In mathematics , the harmonic mean is a kind of average , one of the Pythagorean means .
54-539: The Division of Wannon is an Australian Electoral Division in the state of Victoria . The division was proclaimed in 1900, and was one of the original 65 divisions to be contested at the first Federal election . The division was named after the Wannon River . For the first half-century after Federation , it regularly traded hands between the Australian Labor Party and the conservative parties. However,
108-445: A right triangle with legs a and b and altitude h from the hypotenuse to the right angle, h is half the harmonic mean of a and b . Let t and s ( t > s ) be the sides of the two inscribed squares in a right triangle with hypotenuse c . Then s equals half the harmonic mean of c and t . Let a trapezoid have vertices A, B, C, and D in sequence and have parallel sides AB and CD. Let E be
162-557: A 1955 redistribution removed most of the seat's Labor-friendly territory, and it has been a safe Liberal seat for most of its history since then. The seat's most notable member was Prime Minister Malcolm Fraser , to date the last prime minister from a country seat. His successor, David Hawker , was Speaker of the Australian House of Representatives during the last term of the Howard Government. Hawker retired in 2010 and
216-402: A range of cars one measure will produce the harmonic mean of the other – i.e., converting the mean value of fuel economy expressed in litres per 100 km to miles per gallon will produce the harmonic mean of the fuel economy expressed in miles per gallon. For calculating the average fuel consumption of a fleet of vehicles from the individual fuel consumptions, the harmonic mean should be used if
270-429: A roundtrip journey (see above). In any triangle , the radius of the incircle is one-third of the harmonic mean of the altitudes . For any point P on the minor arc BC of the circumcircle of an equilateral triangle ABC, with distances q and t from B and C respectively, and with the intersection of PA and BC being at a distance y from point P, we have that y is half the harmonic mean of q and t . In
324-414: A set of non-identical numbers is subjected to a mean-preserving spread — that is, two or more elements of the set are "spread apart" from each other while leaving the arithmetic mean unchanged — then the harmonic mean always decreases. For the special case of just two numbers, x 1 {\displaystyle x_{1}} and x 2 {\displaystyle x_{2}} ,
378-444: A speed y , then its average speed is the arithmetic mean of x and y , which in the above example is 40 km/h. Average speed for the entire journey = Total distance traveled / Sum of time for each segment = xt+yt / 2t = x+y / 2 The same principle applies to more than two segments: given a series of sub-trips at different speeds, if each sub-trip covers
432-758: A state are malapportioned. The division is located in the south-west of the state, and encompasses most of the Western District of the state. It adjoins the South Australian border in the west, the Bass Strait coast in the south to Anglesea in the east, and extends north to the Grampians . Following the redistribution on 26 July 2021 the division encompasses the towns of Warrnambool , Colac , Portland , Hamilton , and Ararat . The Budj Bim , Great Otway , Lower Glenelg , and Port Campbell National Parks are in
486-416: A wall at height A and the other leaning against the opposite wall at height B , as shown. The ladders cross at a height of h above the alley floor. Then h is half the harmonic mean of A and B . This result still holds if the walls are slanted but still parallel and the "heights" A , B , and h are measured as distances from the floor along lines parallel to the walls. This can be proved easily using
540-460: A weighted arithmetic mean, high data points are given greater weights than low data points. The weighted harmonic mean, on the other hand, correctly weights each data point. The simple weighted arithmetic mean when applied to non-price normalized ratios such as the P/E is biased upwards and cannot be numerically justified, since it is based on equalized earnings; just as vehicles speeds cannot be averaged for
594-637: Is The harmonic mean H of the positive real numbers x 1 , x 2 , … , x n {\displaystyle x_{1},x_{2},\ldots ,x_{n}} is It is the reciprocal of the arithmetic mean of the reciprocals, and vice versa: where the arithmetic mean is A ( x 1 , x 2 , … , x n ) = 1 n ∑ i = 1 n x i . {\textstyle A(x_{1},x_{2},\ldots ,x_{n})={\tfrac {1}{n}}\sum _{i=1}^{n}x_{i}.} The harmonic mean
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#1732876249248648-467: Is a Schur-concave function, and is greater than or equal to the minimum of its arguments: for positive arguments, min ( x 1 … x n ) ≤ H ( x 1 … x n ) ≤ n min ( x 1 … x n ) {\displaystyle \min(x_{1}\ldots x_{n})\leq H(x_{1}\ldots x_{n})\leq n\min(x_{1}\ldots x_{n})} . Thus,
702-596: Is entitled (called apportionment ) and the boundaries of each electorate, in a process known as redistribution . Such apportionment and redistributions apply to the next federal election, but not to any by-elections. The last apportionment determination was made in July 2020. The resulting redistribution took place during 2021 and was used in the 2022 federal election , which was held on 21 May 2022. Within each state and territory, electoral boundaries are redrawn from time to time. This takes place at least once every 7 years, or when
756-418: Is equal to 2.4 hours, to drain the pool together. This is one-half of the harmonic mean of 6 and 4: 2·6·4 / 6 + 4 = 4.8 . That is, the appropriate average for the two types of pump is the harmonic mean, and with one pair of pumps (two pumps), it takes half this harmonic mean time, while with two pairs of pumps (four pumps) it would take a quarter of this harmonic mean time. In hydrology ,
810-438: Is equivalent to two thin lenses of focal length f hm , their harmonic mean, in series. Expressed as optical power , two thin lenses of optical powers P 1 and P 2 in series is equivalent to two thin lenses of optical power P am , their arithmetic mean, in series. The weighted harmonic mean is the preferable method for averaging multiples, such as the price–earnings ratio (P/E). If these ratios are averaged using
864-447: Is found, invert it so as to find the "true" average trip speed. For each trip segment i, the slowness s i = 1/speed i . Then take the weighted arithmetic mean of the s i 's weighted by their respective distances (optionally with the weights normalized so they sum to 1 by dividing them by trip length). This gives the true average slowness (in time per kilometre). It turns out that this procedure, which can be done with no knowledge of
918-423: Is related to the other Pythagorean means, as seen in the equation below. This can be seen by interpreting the denominator to be the arithmetic mean of the product of numbers n times but each time omitting the j -th term. That is, for the first term, we multiply all n numbers except the first; for the second, we multiply all n numbers except the second; and so on. The numerator, excluding the n , which goes with
972-423: Is the case in most other legislatures around the world. Most divisions are named in honour of prominent historical people, such as former politicians (often Prime Ministers ), explorers, artists and engineers. There is also a preference for retaining names used since Federation. In some cases where a division is named after a geographical locality, the connection to that locality is sometimes tenuous. For instance,
1026-413: Is the most appropriate average for ratios and rates such as speeds, and is normally only used for positive arguments. The harmonic mean is the reciprocal of the arithmetic mean of the reciprocals of the numbers, that is, the generalized f-mean with f ( x ) = 1 x {\displaystyle f(x)={\frac {1}{x}}} . For example, the harmonic mean of 1, 4, and 4
1080-885: The 2022 Australian federal election , based on the 2021 apportionment, there were 151 divisions: 47 in New South Wales, 39 in Victoria, 30 in Queensland, 15 in Western Australia, 10 in South Australia, 5 in Tasmania, 3 in the ACT and 2 in Northern Territory. The divisions of the House of Representatives are unusual in that many of them are not named after geographical features or numbered, as
1134-611: The Division of Werriwa , created in 1901, was named after the Aboriginal word for Lake George in the Canberra region. However, Werriwa has not contained Lake George for many decades, and has steadily moved some 200 km north to the south-western suburbs of Sydney over the past century. The divisions that existed at the 2022 Australian federal election appear in the table below. These divisions no longer exist: Harmonic mean It
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#17328762492481188-714: The Parliament of the Commonwealth . There are currently 151 electorates. Section 24 of the Constitution of Australia specifies that the total number of members of the Australian House of Representatives shall be "as nearly as practicable" twice as many as the number of members of the Australian Senate . The section also requires that electorates be apportioned among the states in proportion to their respective populations; provided that each original state has at least 5 members in
1242-840: The arithmetic mean is always the greatest of the three and the geometric mean is always in between. (If all values in a nonempty data set are equal, the three means are always equal.) It is the special case M −1 of the power mean : H ( x 1 , x 2 , … , x n ) = M − 1 ( x 1 , x 2 , … , x n ) = n x 1 − 1 + x 2 − 1 + ⋯ + x n − 1 {\displaystyle H\left(x_{1},x_{2},\ldots ,x_{n}\right)=M_{-1}\left(x_{1},x_{2},\ldots ,x_{n}\right)={\frac {n}{x_{1}^{-1}+x_{2}^{-1}+\cdots +x_{n}^{-1}}}} Since
1296-414: The gene pool limiting the genetic variation present in the population for many generations to come. When considering fuel economy in automobiles two measures are commonly used – miles per gallon (mpg), and litres per 100 km. As the dimensions of these quantities are the inverse of each other (one is distance per volume, the other volume per distance) when taking the mean value of the fuel economy of
1350-595: The inequality of arithmetic and geometric means , this shows for the n = 2 case that H ≤ G (a property that in fact holds for all n ). It also follows that G = A H {\displaystyle G={\sqrt {AH}}} , meaning the two numbers' geometric mean equals the geometric mean of their arithmetic and harmonic means. For the special case of three numbers, x 1 {\displaystyle x_{1}} , x 2 {\displaystyle x_{2}} and x 3 {\displaystyle x_{3}} ,
1404-476: The recall (true positives per real positive) is often used as an aggregated performance score for the evaluation of algorithms and systems: the F-score (or F-measure). This is used in information retrieval because only the positive class is of relevance , while number of negatives, in general, is large and unknown. It is thus a trade-off as to whether the correct positive predictions should be measured in relation to
1458-458: The weighted harmonic mean is defined by The unweighted harmonic mean can be regarded as the special case where all of the weights are equal. The prime number theorem states that the number of primes less than or equal to n {\displaystyle n} is asymptotically equal to the harmonic mean of the first n {\displaystyle n} natural numbers . In many situations involving rates and ratios ,
1512-406: The ACT and the whole of Jervis Bay Territory ), and the Division of Lingiari (covering part of the Northern Territory and the whole of Christmas Island and Cocos (Keeling) Islands ). The Commonwealth Electoral Act 1918 sets out further provisions. The Australian Electoral Commission (AEC) determines the number of members of the House of Representatives to which each state and territory
1566-938: The House of Representatives, a provision that has given Tasmania higher representation than its population would otherwise justify. There are three electorates in the Australian Capital Territory and even though the Northern Territory should have only one electorate based on their population, parliament have legislated that they receive two (by setting the quota for seat allocation using the harmonic mean for territories, meaning only around 1.3 quotas rather than 1.5 quotas are needed to have two seats). In addition, Section 29 forbids electorate boundaries from crossing state lines, forcing populated areas along state and territory borders to be placed in different electorates, such as Albury in New South Wales being part of
1620-636: The alloy (exclusive of typically minor volume changes due to atom packing effects) is the weighted harmonic mean of the individual densities, weighted by mass, rather than the weighted arithmetic mean as one might at first expect. To use the weighted arithmetic mean, the densities would have to be weighted by volume. Applying dimensional analysis to the problem while labeling the mass units by element and making sure that only like element-masses cancel makes this clear. If one connects two electrical resistors in parallel, one having resistance x (e.g., 60 Ω ) and one having resistance y (e.g., 40 Ω), then
1674-438: The area formula of a trapezoid and area addition formula. In an ellipse , the semi-latus rectum (the distance from a focus to the ellipse along a line parallel to the minor axis) is the harmonic mean of the maximum and minimum distances of the ellipse from a focus. In computer science , specifically information retrieval and machine learning , the harmonic mean of the precision (true positives per predicted positive) and
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1728-1604: The arithmetic mean, is the geometric mean to the power n . Thus the n -th harmonic mean is related to the n -th geometric and arithmetic means. The general formula is H ( x 1 , … , x n ) = ( G ( x 1 , … , x n ) ) n A ( x 2 x 3 ⋯ x n , x 1 x 3 ⋯ x n , … , x 1 x 2 ⋯ x n − 1 ) = ( G ( x 1 , … , x n ) ) n A ( 1 x 1 ∏ i = 1 n x i , 1 x 2 ∏ i = 1 n x i , … , 1 x n ∏ i = 1 n x i ) . {\displaystyle H\left(x_{1},\ldots ,x_{n}\right)={\frac {\left(G\left(x_{1},\ldots ,x_{n}\right)\right)^{n}}{A\left(x_{2}x_{3}\cdots x_{n},x_{1}x_{3}\cdots x_{n},\ldots ,x_{1}x_{2}\cdots x_{n-1}\right)}}={\frac {\left(G\left(x_{1},\ldots ,x_{n}\right)\right)^{n}}{A\left({\frac {1}{x_{1}}}{\prod \limits _{i=1}^{n}x_{i}},{\frac {1}{x_{2}}}{\prod \limits _{i=1}^{n}x_{i}},\ldots ,{\frac {1}{x_{n}}}{\prod \limits _{i=1}^{n}x_{i}}\right)}}.} If
1782-538: The division, as is southern portion of the Grampians National Park . 37°42′11″S 142°26′53″E / 37.703°S 142.448°E / -37.703; 142.448 Divisions of the Australian House of Representatives Electorates (also known as electoral divisions or seats ) of the Australian House of Representatives are single member electoral districts for the lower house of
1836-429: The duration of that portion, while for the harmonic mean, the corresponding weight is the distance. In both cases, the resulting formula reduces to dividing the total distance by the total time.) However, one may avoid the use of the harmonic mean for the case of "weighting by distance". Pose the problem as finding "slowness" of the trip where "slowness" (in hours per kilometre) is the inverse of speed. When trip slowness
1890-489: The effect is the same as if one had used two resistors with the same resistance, both equal to the harmonic mean of x and y (48 Ω): the equivalent resistance, in either case, is 24 Ω (one-half of the harmonic mean). This same principle applies to capacitors in series or to inductors in parallel. However, if one connects the resistors in series, then the average resistance is the arithmetic mean of x and y (50 Ω), with total resistance equal to twice this,
1944-450: The electorate of Farrer , while nearby Wodonga in Victoria is part of the electorate of Indi . The same restriction does not apply to territories, and several current electoral divisions incorporate electors from multiple territories. This is currently the case for the Division of Bean (covering part of the ACT and the whole of Norfolk Island ), the Division of Fenner (covering part of
1998-410: The entire journey = Total distance traveled / Sum of time for each segment = 2 d / d / x + d / y = 2 / 1 / x + 1 / y However, if the vehicle travels for a certain amount of time at a speed x and then the same amount of time at
2052-501: The fleet uses miles per gallon, whereas the arithmetic mean should be used if the fleet uses litres per 100 km. In the USA the CAFE standards (the federal automobile fuel consumption standards) make use of the harmonic mean. In chemistry and nuclear physics the average mass per particle of a mixture consisting of different species (e.g., molecules or isotopes) is given by the harmonic mean of
2106-512: The harmonic mean can be written as: In this special case, the harmonic mean is related to the arithmetic mean A = x 1 + x 2 2 {\displaystyle A={\frac {x_{1}+x_{2}}{2}}} and the geometric mean G = x 1 x 2 , {\displaystyle G={\sqrt {x_{1}x_{2}}},} by Since G A ≤ 1 {\displaystyle {\tfrac {G}{A}}\leq 1} by
2160-538: The harmonic mean can be written as: Three positive numbers H , G , and A are respectively the harmonic, geometric, and arithmetic means of three positive numbers if and only if the following inequality holds If a set of weights w 1 {\displaystyle w_{1}} , ..., w n {\displaystyle w_{n}} is associated to the data set x 1 {\displaystyle x_{1}} , ..., x n {\displaystyle x_{n}} ,
2214-408: The harmonic mean cannot be made arbitrarily large by changing some values to bigger ones (while having at least one value unchanged). The harmonic mean is also concave for positive arguments, an even stronger property than Schur-concavity. For all positive data sets containing at least one pair of nonequal values , the harmonic mean is always the least of the three Pythagorean means, while
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2268-489: The harmonic mean is similarly used to average hydraulic conductivity values for a flow that is perpendicular to layers (e.g., geologic or soil) - flow parallel to layers uses the arithmetic mean. This apparent difference in averaging is explained by the fact that hydrology uses conductivity, which is the inverse of resistivity. In sabermetrics , a baseball player's Power–speed number is the harmonic mean of their home run and stolen base totals. In population genetics ,
2322-411: The harmonic mean is used when calculating the effects of fluctuations in the census population size on the effective population size. The harmonic mean takes into account the fact that events such as population bottleneck increase the rate genetic drift and reduce the amount of genetic variation in the population. This is a result of the fact that following a bottleneck very few individuals contribute to
2376-421: The harmonic mean of a list of numbers tends strongly toward the least elements of the list, it tends (compared to the arithmetic mean) to mitigate the impact of large outliers and aggravate the impact of small ones. The arithmetic mean is often mistakenly used in places calling for the harmonic mean. In the speed example below for instance, the arithmetic mean of 40 is incorrect, and too big. The harmonic mean
2430-462: The harmonic mean provides the correct average . For instance, if a vehicle travels a certain distance d outbound at a speed x (e.g. 60 km/h) and returns the same distance at a speed y (e.g. 20 km/h), then its average speed is the harmonic mean of x and y (30 km/h), not the arithmetic mean (40 km/h). The total travel time is the same as if it had traveled the whole distance at that average speed. This can be proven as follows: Average speed for
2484-402: The harmonic mean, amounts to the same mathematical operations as one would use in solving this problem by using the harmonic mean. Thus it illustrates why the harmonic mean works in this case. Similarly, if one wishes to estimate the density of an alloy given the densities of its constituent elements and their mass fractions (or, equivalently, percentages by mass), then the predicted density of
2538-406: The intersection of the diagonals , and let F be on side DA and G be on side BC such that FEG is parallel to AB and CD. Then FG is the harmonic mean of AB and DC. (This is provable using similar triangles.) One application of this trapezoid result is in the crossed ladders problem , where two ladders lie oppositely across an alley, each with feet at the base of one sidewall, with one leaning against
2592-476: The number of predicted positives or the number of real positives, so it is measured versus a putative number of positives that is an arithmetic mean of the two possible denominators. A consequence arises from basic algebra in problems where people or systems work together. As an example, if a gas-powered pump can drain a pool in 4 hours and a battery-powered pump can drain the same pool in 6 hours, then it will take both pumps 6·4 / 6 + 4 , which
2646-448: The number of voters vary by more than 3.5% from the average projected enrolment 3.5 years into the future. However, due to various reasons, larger seats like Cowper in New South Wales contain 80% more electors than that of smaller seats like Solomon in the Northern Territory . In 2018, seats in Victoria, Tasmania and South Australia were also abolished, in order to make way for seats in similar locations but with different names. At
2700-400: The same distance , then the average speed is the harmonic mean of all the sub-trip speeds; and if each sub-trip takes the same amount of time , then the average speed is the arithmetic mean of all the sub-trip speeds. (If neither is the case, then a weighted harmonic mean or weighted arithmetic mean is needed. For the arithmetic mean, the speed of each portion of the trip is weighted by
2754-404: The state's entitlement to the number of members of the House of Representatives changes. Boundaries are drawn by a Redistribution Committee, and redistributions within a state are on the basis of the number of enrolled voters , rather than total residents or "population". The number of enrolled voters in each division cannot vary by more than 10% from the average across a state or territory, nor can
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#17328762492482808-422: The sum of x and y (100 Ω). This principle applies to capacitors in parallel or to inductors in series. As with the previous example, the same principle applies when more than two resistors, capacitors or inductors are connected, provided that all are in parallel or all are in series. The "conductivity effective mass" of a semiconductor is also defined as the harmonic mean of the effective masses along
2862-409: The three crystallographic directions. As for other optic equations , the thin lens equation 1 / f = 1 / u + 1 / v can be rewritten such that the focal length f is one-half of the harmonic mean of the distances of the subject u and object v from the lens. Two thin lenses of focal length f 1 and f 2 in series
2916-531: Was succeeded by Dan Tehan . Since 1984, federal electoral division boundaries in Australia have been determined at redistributions by a redistribution committee appointed by the Australian Electoral Commission . Redistributions occur for the boundaries of divisions in a particular state, and they occur every seven years, or sooner if a state's representation entitlement changes or when divisions of
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