendency § Solutions to variational problems
82:
would be higher by including personal incomes from a few
4333:{\displaystyle y=f^{-1}\left({\frac {1}{n}}\left\right)}
3903:{\displaystyle {\frac {1}{2}}\left(\max x+\min x\right)}
113:
is at least as large as the corresponding entry on list
4992:
are frequently used when referring to things that have
54:
to be taken as the average might be another measure of
5439:
4768:
there seem to be no direct record of the calculation.
261:
4649:(1 − 10%) × (1 + 60%) = (1 − 0.1) × (1 + 0.6) = (1 +
4631:
4211:
4083:
4061:
4037:
3917:
3864:
3726:
3503:
3363:
3272:
3097:
3019:
2844:
2678:
2560:
2408:
2289:
2185:
2079:
1948:
1824:
1734:
1547:
1437:
1289:
1056:
1022:
877:
847:
742:
643:
521:
260:
4879:
5487:"Framed for Lying: Statistics as In/Artistic Proof"
4984:is a rarity or non-existent, while the forms عواري
27:
Number taken as representative of a list of numbers
5440:
4672:, that is the solution of the following equation:
4332:
4194:One can create one's own average metric using the
4154:
4067:
4047:
4004:
3902:
3810:
3711:
3477:
3348:
3250:
3082:
2997:
2829:
2656:
2545:
2384:
2274:
2163:
2064:
1926:
1809:
1712:
1532:
1415:
1264:
1037:
999:
862:
813:
717:
628:
321:
4409:in each argument, and symmetric (invariant under
5536:
5418:A Collection of English Words not Generally Used
4861:
3889:
3880:
3837:A special case of the truncated mean, using the
5480:
5478:
5171:
5338:
5336:
4745:was extended from 2 to n cases for the use of
5239:Time Series Analysis: Forecasting and Control
5237:Box, George E.P.; Jenkins, Gwilym M. (1976).
5000:(book links are clickable on righthand side).
109:have the same length, and each entry of list
5475:
5323:
5321:
5285:
5283:
3969:
3951:
3245:
3227:
2992:
2974:
1707:
1689:
5333:
4980:, but in practice in medieval Arabic texts
4647:, can be obtained by solving the equation:
4519:provides the arithmetic mean. The function
4359:, and the geometric mean is another, using
5236:
5502:
5318:
5280:
5221:
5008:
5006:
3975:
3938:
3931:
3747:
3740:
3394:
3220:
3216:
3128:
3112:
2967:
2963:
2875:
2859:
2591:
2575:
2310:
2303:
2100:
2093:
1855:
1839:
1682:
1678:
1578:
1562:
1310:
1303:
1130:
1117:
1093:
1080:
1025:
914:
901:
850:
763:
756:
664:
657:
5289:
5040:at Genoa is in the downloadable lexicon
418:
29:
5484:
5197:
5195:
4179:Other more sophisticated averages are:
1281:The most frequent value in the data set
191:
14:
5537:
5261:
5003:
4174:
5201:
5114:, and (#2) among the Latins the word
89:
5469:participating institution membership
5192:
5100:Hensleigh Wedgwood year 1882 page 11
4945:meaning "blind in one eye" and عوار
4776:The root is found in Arabic as عوار
506:As solution to optimization problem
218:as estimates of central tendency in
155:
5411:
482:
24:
5084:Blount's dictionary (1707 edition)
5080:Kersey-Phillips' dictionary (1706)
4632:Average percentage return and CAGR
4098:
4040:
3217:
2964:
1679:
368:Most frequent value in a data set
214:are often used in addition to the
25:
5566:
5523:
5127:is phonetically a good match for
4741:The first recorded time that the
4687:
4401:) of a list of arguments that is
4171:explains the symbols used below.
4031:A representative object of a set
5241:(revised ed.). Holden-Day.
5046:, by Sergio Aprosio, year 2001,
4882:
1038:{\displaystyle \mathbb {R} ^{d}}
863:{\displaystyle \mathbb {R} ^{d}}
198:Mean § Statistical location
5433:
5405:
5378:
5104:Walter Skeat year 1888 page 781
4674:(1 − 0.23) × (1 + 0.13) = (1 +
4413:of the arguments). The average
3466:
3194:
2941:
1656:
5485:Libertz, Daniel (2018-12-31).
5347:
5255:
5230:
5165:
4935:
4317:
4304:
4289:
4276:
4267:
4254:
4191:, with their generalizations.
4149:
4130:
4048:{\displaystyle {\mathcal {X}}}
3998:
3977:
3799:
3779:
3454:
3450:
3437:
3428:
3422:
3416:
3388:
3382:
3338:
3325:
3182:
3150:
2929:
2897:
2645:
2613:
2373:
2353:
2152:
2132:
1644:
1640:
1627:
1615:
1609:
1600:
1224:
1212:
1206:
1185:
1175:
1106:
1069:
987:
981:
968:
952:
942:
937:
890:
807:
786:
706:
686:
623:
591:
528:
268:
184:are known collectively as the
97:Another universal property is
13:
1:
5158:
4862:Averages as a rhetorical tool
4169:table of mathematical symbols
168:Mean § Pythagorean means
125:. Also, all averages satisfy
5209:Glasgow Mathematical Journal
4771:
1045:—a point that maximizes the
468:
399:
396:
388:
381:
370:
362:
355:
348:
340:
333:
330:
248:
7:
4875:
4680:, giving an average return
4638:Compound annual growth rate
117:, then the average of list
74:. For example, the average
10:
5571:
5096:Martin's dictionary (1749)
5092:Bailey's dictionary (1726)
5088:Hatton's dictionary (1712)
4898:Average absolute deviation
4731:
4691:
4635:
486:
472:
454:
412:
195:
165:
159:
144:. Also, for some types of
101:: if two lists of numbers
5448:Oxford English Dictionary
5391:Dictionary.com Unabridged
5223:10.1017/s0017089500002135
5054:, usually in the plurals
4736:
4725:digital signal processing
450:
121:is at least that of list
4941:Medieval Arabic had عور
4928:
4869:University of Pittsburgh
4824:Marine damage is either
4706:: one chooses a number
437:log-normal distributions
134:weighted arithmetic mean
5453:Oxford University Press
4867:as In/Artistic Proof",
4788:) = "of or relating to
503:Equation or description
489:Mean § Other means
408:
138:weighted geometric mean
5264:Adaptive Filter Theory
5262:Haykin, Simon (1986).
4964:Arabic-English Lexicon
4758:clarify the meaning):
4334:
4156:
4126:
4069:
4049:
4006:
3904:
3812:
3768:
3713:
3570:
3527:
3479:
3415:
3350:
3321:
3252:
3149:
3084:
3056:
2999:
2896:
2831:
2712:
2658:
2612:
2547:
2441:
2386:
2331:
2276:
2247:
2209:
2165:
2121:
2066:
1928:
1876:
1811:
1714:
1599:
1534:
1461:
1417:
1331:
1266:
1151:
1039:
1001:
935:
864:
815:
784:
719:
685:
630:
567:
443:
323:
307:
220:descriptive statistics
35:
5149:see TLIO (in Italian)
5052:StoriaPatriaGenova.it
5012:The Arabic origin of
4966:, pages 2193 and 2195
4903:Central limit theorem
4335:
4157:
4106:
4070:
4050:
4007:
3905:
3813:
3748:
3714:
3550:
3507:
3480:
3395:
3351:
3301:
3263:Quasi-arithmetic mean
3253:
3129:
3085:
3036:
3000:
2876:
2832:
2692:
2659:
2592:
2548:
2421:
2387:
2311:
2277:
2227:
2189:
2166:
2101:
2067:
1929:
1856:
1812:
1715:
1579:
1535:
1441:
1418:
1311:
1267:
1131:
1040:
1002:
915:
865:
816:
764:
720:
665:
631:
547:
422:
324:
287:
33:
5545:Arithmetic functions
5202:Bibby, John (1974).
5066:(ablative plural of
4684:of 0.0600 or 6.00%.
4209:
4081:
4059:
4035:
3915:
3862:
3724:
3501:
3361:
3270:
3095:
3017:
2842:
2676:
2558:
2406:
2287:
2183:
2077:
1946:
1822:
1732:
1545:
1435:
1287:
1054:
1020:
875:
845:
740:
641:
519:
331:(1+2+2+3+4+7+9) / 7
258:
192:Statistical location
5504:10.29107/rr2018.4.1
5451:(Online ed.).
5421:. London: H. Bruges
5070:), as published in
4407:strictly increasing
4175:Miscellaneous types
3839:interquartile range
3180:
3071:
2927:
2813:
2789:
2771:
2727:
2643:
2535:
2511:
2493:
2456:
2352:
2268:
2224:
2008:
1984:
1966:
1939:Contraharmonic mean
231:
5555:Summary statistics
5072:Blancard year 1884
5043:Vocabolario Ligure
5032:2019-01-06 at the
4956:2013-10-29 at the
4890:Mathematics portal
4854:haven, and Arabic
4826:particular average
4330:
4152:
4104:
4065:
4045:
4002:
3973:
3936:
3900:
3833:Interquartile mean
3808:
3745:
3709:
3475:
3392:
3346:
3248:
3166:
3126:
3080:
3057:
2995:
2913:
2873:
2827:
2799:
2775:
2757:
2713:
2654:
2629:
2589:
2543:
2521:
2497:
2479:
2442:
2382:
2332:
2308:
2272:
2248:
2210:
2161:
2098:
2062:
1994:
1970:
1952:
1924:
1853:
1807:
1710:
1576:
1530:
1413:
1404:
1308:
1262:
1253:
1128:
1091:
1035:
997:
912:
860:
811:
761:
715:
662:
626:
444:
319:
318:
229:
127:linear homogeneity
90:General properties
36:
5467:(Subscription or
5266:. Prentice-Hall.
4392:, ...,
4244:
4165:
4164:
4085:
4068:{\displaystyle d}
3940:
3919:
3873:
3728:
3707:
3582:
3470:
3365:
3299:
3198:
3099:
3078:
3031:
2945:
2846:
2825:
2750:
2734:
2690:
2562:
2541:
2472:
2457:
2419:
2291:
2270:
2081:
2060:
1911:
1891:
1826:
1805:
1802:
1776:
1756:
1660:
1549:
1528:
1478:
1384:
1353:
1291:
1249:
1227:
1209:
1188:
1173:
1109:
1095:
1072:
1058:
971:
955:
893:
879:
837:extension of the
744:
645:
589:
545:
531:
415:Mode (statistics)
406:
405:
285:
271:
186:Pythagorean means
162:Pythagorean means
156:Pythagorean means
34:Average of chords
16:(Redirected from
5562:
5517:
5516:
5506:
5482:
5473:
5472:
5464:
5462:
5460:
5455:. September 2019
5444:
5437:
5431:
5430:
5428:
5426:
5409:
5403:
5402:
5400:
5399:
5382:
5376:
5375:
5373:
5372:
5366:
5360:. Archived from
5359:
5351:
5345:
5340:
5331:
5325:
5316:
5315:
5298:(1/2): 130–135.
5287:
5278:
5277:
5259:
5253:
5252:
5234:
5228:
5227:
5225:
5199:
5190:
5189:
5169:
5152:
5010:
5001:
4939:
4892:
4887:
4886:
4717:weighted average
4679:
4659:. The value of
4658:
4627:
4601:
4571:
4548:
4518:
4494:
4464:
4435:
4339:
4337:
4336:
4331:
4329:
4325:
4324:
4320:
4316:
4315:
4288:
4287:
4266:
4265:
4245:
4237:
4230:
4229:
4161:
4159:
4158:
4153:
4148:
4147:
4125:
4120:
4105:
4103:
4102:
4101:
4074:
4072:
4071:
4066:
4054:
4052:
4051:
4046:
4044:
4043:
4011:
4009:
4008:
4003:
4001:
3996:
3995:
3980:
3974:
3972:
3937:
3935:
3934:
3909:
3907:
3906:
3901:
3899:
3895:
3874:
3866:
3817:
3815:
3814:
3809:
3807:
3806:
3797:
3796:
3778:
3777:
3767:
3762:
3746:
3744:
3743:
3718:
3716:
3715:
3710:
3708:
3706:
3705:
3704:
3686:
3685:
3673:
3672:
3662:
3661:
3660:
3651:
3650:
3632:
3631:
3622:
3621:
3609:
3608:
3599:
3598:
3588:
3583:
3581:
3580:
3579:
3569:
3564:
3548:
3547:
3546:
3537:
3536:
3526:
3521:
3505:
3484:
3482:
3481:
3476:
3471:
3468:
3462:
3461:
3449:
3448:
3414:
3409:
3393:
3391:
3355:
3353:
3352:
3347:
3345:
3341:
3337:
3336:
3320:
3315:
3300:
3292:
3285:
3284:
3257:
3255:
3254:
3249:
3209:
3208:
3199:
3196:
3190:
3189:
3179:
3174:
3162:
3161:
3148:
3143:
3127:
3125:
3124:
3123:
3115:
3089:
3087:
3086:
3081:
3079:
3077:
3072:
3070:
3065:
3055:
3050:
3032:
3024:
3021:
3010:Generalized mean
3004:
3002:
3001:
2996:
2956:
2955:
2946:
2943:
2937:
2936:
2926:
2921:
2909:
2908:
2895:
2890:
2874:
2872:
2871:
2870:
2862:
2836:
2834:
2833:
2828:
2826:
2824:
2819:
2818:
2814:
2812:
2807:
2788:
2783:
2770:
2765:
2751:
2743:
2740:
2735:
2733:
2728:
2726:
2721:
2711:
2706:
2691:
2683:
2680:
2663:
2661:
2660:
2655:
2653:
2652:
2642:
2637:
2625:
2624:
2611:
2606:
2590:
2588:
2587:
2586:
2578:
2552:
2550:
2549:
2544:
2542:
2540:
2536:
2534:
2529:
2510:
2505:
2492:
2487:
2473:
2465:
2463:
2458:
2455:
2450:
2440:
2435:
2420:
2412:
2410:
2391:
2389:
2388:
2383:
2381:
2380:
2371:
2370:
2351:
2340:
2330:
2325:
2309:
2307:
2306:
2281:
2279:
2278:
2273:
2271:
2269:
2267:
2256:
2246:
2241:
2225:
2223:
2218:
2208:
2203:
2187:
2170:
2168:
2167:
2162:
2160:
2159:
2150:
2149:
2131:
2130:
2120:
2115:
2099:
2097:
2096:
2071:
2069:
2068:
2063:
2061:
2059:
2058:
2057:
2056:
2037:
2036:
2035:
2022:
2021:
2020:
2009:
2007:
2002:
1983:
1978:
1965:
1960:
1950:
1933:
1931:
1930:
1925:
1923:
1922:
1917:
1913:
1912:
1910:
1909:
1897:
1892:
1884:
1875:
1870:
1854:
1852:
1851:
1850:
1842:
1816:
1814:
1813:
1808:
1806:
1804:
1803:
1801:
1800:
1788:
1777:
1775:
1774:
1762:
1757:
1755:
1754:
1742:
1736:
1719:
1717:
1716:
1711:
1671:
1670:
1661:
1658:
1652:
1651:
1639:
1638:
1598:
1593:
1577:
1575:
1574:
1573:
1565:
1539:
1537:
1536:
1531:
1529:
1527:
1522:
1521:
1520:
1508:
1507:
1495:
1494:
1484:
1479:
1477:
1472:
1471:
1470:
1460:
1455:
1439:
1422:
1420:
1419:
1414:
1412:
1408:
1407:
1401:
1400:
1385:
1382:
1370:
1369:
1354:
1351:
1330:
1325:
1309:
1307:
1306:
1271:
1269:
1268:
1263:
1261:
1257:
1256:
1250:
1247:
1229:
1228:
1220:
1211:
1210:
1202:
1196:
1195:
1190:
1189:
1181:
1174:
1171:
1150:
1145:
1129:
1127:
1126:
1125:
1120:
1111:
1110:
1102:
1092:
1090:
1089:
1088:
1083:
1074:
1073:
1065:
1044:
1042:
1041:
1036:
1034:
1033:
1028:
1006:
1004:
1003:
998:
996:
995:
990:
984:
979:
978:
973:
972:
964:
957:
956:
948:
945:
940:
934:
929:
913:
911:
910:
909:
904:
895:
894:
886:
869:
867:
866:
861:
859:
858:
853:
826:Geometric median
820:
818:
817:
812:
810:
805:
804:
789:
783:
778:
762:
760:
759:
724:
722:
721:
716:
714:
713:
704:
703:
684:
679:
663:
661:
660:
635:
633:
632:
627:
622:
621:
603:
602:
590:
582:
577:
576:
566:
561:
546:
538:
533:
532:
524:
497:
496:
483:Summary of types
328:
326:
325:
320:
317:
316:
306:
301:
286:
278:
273:
272:
264:
232:
228:
56:central tendency
21:
5570:
5569:
5565:
5564:
5563:
5561:
5560:
5559:
5535:
5534:
5526:
5521:
5520:
5483:
5476:
5466:
5458:
5456:
5438:
5434:
5424:
5422:
5410:
5406:
5397:
5395:
5384:
5383:
5379:
5370:
5368:
5364:
5357:
5353:
5352:
5348:
5341:
5334:
5326:
5319:
5304:10.2307/2333051
5288:
5281:
5274:
5260:
5256:
5249:
5235:
5231:
5200:
5193:
5170:
5166:
5161:
5156:
5155:
5076:NED (year 1888)
5034:Wayback Machine
5011:
5004:
4958:Wayback Machine
4940:
4936:
4931:
4918:Population mean
4913:Law of averages
4888:
4881:
4878:
4864:
4837:draught animals
4830:general average
4774:
4743:arithmetic mean
4739:
4734:
4696:
4690:
4673:
4648:
4640:
4634:
4625:
4616:
4609:
4602:
4599:
4590:
4583:
4573:
4570:
4561:
4555:
4549:
4546:
4537:
4530:
4520:
4517:
4508:
4501:
4495:
4492:
4483:
4476:
4466:
4462:
4453:
4446:
4436:
4418:
4400:
4391:
4384:
4311:
4307:
4283:
4279:
4261:
4257:
4250:
4246:
4236:
4235:
4231:
4222:
4218:
4210:
4207:
4206:
4189:normalized mean
4177:
4143:
4139:
4121:
4110:
4097:
4096:
4089:
4084:
4082:
4079:
4078:
4060:
4057:
4056:
4039:
4038:
4036:
4033:
4032:
4017:Winsorized mean
3997:
3991:
3987:
3976:
3944:
3939:
3930:
3923:
3918:
3916:
3913:
3912:
3879:
3875:
3865:
3863:
3860:
3859:
3802:
3798:
3792:
3788:
3773:
3769:
3763:
3752:
3739:
3732:
3727:
3725:
3722:
3721:
3700:
3696:
3681:
3677:
3668:
3664:
3663:
3656:
3652:
3646:
3642:
3627:
3623:
3617:
3613:
3604:
3600:
3594:
3590:
3589:
3587:
3575:
3571:
3565:
3554:
3549:
3542:
3538:
3532:
3528:
3522:
3511:
3506:
3504:
3502:
3499:
3498:
3467:
3457:
3453:
3444:
3440:
3410:
3399:
3369:
3364:
3362:
3359:
3358:
3332:
3328:
3316:
3305:
3291:
3290:
3286:
3277:
3273:
3271:
3268:
3267:
3204:
3200:
3195:
3185:
3181:
3175:
3170:
3157:
3153:
3144:
3133:
3116:
3111:
3110:
3103:
3098:
3096:
3093:
3092:
3073:
3066:
3061:
3051:
3040:
3023:
3022:
3020:
3018:
3015:
3014:
2951:
2947:
2942:
2932:
2928:
2922:
2917:
2904:
2900:
2891:
2880:
2863:
2858:
2857:
2850:
2845:
2843:
2840:
2839:
2820:
2808:
2803:
2784:
2779:
2766:
2761:
2756:
2752:
2742:
2741:
2739:
2729:
2722:
2717:
2707:
2696:
2682:
2681:
2679:
2677:
2674:
2673:
2648:
2644:
2638:
2633:
2620:
2616:
2607:
2596:
2579:
2574:
2573:
2566:
2561:
2559:
2556:
2555:
2530:
2525:
2506:
2501:
2488:
2483:
2478:
2474:
2464:
2462:
2451:
2446:
2436:
2425:
2411:
2409:
2407:
2404:
2403:
2399:
2376:
2372:
2366:
2362:
2341:
2336:
2326:
2315:
2302:
2295:
2290:
2288:
2285:
2284:
2257:
2252:
2242:
2231:
2226:
2219:
2214:
2204:
2193:
2188:
2186:
2184:
2181:
2180:
2155:
2151:
2145:
2141:
2126:
2122:
2116:
2105:
2092:
2085:
2080:
2078:
2075:
2074:
2052:
2048:
2047:
2031:
2027:
2026:
2016:
2012:
2011:
2010:
2003:
1998:
1979:
1974:
1961:
1956:
1951:
1949:
1947:
1944:
1943:
1918:
1905:
1901:
1896:
1883:
1882:
1878:
1877:
1871:
1860:
1843:
1838:
1837:
1830:
1825:
1823:
1820:
1819:
1796:
1792:
1787:
1770:
1766:
1761:
1750:
1746:
1741:
1740:
1735:
1733:
1730:
1729:
1666:
1662:
1657:
1647:
1643:
1634:
1630:
1594:
1583:
1566:
1561:
1560:
1553:
1548:
1546:
1543:
1542:
1523:
1516:
1512:
1503:
1499:
1490:
1486:
1485:
1483:
1473:
1466:
1462:
1456:
1445:
1440:
1438:
1436:
1433:
1432:
1403:
1402:
1396:
1392:
1381:
1372:
1371:
1365:
1361:
1350:
1337:
1336:
1332:
1326:
1315:
1302:
1295:
1290:
1288:
1285:
1284:
1252:
1251:
1248: otherwise
1246:
1237:
1236:
1219:
1218:
1201:
1200:
1191:
1180:
1179:
1178:
1170:
1157:
1156:
1152:
1146:
1135:
1121:
1116:
1115:
1101:
1100:
1099:
1094:
1084:
1079:
1078:
1064:
1063:
1062:
1057:
1055:
1052:
1051:
1029:
1024:
1023:
1021:
1018:
1017:
991:
986:
985:
980:
974:
963:
962:
961:
947:
946:
941:
936:
930:
919:
905:
900:
899:
885:
884:
883:
878:
876:
873:
872:
854:
849:
848:
846:
843:
842:
806:
800:
796:
785:
779:
768:
755:
748:
743:
741:
738:
737:
709:
705:
699:
695:
680:
669:
656:
649:
644:
642:
639:
638:
617:
613:
598:
594:
581:
572:
568:
562:
551:
537:
523:
522:
520:
517:
516:
512:Arithmetic mean
495:
485:
477:
471:
459:
453:
439:with different
425:arithmetic mean
417:
411:
312:
308:
302:
291:
277:
263:
262:
259:
256:
255:
250:Arithmetic mean
200:
194:
174:arithmetic mean
170:
164:
158:
142:weighted median
92:
76:personal income
48:arithmetic mean
28:
23:
22:
15:
12:
11:
5:
5568:
5558:
5557:
5552:
5547:
5533:
5532:
5525:
5524:External links
5522:
5519:
5518:
5474:
5442:"average, n.2"
5432:
5404:
5377:
5346:
5332:
5317:
5279:
5272:
5254:
5247:
5229:
5191:
5163:
5162:
5160:
5157:
5154:
5153:
5137:ʿawār | ʿawārī
5125:ʿawār | ʿawārī
5002:
4933:
4932:
4930:
4927:
4926:
4925:
4920:
4915:
4910:
4908:Expected value
4905:
4900:
4894:
4893:
4877:
4874:
4863:
4860:
4773:
4770:
4765:
4764:
4738:
4735:
4733:
4730:
4704:moving average
4694:Moving average
4692:Main article:
4689:
4688:Moving average
4686:
4636:Main article:
4633:
4630:
4621:
4614:
4607:
4595:
4588:
4581:
4566:
4559:
4553:
4542:
4535:
4528:
4513:
4506:
4499:
4488:
4481:
4474:
4458:
4451:
4444:
4396:
4389:
4382:
4341:
4340:
4328:
4323:
4319:
4314:
4310:
4306:
4303:
4300:
4297:
4294:
4291:
4286:
4282:
4278:
4275:
4272:
4269:
4264:
4260:
4256:
4253:
4249:
4243:
4240:
4234:
4228:
4225:
4221:
4217:
4214:
4176:
4173:
4163:
4162:
4151:
4146:
4142:
4138:
4135:
4132:
4129:
4124:
4119:
4116:
4113:
4109:
4100:
4095:
4092:
4088:
4076:
4064:
4042:
4029:
4023:
4022:
4019:
4013:
4012:
4000:
3994:
3990:
3986:
3983:
3979:
3971:
3968:
3965:
3962:
3959:
3956:
3953:
3950:
3947:
3943:
3933:
3929:
3926:
3922:
3910:
3898:
3894:
3891:
3888:
3885:
3882:
3878:
3872:
3869:
3857:
3851:
3850:
3835:
3829:
3828:
3825:
3823:Truncated mean
3819:
3818:
3805:
3801:
3795:
3791:
3787:
3784:
3781:
3776:
3772:
3766:
3761:
3758:
3755:
3751:
3742:
3738:
3735:
3731:
3719:
3703:
3699:
3695:
3692:
3689:
3684:
3680:
3676:
3671:
3667:
3659:
3655:
3649:
3645:
3641:
3638:
3635:
3630:
3626:
3620:
3616:
3612:
3607:
3603:
3597:
3593:
3586:
3578:
3574:
3568:
3563:
3560:
3557:
3553:
3545:
3541:
3535:
3531:
3525:
3520:
3517:
3514:
3510:
3496:
3490:
3489:
3474:
3465:
3460:
3456:
3452:
3447:
3443:
3439:
3436:
3433:
3430:
3427:
3424:
3421:
3418:
3413:
3408:
3405:
3402:
3398:
3390:
3387:
3384:
3381:
3378:
3375:
3372:
3368:
3356:
3344:
3340:
3335:
3331:
3327:
3324:
3319:
3314:
3311:
3308:
3304:
3298:
3295:
3289:
3283:
3280:
3276:
3265:
3259:
3258:
3247:
3244:
3241:
3238:
3235:
3232:
3229:
3226:
3223:
3219:
3215:
3212:
3207:
3203:
3193:
3188:
3184:
3178:
3173:
3169:
3165:
3160:
3156:
3152:
3147:
3142:
3139:
3136:
3132:
3122:
3119:
3114:
3109:
3106:
3102:
3090:
3076:
3069:
3064:
3060:
3054:
3049:
3046:
3043:
3039:
3035:
3030:
3027:
3012:
3006:
3005:
2994:
2991:
2988:
2985:
2982:
2979:
2976:
2973:
2970:
2966:
2962:
2959:
2954:
2950:
2940:
2935:
2931:
2925:
2920:
2916:
2912:
2907:
2903:
2899:
2894:
2889:
2886:
2883:
2879:
2869:
2866:
2861:
2856:
2853:
2849:
2837:
2823:
2817:
2811:
2806:
2802:
2798:
2795:
2792:
2787:
2782:
2778:
2774:
2769:
2764:
2760:
2755:
2749:
2746:
2738:
2732:
2725:
2720:
2716:
2710:
2705:
2702:
2699:
2695:
2689:
2686:
2671:
2665:
2664:
2651:
2647:
2641:
2636:
2632:
2628:
2623:
2619:
2615:
2610:
2605:
2602:
2599:
2595:
2585:
2582:
2577:
2572:
2569:
2565:
2553:
2539:
2533:
2528:
2524:
2520:
2517:
2514:
2509:
2504:
2500:
2496:
2491:
2486:
2482:
2477:
2471:
2468:
2461:
2454:
2449:
2445:
2439:
2434:
2431:
2428:
2424:
2418:
2415:
2401:
2397:Quadratic mean
2393:
2392:
2379:
2375:
2369:
2365:
2361:
2358:
2355:
2350:
2347:
2344:
2339:
2335:
2329:
2324:
2321:
2318:
2314:
2305:
2301:
2298:
2294:
2282:
2266:
2263:
2260:
2255:
2251:
2245:
2240:
2237:
2234:
2230:
2222:
2217:
2213:
2207:
2202:
2199:
2196:
2192:
2178:
2172:
2171:
2158:
2154:
2148:
2144:
2140:
2137:
2134:
2129:
2125:
2119:
2114:
2111:
2108:
2104:
2095:
2091:
2088:
2084:
2072:
2055:
2051:
2046:
2043:
2040:
2034:
2030:
2025:
2019:
2015:
2006:
2001:
1997:
1993:
1990:
1987:
1982:
1977:
1973:
1969:
1964:
1959:
1955:
1941:
1935:
1934:
1921:
1916:
1908:
1904:
1900:
1895:
1890:
1887:
1881:
1874:
1869:
1866:
1863:
1859:
1849:
1846:
1841:
1836:
1833:
1829:
1817:
1799:
1795:
1791:
1786:
1783:
1780:
1773:
1769:
1765:
1760:
1753:
1749:
1745:
1739:
1727:
1721:
1720:
1709:
1706:
1703:
1700:
1697:
1694:
1691:
1688:
1685:
1681:
1677:
1674:
1669:
1665:
1655:
1650:
1646:
1642:
1637:
1633:
1629:
1626:
1623:
1620:
1617:
1614:
1611:
1608:
1605:
1602:
1597:
1592:
1589:
1586:
1582:
1572:
1569:
1564:
1559:
1556:
1552:
1540:
1526:
1519:
1515:
1511:
1506:
1502:
1498:
1493:
1489:
1482:
1476:
1469:
1465:
1459:
1454:
1451:
1448:
1444:
1430:
1428:Geometric mean
1424:
1423:
1411:
1406:
1399:
1395:
1391:
1388:
1383: if
1380:
1377:
1374:
1373:
1368:
1364:
1360:
1357:
1352: if
1349:
1346:
1343:
1342:
1340:
1335:
1329:
1324:
1321:
1318:
1314:
1305:
1301:
1298:
1294:
1282:
1279:
1273:
1272:
1260:
1255:
1245:
1242:
1239:
1238:
1235:
1232:
1226:
1223:
1217:
1214:
1208:
1205:
1199:
1194:
1187:
1184:
1177:
1172: if
1169:
1166:
1163:
1162:
1160:
1155:
1149:
1144:
1141:
1138:
1134:
1124:
1119:
1114:
1108:
1105:
1098:
1087:
1082:
1077:
1071:
1068:
1061:
1049:
1032:
1027:
1014:
1008:
1007:
994:
989:
983:
977:
970:
967:
960:
954:
951:
944:
939:
933:
928:
925:
922:
918:
908:
903:
898:
892:
889:
882:
870:
857:
852:
841:for points in
828:
822:
821:
809:
803:
799:
795:
792:
788:
782:
777:
774:
771:
767:
758:
754:
751:
747:
735:
732:
726:
725:
712:
708:
702:
698:
694:
691:
688:
683:
678:
675:
672:
668:
659:
655:
652:
648:
636:
625:
620:
616:
612:
609:
606:
601:
597:
593:
588:
585:
580:
575:
571:
565:
560:
557:
554:
550:
544:
541:
536:
530:
527:
514:
508:
507:
504:
501:
484:
481:
473:Main article:
470:
467:
455:Main article:
452:
449:
423:Comparison of
413:Main article:
410:
407:
404:
403:
398:
395:
392:
386:
385:
380:
369:
366:
360:
359:
354:
347:
344:
338:
337:
332:
329:
315:
311:
305:
300:
297:
294:
290:
284:
281:
276:
270:
267:
252:
246:
245:
242:
239:
236:
193:
190:
178:geometric mean
160:Main article:
157:
154:
146:moving average
91:
88:
72:geometric mean
58:, such as the
26:
9:
6:
4:
3:
2:
5567:
5556:
5553:
5551:
5548:
5546:
5543:
5542:
5540:
5531:
5528:
5527:
5514:
5510:
5505:
5500:
5496:
5492:
5491:Res Rhetorica
5488:
5481:
5479:
5470:
5454:
5450:
5449:
5443:
5436:
5420:
5419:
5414:
5408:
5394:(Online). n.d
5393:
5392:
5387:
5381:
5367:on 2016-03-04
5363:
5356:
5350:
5344:
5339:
5337:
5330:
5324:
5322:
5313:
5309:
5305:
5301:
5297:
5293:
5286:
5284:
5275:
5269:
5265:
5258:
5250:
5244:
5240:
5233:
5224:
5219:
5215:
5211:
5210:
5205:
5198:
5196:
5187:
5183:
5179:
5175:
5168:
5164:
5150:
5146:
5142:
5138:
5134:
5130:
5126:
5121:
5117:
5113:
5109:
5105:
5101:
5097:
5093:
5089:
5085:
5081:
5077:
5073:
5069:
5065:
5061:
5057:
5053:
5049:
5045:
5044:
5039:
5035:
5031:
5028:
5026:
5021:
5020:
5015:
5009:
5007:
4999:
4995:
4991:
4987:
4983:
4979:
4975:
4971:
4967:
4965:
4959:
4955:
4952:
4948:
4944:
4938:
4934:
4924:
4921:
4919:
4916:
4914:
4911:
4909:
4906:
4904:
4901:
4899:
4896:
4895:
4891:
4885:
4880:
4873:
4870:
4859:
4857:
4853:
4848:
4846:
4845:Domesday Book
4840:
4838:
4833:
4831:
4827:
4822:
4820:
4815:
4811:
4807:
4803:
4799:
4795:
4791:
4787:
4783:
4779:
4769:
4761:
4760:
4759:
4755:
4753:
4748:
4744:
4729:
4726:
4722:
4718:
4713:
4709:
4705:
4701:
4695:
4685:
4683:
4677:
4671:
4665:
4662:
4656:
4652:
4646:
4639:
4629:
4624:
4620:
4613:
4606:
4598:
4594:
4587:
4580:
4576:
4569:
4565:
4558:
4552:
4545:
4541:
4534:
4527:
4523:
4516:
4512:
4505:
4498:
4491:
4487:
4480:
4473:
4469:
4461:
4457:
4450:
4443:
4439:
4433:
4429:
4425:
4421:
4416:
4412:
4408:
4404:
4399:
4395:
4388:
4381:
4377:
4372:
4370:
4367:) = log
4366:
4362:
4358:
4354:
4350:
4346:
4326:
4321:
4312:
4308:
4301:
4298:
4295:
4292:
4284:
4280:
4273:
4270:
4262:
4258:
4251:
4247:
4241:
4238:
4232:
4226:
4223:
4219:
4215:
4212:
4205:
4204:
4203:
4201:
4199:
4192:
4190:
4186:
4182:
4172:
4170:
4144:
4140:
4136:
4133:
4127:
4122:
4117:
4114:
4111:
4107:
4093:
4090:
4086:
4077:
4062:
4030:
4028:
4025:
4024:
4020:
4018:
4015:
4014:
3992:
3988:
3984:
3981:
3966:
3963:
3960:
3957:
3954:
3948:
3945:
3941:
3927:
3924:
3920:
3911:
3896:
3892:
3886:
3883:
3876:
3870:
3867:
3858:
3856:
3853:
3852:
3848:
3844:
3840:
3836:
3834:
3831:
3830:
3826:
3824:
3821:
3820:
3803:
3793:
3789:
3785:
3782:
3774:
3770:
3764:
3759:
3756:
3753:
3749:
3736:
3733:
3729:
3720:
3701:
3697:
3693:
3690:
3687:
3682:
3678:
3674:
3669:
3665:
3657:
3653:
3647:
3643:
3639:
3636:
3633:
3628:
3624:
3618:
3614:
3610:
3605:
3601:
3595:
3591:
3584:
3576:
3572:
3566:
3561:
3558:
3555:
3551:
3543:
3539:
3533:
3529:
3523:
3518:
3515:
3512:
3508:
3497:
3495:
3494:Weighted mean
3492:
3491:
3488:
3472:
3463:
3458:
3445:
3441:
3434:
3431:
3425:
3419:
3411:
3406:
3403:
3400:
3396:
3385:
3379:
3376:
3373:
3370:
3366:
3357:
3342:
3333:
3329:
3322:
3317:
3312:
3309:
3306:
3302:
3296:
3293:
3287:
3281:
3278:
3274:
3266:
3264:
3261:
3260:
3242:
3239:
3236:
3233:
3230:
3224:
3221:
3213:
3210:
3205:
3201:
3191:
3186:
3176:
3171:
3167:
3163:
3158:
3154:
3145:
3140:
3137:
3134:
3130:
3120:
3117:
3107:
3104:
3100:
3091:
3074:
3067:
3062:
3058:
3052:
3047:
3044:
3041:
3037:
3033:
3028:
3025:
3013:
3011:
3008:
3007:
2989:
2986:
2983:
2980:
2977:
2971:
2968:
2960:
2957:
2952:
2948:
2938:
2933:
2923:
2918:
2914:
2910:
2905:
2901:
2892:
2887:
2884:
2881:
2877:
2867:
2864:
2854:
2851:
2847:
2838:
2821:
2815:
2809:
2804:
2800:
2796:
2793:
2790:
2785:
2780:
2776:
2772:
2767:
2762:
2758:
2753:
2747:
2744:
2736:
2730:
2723:
2718:
2714:
2708:
2703:
2700:
2697:
2693:
2687:
2684:
2672:
2670:
2667:
2666:
2649:
2639:
2634:
2630:
2626:
2621:
2617:
2608:
2603:
2600:
2597:
2593:
2583:
2580:
2570:
2567:
2563:
2554:
2537:
2531:
2526:
2522:
2518:
2515:
2512:
2507:
2502:
2498:
2494:
2489:
2484:
2480:
2475:
2469:
2466:
2459:
2452:
2447:
2443:
2437:
2432:
2429:
2426:
2422:
2416:
2413:
2402:
2398:
2395:
2394:
2377:
2367:
2363:
2359:
2356:
2348:
2345:
2342:
2337:
2333:
2327:
2322:
2319:
2316:
2312:
2299:
2296:
2292:
2283:
2264:
2261:
2258:
2253:
2249:
2243:
2238:
2235:
2232:
2228:
2220:
2215:
2211:
2205:
2200:
2197:
2194:
2190:
2179:
2177:
2174:
2173:
2156:
2146:
2142:
2138:
2135:
2127:
2123:
2117:
2112:
2109:
2106:
2102:
2089:
2086:
2082:
2073:
2053:
2049:
2044:
2041:
2038:
2032:
2028:
2023:
2017:
2013:
2004:
1999:
1995:
1991:
1988:
1985:
1980:
1975:
1971:
1967:
1962:
1957:
1953:
1942:
1940:
1937:
1936:
1919:
1914:
1906:
1902:
1898:
1893:
1888:
1885:
1879:
1872:
1867:
1864:
1861:
1857:
1847:
1844:
1834:
1831:
1827:
1818:
1797:
1793:
1789:
1784:
1781:
1778:
1771:
1767:
1763:
1758:
1751:
1747:
1743:
1737:
1728:
1726:
1725:Harmonic mean
1723:
1722:
1704:
1701:
1698:
1695:
1692:
1686:
1683:
1675:
1672:
1667:
1663:
1653:
1648:
1635:
1631:
1624:
1621:
1618:
1612:
1606:
1603:
1595:
1590:
1587:
1584:
1580:
1570:
1567:
1557:
1554:
1550:
1541:
1524:
1517:
1513:
1509:
1504:
1500:
1496:
1491:
1487:
1480:
1474:
1467:
1463:
1457:
1452:
1449:
1446:
1442:
1431:
1429:
1426:
1425:
1409:
1397:
1393:
1389:
1386:
1378:
1375:
1366:
1362:
1358:
1355:
1347:
1344:
1338:
1333:
1327:
1322:
1319:
1316:
1312:
1299:
1296:
1292:
1283:
1280:
1278:
1275:
1274:
1258:
1243:
1240:
1233:
1230:
1221:
1215:
1203:
1197:
1192:
1182:
1167:
1164:
1158:
1153:
1147:
1142:
1139:
1136:
1132:
1122:
1112:
1103:
1096:
1085:
1075:
1066:
1059:
1050:
1048:
1030:
1015:
1013:
1010:
1009:
992:
975:
965:
958:
949:
931:
926:
923:
920:
916:
906:
896:
887:
880:
871:
855:
840:
836:
833:
829:
827:
824:
823:
801:
797:
793:
790:
780:
775:
772:
769:
765:
752:
749:
745:
736:
733:
731:
728:
727:
710:
700:
696:
692:
689:
681:
676:
673:
670:
666:
653:
650:
646:
637:
618:
614:
610:
607:
604:
599:
595:
586:
583:
578:
573:
569:
563:
558:
555:
552:
548:
542:
539:
534:
525:
515:
513:
510:
509:
505:
502:
499:
498:
494:
490:
480:
476:
466:
462:
458:
448:
442:
438:
434:
430:
426:
421:
416:
402:
393:
391:
387:
384:
379:, 3, 4, 7, 9
378:
374:
367:
365:
361:
358:
352:
345:
343:
339:
336:
313:
309:
303:
298:
295:
292:
288:
282:
279:
274:
265:
253:
251:
247:
243:
240:
237:
234:
233:
227:
225:
221:
217:
213:
209:
205:
199:
189:
187:
183:
182:harmonic mean
179:
175:
169:
163:
153:
151:
147:
143:
139:
135:
130:
128:
124:
120:
116:
112:
108:
104:
100:
95:
87:
85:
81:
77:
73:
69:
65:
61:
57:
53:
49:
45:
42:language, an
41:
32:
19:
5494:
5490:
5459:September 5,
5457:. Retrieved
5446:
5435:
5423:. Retrieved
5417:
5407:
5396:. Retrieved
5389:
5380:
5369:. Retrieved
5362:the original
5349:
5295:
5291:
5263:
5257:
5238:
5232:
5213:
5207:
5177:
5173:
5167:
5144:
5140:
5136:
5132:
5128:
5124:
5119:
5115:
5111:
5107:
5067:
5063:
5059:
5055:
5047:
5042:
5037:
5024:
5018:
5013:
4993:
4989:
4985:
4981:
4977:
4973:
4969:
4963:
4946:
4942:
4937:
4865:
4855:
4851:
4849:
4841:
4834:
4825:
4823:
4818:
4813:
4809:
4805:
4801:
4797:
4793:
4789:
4785:
4784:(also عوارة
4781:
4777:
4775:
4766:
4756:
4740:
4711:
4707:
4703:
4697:
4681:
4675:
4669:
4666:
4660:
4654:
4650:
4644:
4641:
4622:
4618:
4611:
4604:
4596:
4592:
4585:
4578:
4574:
4567:
4563:
4556:
4550:
4543:
4539:
4532:
4525:
4521:
4514:
4510:
4503:
4496:
4489:
4485:
4478:
4471:
4467:
4459:
4455:
4448:
4441:
4437:
4431:
4427:
4423:
4419:
4414:
4397:
4393:
4386:
4379:
4375:
4373:
4368:
4364:
4360:
4356:
4352:
4348:
4344:
4342:
4197:
4196:generalized
4193:
4178:
4166:
1012:Tukey median
478:
463:
460:
445:
400:
382:
376:
372:
356:
350:
334:
238:Description
201:
185:
171:
131:
122:
118:
114:
110:
106:
102:
99:monotonicity
96:
93:
84:billionaires
43:
37:
4998:AlWaraq.net
4951:Baheth.info
4923:Sample mean
4839:("avers").
4812:and French
4700:time series
4411:permutation
3847:percentiles
2176:Lehmer mean
1047:Tukey depth
150:permutation
5539:Categories
5471:required.)
5398:2023-05-25
5371:2018-11-27
5292:Biometrika
5273:0130040525
5248:0816211043
5159:References
5110:, Catalan
5027:@ CNRTL.fr
4988:and عوارة
4808:, Catalan
4747:estimation
4403:continuous
2669:Cubic mean
487:See also:
397:(1+9) / 2
353:, 4, 7, 9
210:, and the
196:See also:
166:See also:
5513:2392-3113
5413:Ray, John
5386:"average"
5216:: 63–65.
5186:1886-516X
5180:: 69–84.
5019:Glossaire
4772:Etymology
4752:mid-range
4721:filtering
4653:) × (1 +
4296:⋯
4224:−
4185:trimedian
4108:∑
4094:∈
3985:−
3961:…
3949:∈
3928:∈
3786:−
3750:∑
3737:∈
3691:⋯
3637:⋯
3552:∑
3509:∑
3487:monotonic
3432:−
3397:∑
3380:
3374:∈
3303:∑
3279:−
3237:…
3225:∈
3218:∀
3211:≥
3164:−
3131:∑
3118:≥
3108:∈
3038:∑
3034:⋅
2984:…
2972:∈
2965:∀
2958:≥
2911:−
2878:∑
2865:≥
2855:∈
2794:⋯
2694:∑
2627:−
2594:∑
2581:≥
2571:∈
2516:⋯
2423:∑
2360:−
2346:−
2313:∑
2300:∈
2262:−
2229:∑
2191:∑
2139:−
2103:∑
2090:∈
2042:⋯
1989:⋯
1894:−
1858:∑
1845:≠
1835:∈
1782:⋯
1699:…
1687:∈
1680:∀
1625:
1619:−
1607:
1581:∑
1558:∈
1510:⋯
1497:⋅
1443:∏
1390:≠
1313:∑
1300:∈
1231:≥
1225:→
1216:⋅
1207:→
1198:−
1186:→
1133:∑
1113:∈
1107:→
1076:∈
1070:→
969:→
959:−
953:→
917:∑
897:∈
891:→
835:invariant
794:−
766:∑
753:∈
693:−
667:∑
654:∈
608:⋯
549:∑
529:¯
475:Mid-range
469:Mid-range
390:Mid-range
349:1, 2, 2,
289:∑
269:¯
212:mid-range
60:mid-range
52:statistic
18:Averaging
5415:(1674).
5030:Archived
4954:Archived
4876:See also
4847:(1085).
4698:Given a
4668:return,
4617:+ ··· +
4509:+ ··· +
3855:Midrange
3469:if
3197:if
2944:if
2400:(or RMS)
1659:if
832:rotation
441:skewness
241:Example
180:and the
140:and the
40:ordinary
5312:2333051
5064:avariis
5060:avarias
5056:avariis
4982:ʿawārīa
4974:ʿAwārīa
4970:ʿawārīa
4962:Lane's
4732:History
4591:, ...,
4538:, ...,
4484:, ...,
4454:, ...,
4430:, ...,
4385:,
4181:trimean
3843:deciles
435:of two
244:Result
44:average
5511:
5425:18 May
5310:
5270:
5245:
5184:
5145:avaria
5141:avaria
5129:avaria
5120:avaria
5116:avaria
5112:avaria
5108:avaria
5068:avaria
5048:avaria
5038:avaria
5025:avarie
5014:avaria
4990:ʿawāra
4986:ʿawārī
4819:avaria
4814:avarie
4810:avaria
4806:avaria
4802:avarie
4798:avaria
4794:avaria
4786:ʿawāra
4782:ʿawārī
4737:Origin
4355:) = 1/
4343:where
4187:, and
4087:argmin
4027:Medoid
3921:argmin
3730:argmin
3367:argmin
3101:argmin
2848:argmin
2564:argmin
2293:argmin
2083:argmin
1828:argmin
1551:argmin
1293:argmax
1060:argmax
881:argmin
839:median
746:argmin
730:Median
647:argmin
491:, and
457:Median
451:Median
429:median
342:Median
208:median
206:, the
176:, the
136:, the
64:median
5550:Means
5497:(4).
5465:
5365:(PDF)
5358:(PDF)
5329:1971.
5308:JSTOR
4994:ʿawār
4978:ʿawār
4947:ʿawār
4929:Notes
4856:ʿawâr
4852:hafen
4790:ʿawār
4778:ʿawār
4723:. In
4200:-mean
235:Type
5509:ISSN
5461:2019
5427:2015
5268:ISBN
5243:ISBN
5182:ISSN
5102:and
5058:and
4943:ʿawr
4167:The
1673:>
1568:>
1277:Mode
500:Name
433:mode
431:and
409:Mode
364:Mode
216:mean
204:mode
202:The
172:The
105:and
80:mean
68:mode
5499:doi
5300:doi
5218:doi
5133:-ia
4600:) =
4562:···
4547:) =
4493:) =
4434:) =
3942:max
3890:min
3881:max
3845:or
3485:is
3377:dom
1097:min
371:1,
70:or
38:In
5541::
5507:.
5493:.
5489:.
5477:^
5445:.
5388:.
5335:^
5320:^
5306:.
5296:45
5294:.
5282:^
5214:15
5212:.
5206:.
5194:^
5176:.
5094:,
5090:,
5086:,
5082:,
5005:^
4972:.
4626:))
4584:,
4531:,
4477:,
4447:,
4426:,
4405:,
4371:.
4202::
4183:,
1622:ln
1604:ln
830:A
427:,
375:,
226:.
188:.
66:,
62:,
5515:.
5501::
5495:5
5463:.
5429:.
5401:.
5374:.
5314:.
5302::
5276:.
5251:.
5226:.
5220::
5188:.
5178:9
4712:n
4708:n
4682:R
4678:)
4676:R
4670:R
4661:R
4657:)
4655:R
4651:R
4645:R
4623:n
4619:x
4615:2
4612:x
4610:+
4608:1
4605:x
4603:(
4597:n
4593:x
4589:2
4586:x
4582:1
4579:x
4577:(
4575:g
4568:n
4564:x
4560:2
4557:x
4554:1
4551:x
4544:n
4540:x
4536:2
4533:x
4529:1
4526:x
4524:(
4522:g
4515:n
4511:x
4507:2
4504:x
4502:+
4500:1
4497:x
4490:n
4486:x
4482:2
4479:x
4475:1
4472:x
4470:(
4468:g
4463:)
4460:n
4456:x
4452:2
4449:x
4445:1
4442:x
4440:(
4438:g
4432:y
4428:y
4424:y
4422:(
4420:g
4415:y
4398:n
4394:x
4390:2
4387:x
4383:1
4380:x
4378:(
4376:g
4369:x
4365:x
4363:(
4361:f
4357:x
4353:x
4351:(
4349:f
4345:f
4327:)
4322:]
4318:)
4313:n
4309:x
4305:(
4302:f
4299:+
4293:+
4290:)
4285:2
4281:x
4277:(
4274:f
4271:+
4268:)
4263:1
4259:x
4255:(
4252:f
4248:[
4242:n
4239:1
4233:(
4227:1
4220:f
4216:=
4213:y
4198:f
4150:)
4145:i
4141:x
4137:,
4134:y
4131:(
4128:d
4123:n
4118:1
4115:=
4112:i
4099:X
4091:y
4075:.
4063:d
4041:X
3999:|
3993:i
3989:x
3982:x
3978:|
3970:}
3967:n
3964:,
3958:,
3955:1
3952:{
3946:i
3932:R
3925:x
3897:)
3893:x
3887:+
3884:x
3877:(
3871:2
3868:1
3804:2
3800:)
3794:i
3790:x
3783:x
3780:(
3775:i
3771:w
3765:n
3760:1
3757:=
3754:i
3741:R
3734:x
3702:n
3698:w
3694:+
3688:+
3683:2
3679:w
3675:+
3670:1
3666:w
3658:n
3654:x
3648:n
3644:w
3640:+
3634:+
3629:2
3625:x
3619:2
3615:w
3611:+
3606:1
3602:x
3596:1
3592:w
3585:=
3577:i
3573:w
3567:n
3562:1
3559:=
3556:i
3544:i
3540:x
3534:i
3530:w
3524:n
3519:1
3516:=
3513:i
3473:f
3464:,
3459:2
3455:)
3451:)
3446:i
3442:x
3438:(
3435:f
3429:)
3426:x
3423:(
3420:f
3417:(
3412:n
3407:1
3404:=
3401:i
3389:)
3386:f
3383:(
3371:x
3343:)
3339:)
3334:k
3330:x
3326:(
3323:f
3318:n
3313:1
3310:=
3307:k
3297:n
3294:1
3288:(
3282:1
3275:f
3246:}
3243:n
3240:,
3234:,
3231:1
3228:{
3222:i
3214:0
3206:i
3202:x
3192:,
3187:2
3183:)
3177:p
3172:i
3168:x
3159:p
3155:x
3151:(
3146:n
3141:1
3138:=
3135:i
3121:0
3113:R
3105:x
3075:p
3068:p
3063:i
3059:x
3053:n
3048:1
3045:=
3042:i
3029:n
3026:1
2993:}
2990:n
2987:,
2981:,
2978:1
2975:{
2969:i
2961:0
2953:i
2949:x
2939:,
2934:2
2930:)
2924:3
2919:i
2915:x
2906:3
2902:x
2898:(
2893:n
2888:1
2885:=
2882:i
2868:0
2860:R
2852:x
2822:3
2816:)
2810:3
2805:n
2801:x
2797:+
2791:+
2786:3
2781:2
2777:x
2773:+
2768:3
2763:1
2759:x
2754:(
2748:n
2745:1
2737:=
2731:3
2724:3
2719:i
2715:x
2709:n
2704:1
2701:=
2698:i
2688:n
2685:1
2650:2
2646:)
2640:2
2635:i
2631:x
2622:2
2618:x
2614:(
2609:n
2604:1
2601:=
2598:i
2584:0
2576:R
2568:x
2538:)
2532:2
2527:n
2523:x
2519:+
2513:+
2508:2
2503:2
2499:x
2495:+
2490:2
2485:1
2481:x
2476:(
2470:n
2467:1
2460:=
2453:2
2448:i
2444:x
2438:n
2433:1
2430:=
2427:i
2417:n
2414:1
2378:2
2374:)
2368:i
2364:x
2357:x
2354:(
2349:1
2343:p
2338:i
2334:x
2328:n
2323:1
2320:=
2317:i
2304:R
2297:x
2265:1
2259:p
2254:i
2250:x
2244:n
2239:1
2236:=
2233:i
2221:p
2216:i
2212:x
2206:n
2201:1
2198:=
2195:i
2157:2
2153:)
2147:i
2143:x
2136:x
2133:(
2128:i
2124:x
2118:n
2113:1
2110:=
2107:i
2094:R
2087:x
2054:n
2050:x
2045:+
2039:+
2033:2
2029:x
2024:+
2018:1
2014:x
2005:2
2000:n
1996:x
1992:+
1986:+
1981:2
1976:2
1972:x
1968:+
1963:2
1958:1
1954:x
1920:2
1915:)
1907:i
1903:x
1899:1
1889:x
1886:1
1880:(
1873:n
1868:1
1865:=
1862:i
1848:0
1840:R
1832:x
1798:n
1794:x
1790:1
1785:+
1779:+
1772:2
1768:x
1764:1
1759:+
1752:1
1748:x
1744:1
1738:n
1708:}
1705:n
1702:,
1696:,
1693:1
1690:{
1684:i
1676:0
1668:i
1664:x
1654:,
1649:2
1645:)
1641:)
1636:i
1632:x
1628:(
1616:)
1613:x
1610:(
1601:(
1596:n
1591:1
1588:=
1585:i
1571:0
1563:R
1555:x
1525:n
1518:n
1514:x
1505:2
1501:x
1492:1
1488:x
1481:=
1475:n
1468:i
1464:x
1458:n
1453:1
1450:=
1447:i
1410:)
1398:i
1394:x
1387:x
1379:,
1376:0
1367:i
1363:x
1359:=
1356:x
1348:,
1345:1
1339:{
1334:(
1328:n
1323:1
1320:=
1317:i
1304:R
1297:x
1259:)
1244:,
1241:0
1234:0
1222:u
1213:)
1204:x
1193:i
1183:x
1176:(
1168:,
1165:1
1159:{
1154:(
1148:n
1143:1
1140:=
1137:i
1123:d
1118:R
1104:u
1086:d
1081:R
1067:x
1031:d
1026:R
993:2
988:|
982:|
976:i
966:x
950:x
943:|
938:|
932:n
927:1
924:=
921:i
907:d
902:R
888:x
856:d
851:R
808:|
802:i
798:x
791:x
787:|
781:n
776:1
773:=
770:i
757:R
750:x
711:2
707:)
701:i
697:x
690:x
687:(
682:n
677:1
674:=
671:i
658:R
651:x
624:)
619:n
615:x
611:+
605:+
600:1
596:x
592:(
587:n
584:1
579:=
574:i
570:x
564:n
559:1
556:=
553:i
543:n
540:1
535:=
526:x
401:5
383:2
377:2
373:2
357:3
351:3
335:4
314:i
310:x
304:n
299:1
296:=
293:i
283:n
280:1
275:=
266:x
123:B
119:A
115:B
111:A
107:B
103:A
20:)
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