In
mathematics, the harmonic mean is one of several kinds of
average
In ordinary language, an average is a single number taken as representative of a list of numbers, usually the sum of the numbers divided by how many numbers are in the list (the arithmetic mean). For example, the average of the numbers 2, 3, 4, 7 ...
, and in particular, one of the
Pythagorean means
In mathematics, the three classical Pythagorean means are the arithmetic mean (AM), the geometric mean (GM), and the harmonic mean (HM). These means were studied with proportions by Pythagoreans and later generations of Greek mathematicians b ...
. It is sometimes appropriate for situations when the average
rate is desired.
The harmonic mean can be expressed as the
reciprocal
Reciprocal may refer to:
In mathematics
* Multiplicative inverse, in mathematics, the number 1/''x'', which multiplied by ''x'' gives the product 1, also known as a ''reciprocal''
* Reciprocal polynomial, a polynomial obtained from another pol ...
of the
arithmetic mean of the reciprocals of the given set of observations. As a simple example, the harmonic mean of 1, 4, and 4 is
:
Definition
The harmonic mean ''H'' of the positive
real number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
s
is defined to be
:
The third formula in the above equation expresses the harmonic mean as the reciprocal of the arithmetic mean of the reciprocals.
From the following formula:
:
it is more apparent that the harmonic mean is related to the
arithmetic and
geometric means. It is the reciprocal
dual of the
arithmetic mean for positive inputs:
:
The harmonic mean is a
Schur-concave In mathematics, a Schur-convex function, also known as S-convex, isotonic function and order-preserving function is a function (mathematics), function f: \mathbb^d\rightarrow \mathbb that for all x,y\in \mathbb^d such that x is majorization, majori ...
function, and dominated by the minimum of its arguments, in the sense that for any positive set of arguments,
. Thus, the harmonic mean cannot be made
arbitrarily large In mathematics, the phrases arbitrarily large, arbitrarily small and arbitrarily long are used in statements to make clear of the fact that an object is large, small and long with little limitation or restraint, respectively. The use of "arbitrarily ...
by changing some values to bigger ones (while having at least one value unchanged).
The harmonic mean is also
concave
Concave or concavity may refer to:
Science and technology
* Concave lens
* Concave mirror
Mathematics
* Concave function, the negative of a convex function
* Concave polygon, a polygon which is not convex
* Concave set
* The concavity of a ...
, which is an even stronger property than Schur-concavity.
One has to take care to only use positive numbers though, since the mean fails to be concave if negative values are used.
Relationship with other means
The harmonic mean is one of the three
Pythagorean means
In mathematics, the three classical Pythagorean means are the arithmetic mean (AM), the geometric mean (GM), and the harmonic mean (HM). These means were studied with proportions by Pythagoreans and later generations of Greek mathematicians b ...
. For all ''positive'' data sets ''containing at least one pair of nonequal values'', the harmonic mean is always the least of the three means, while the
arithmetic mean is always the greatest of the three and the
geometric mean is always in between. (If all values in a nonempty dataset are equal, the three means are always equal to one another; e.g., the harmonic, geometric, and arithmetic means of are all 2.)
It is the special case ''M''
−1 of the
power mean
Power most often refers to:
* Power (physics), meaning "rate of doing work"
** Engine power, the power put out by an engine
** Electric power
* Power (social and political), the ability to influence people or events
** Abusive power
Power may ...
:
:
Since 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 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 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
:
If a set of non-identical numbers is subjected to a
mean-preserving spread In probability and statistics, a mean-preserving spread (MPS) is a change from one probability distribution A to another probability distribution B, where B is formed by spreading out one or more portions of A's probability density function or pr ...
— 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.
Harmonic mean of two or three numbers
Two numbers
For the special case of just two numbers,
and
, the harmonic mean can be written
:
or
In this special case, the harmonic mean is related to the
arithmetic mean and the
geometric mean by
:
Since
by the
inequality of arithmetic and geometric means
In mathematics, the inequality of arithmetic and geometric means, or more briefly the AM–GM inequality, states that the arithmetic mean of a list of non-negative real numbers is greater than or equal to the geometric mean of the same list; and ...
, this shows for the ''n'' = 2 case that ''H'' ≤ ''G'' (a property that in fact holds for all ''n''). It also follows that
, meaning the two numbers' geometric mean equals the geometric mean of their arithmetic and harmonic means.
Three numbers
For the special case of three numbers,
,
and
, the harmonic mean can be written
:
Three positive numbers ''H'', ''G'', and ''A'' are respectively the harmonic, geometric, and arithmetic means of three positive numbers
if and only if
In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false.
The connective is b ...
[''Inequalities proposed in “]Crux Mathematicorum
''Crux Mathematicorum'' is a scientific journal of mathematics published by the Canadian Mathematical Society. It contains mathematical problems for secondary school and undergraduate students. , its editor-in-chief is Kseniya Garaschuk.
The journ ...
”'', . the following inequality holds
:
Weighted harmonic mean
If a set of
weights , ...,
is associated to the dataset
, ...,
, the weighted harmonic mean is defined by
[Ferger F (1931) The nature and use of the harmonic mean. Journal of the
American Statistical Association 26(173) 36-40]
:
The unweighted harmonic mean can be regarded as the special case where all of the weights are equal.
Examples
In physics
Average speed
In many situations involving
rates and
ratio
In mathematics, a ratio shows how many times one number contains another. For example, if there are eight oranges and six lemons in a bowl of fruit, then the ratio of oranges to lemons is eight to six (that is, 8:6, which is equivalent to the ...
s, the harmonic mean provides the correct
average
In ordinary language, an average is a single number taken as representative of a list of numbers, usually the sum of the numbers divided by how many numbers are in the list (the arithmetic mean). For example, the average of the numbers 2, 3, 4, 7 ...
. 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 the entire journey
=
However, if the vehicle travels for a certain amount of ''time'' at a speed ''x'' and then the same amount of time at 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
The same principle applies to more than two segments: given a series of sub-trips at different speeds, if each sub-trip covers 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
In mathematics, the harmonic mean is one of several kinds of average, and in particular, one of the Pythagorean means. It is sometimes appropriate for situations when the average rate is desired.
The harmonic mean can be expressed as the recipro ...
or
weighted arithmetic mean
The weighted arithmetic mean is similar to an ordinary arithmetic mean (the most common type of average), except that instead of each of the data points contributing equally to the final average, some data points contribute more than others. The ...
is needed. For the arithmetic mean, the speed of each portion of the trip is weighted by 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 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 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.
Density
Similarly, if one wishes to estimate the density of an
alloy
An alloy is a mixture of chemical elements of which at least one is a metal. Unlike chemical compounds with metallic bases, an alloy will retain all the properties of a metal in the resulting material, such as electrical conductivity, ductilit ...
given the densities of its constituent elements and their mass fractions (or, equivalently, percentages by mass), then the predicted density of 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
In engineering and science, dimensional analysis is the analysis of the relationships between different physical quantities by identifying their base quantities (such as length, mass, time, and electric current) and units of measure (such as mi ...
to the problem while labeling the mass units by element and making sure that only like element-masses cancel makes this clear.
Electricity
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 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
capacitor
A capacitor is a device that stores electrical energy in an electric field by virtue of accumulating electric charges on two close surfaces insulated from each other. It is a passive electronic component with two terminals.
The effect of ...
s in series or to
inductor
An inductor, also called a coil, choke, or reactor, is a passive two-terminal electrical component that stores energy in a magnetic field when electric current flows through it. An inductor typically consists of an insulated wire wound into a c ...
s 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, the sum of ''x'' and ''y'' (100 Ω). This principle applies to
capacitor
A capacitor is a device that stores electrical energy in an electric field by virtue of accumulating electric charges on two close surfaces insulated from each other. It is a passive electronic component with two terminals.
The effect of ...
s in parallel or to
inductor
An inductor, also called a coil, choke, or reactor, is a passive two-terminal electrical component that stores energy in a magnetic field when electric current flows through it. An inductor typically consists of an insulated wire wound into a c ...
s 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 the three crystallographic directions.
Optics
As for other
optic equation
In number theory, the optic equation is an equation that requires the sum of the reciprocals of two positive integers ''a'' and ''b'' to equal the reciprocal of a third positive integer ''c'':Dickson, L. E., ''History of the Theory of Numbers, V ...
s, the
thin lens equation = + 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.
In finance
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 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 a roundtrip journey (see above).
For example, consider two firms, one with a
market capitalization of $150 billion and earnings of $5 billion (P/E of 30) and one with a market capitalization of $1 billion and earnings of $1 million (P/E of 1000). Consider an
index made of the two stocks, with 30% invested in the first and 70% invested in the second. We want to calculate the P/E ratio of this index.
Using the weighted arithmetic mean (incorrect):
:
Using the weighted harmonic mean (correct):
:
Thus, the correct P/E of 93.46 of this index can only be found using the weighted harmonic mean, while the weighted arithmetic mean will significantly overestimate it.
In geometry
In any
triangle
A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC.
In Euclidean geometry, any three points, when non- colline ...
, the radius of the
incircle
In geometry, the incircle or inscribed circle of a triangle is the largest circle that can be contained in the triangle; it touches (is tangent to) the three sides. The center of the incircle is a triangle center called the triangle's incenter. ...
is one-third of the harmonic mean of the
altitudes
Altitude or height (also sometimes known as depth) is a distance measurement, usually in the vertical or "up" direction, between a reference datum and a point or object. The exact definition and reference datum varies according to the context ...
.
For any point P on the
minor arc BC of the
circumcircle
In geometry, the circumscribed circle or circumcircle of a polygon is a circle that passes through all the vertices of the polygon. The center of this circle is called the circumcenter and its radius is called the circumradius.
Not every polyg ...
of an
equilateral triangle
In geometry, an equilateral triangle is a triangle in which all three sides have the same length. In the familiar Euclidean geometry, an equilateral triangle is also equiangular; that is, all three internal angles are also congruent to each oth ...
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 a
right triangle
A right triangle (American English) or right-angled triangle ( British), or more formally an orthogonal triangle, formerly called a rectangled triangle ( grc, ὀρθόσγωνία, lit=upright angle), is a triangle in which one angle is a right a ...
with legs ''a'' and ''b'' and
altitude
Altitude or height (also sometimes known as depth) is a distance measurement, usually in the vertical or "up" direction, between a reference datum and a point or object. The exact definition and reference datum varies according to the context ...
''h'' from the
hypotenuse
In geometry, a hypotenuse is the longest side of a right-angled triangle, the side opposite the right angle. The length of the hypotenuse can be found using the Pythagorean theorem, which states that the square of the length of the hypotenuse e ...
to the right angle, is half the harmonic mean of and .
Let ''t'' and ''s'' (''t'' > ''s'') be the sides of the two
inscribed squares in a right triangle with hypotenuse ''c''. Then equals half the harmonic mean of and .
Let a
trapezoid
A quadrilateral with at least one pair of parallel sides is called a trapezoid () in American and Canadian English. In British and other forms of English, it is called a trapezium ().
A trapezoid is necessarily a convex quadrilateral in Eu ...
have vertices A, B, C, and D in sequence and have parallel sides AB and CD. Let E be the intersection of the
diagonal
In geometry, a diagonal is a line segment joining two vertices of a polygon or polyhedron, when those vertices are not on the same edge. Informally, any sloping line is called diagonal. The word ''diagonal'' derives from the ancient Greek δ ...
s, 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 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 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 other sciences
In
computer science
Computer science is the study of computation, automation, and information. Computer science spans theoretical disciplines (such as algorithms, theory of computation, information theory, and automation) to practical disciplines (includi ...
, specifically
information retrieval and
machine learning
Machine learning (ML) is a field of inquiry devoted to understanding and building methods that 'learn', that is, methods that leverage data to improve performance on some set of tasks. It is seen as a part of artificial intelligence.
Machine ...
, the harmonic mean of the
precision
Precision, precise or precisely may refer to:
Science, and technology, and mathematics Mathematics and computing (general)
* Accuracy and precision, measurement deviation from true value and its scatter
* Significant figures, the number of digit ...
(true positives per predicted positive) and the
recall
Recall may refer to:
* Recall (bugle call), a signal to stop
* Recall (information retrieval), a statistical measure
* ''ReCALL'' (journal), an academic journal about computer-assisted language learning
* Recall (memory)
* ''Recall'' (Overwatch ...
(true positives per real positive) is often used as an aggregated performance score for the evaluation of algorithms and systems: the
F-score
In statistical analysis of binary classification, the F-score or F-measure is a measure of a test's accuracy. It is calculated from the precision and recall of the test, where the precision is the number of true positive results divided by the n ...
(or F-measure). This is used in information retrieval because only the positive class is of
relevance
Relevance is the concept of one topic being connected to another topic in a way that makes it useful to consider the second topic when considering the first. The concept of relevance is studied in many different fields, including cognitive sci ...
, 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 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 , which is equal to 2.4 hours, to drain the pool together. This is one-half of the harmonic mean of 6 and 4: . 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
Hydrology () is the scientific study of the movement, distribution, and management of water on Earth and other planets, including the water cycle, water resources, and environmental watershed sustainability. A practitioner of hydrology is call ...
, the harmonic mean is similarly used to average
hydraulic conductivity
Hydraulic conductivity, symbolically represented as (unit: m/s), is a property of porous materials, soils and rocks, that describes the ease with which a fluid (usually water) can move through the pore space, or fractures network. It depends on ...
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
Sabermetrics, or originally SABRmetrics, is the empirical analysis of baseball, especially baseball statistics that measure in-game activity. Sabermetricians collect and summarize the relevant data from this in-game activity to answer specific ques ...
, a player's
Power–speed number
Power–speed number or power/speed number (PSN) is a sabermetrics baseball statistic developed by baseball author and analyst Bill James which combines a player's home run and stolen base numbers into one number.http://www.chesscafe.com/text/hei ...
is the harmonic mean of their
home run
In baseball, a home run (abbreviated HR) is scored when the ball is hit in such a way that the batter is able to circle the bases and reach home plate safely in one play without any errors being committed by the defensive team. A home run i ...
and
stolen base totals.
In
population genetics
Population genetics is a subfield of genetics that deals with genetic differences within and between populations, and is a part of evolutionary biology. Studies in this branch of biology examine such phenomena as Adaptation (biology), adaptation, ...
, 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
Bottleneck literally refers to the narrowed portion (neck) of a bottle near its opening, which limit the rate of outflow, and may describe any object of a similar shape. The literal neck of a bottle was originally used to play what is now known as ...
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 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 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 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
Nuclear physics is the field of physics that studies atomic nuclei and their constituents and interactions, in addition to the study of other forms of nuclear matter.
Nuclear physics should not be confused with atomic physics, which studies the ...
the average mass per particle of a mixture consisting of different species (e.g., molecules or isotopes) is given by the harmonic mean of the individual species' masses weighted by their respective mass fraction.
Beta distribution
The harmonic mean of a
beta distribution
In probability theory and statistics, the beta distribution is a family of continuous probability distributions defined on the interval , 1in terms of two positive parameters, denoted by ''alpha'' (''α'') and ''beta'' (''β''), that appear as ...
with shape parameters ''α'' and ''β'' is:
:
The harmonic mean with ''α'' < 1 is undefined because its defining expression is not bounded in
, 1
The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline o ...
Letting ''α'' = ''β''
:
showing that for ''α'' = ''β'' the harmonic mean ranges from 0 for ''α'' = ''β'' = 1, to 1/2 for ''α'' = ''β'' → ∞.
The following are the limits with one parameter finite (non-zero) and the other parameter approaching these limits:
:
With the geometric mean the harmonic mean may be useful in maximum likelihood estimation in the four parameter case.
A second harmonic mean (''H''
1 − X) also exists for this distribution
:
This harmonic mean with ''β'' < 1 is undefined because its defining expression is not bounded in
0, 1
Letting ''α'' = ''β'' in the above expression
:
showing that for ''α'' = ''β'' the harmonic mean ranges from 0, for ''α'' = ''β'' = 1, to 1/2, for ''α'' = ''β'' → ∞.
The following are the limits with one parameter finite (non zero) and the other approaching these limits:
:
Although both harmonic means are asymmetric, when ''α'' = ''β'' the two means are equal.
Lognormal distribution
The harmonic mean ( ''H'' ) of the
lognormal distribution
In probability theory, a log-normal (or lognormal) distribution is a continuous probability distribution of a random variable whose logarithm is normally distributed. Thus, if the random variable is log-normally distributed, then has a normal ...
of a random variable ''X'' is
[Aitchison J, Brown JAC (1969). The lognormal distribution with special reference to its uses in economics. Cambridge University Press, New York]
:
where ''μ'' and ''σ''
2 are the parameters of the distribution, i.e. the mean and variance of the distribution of the natural logarithm of ''X''.
The harmonic and arithmetic means of the distribution are related by
:
where ''C''
v and ''μ''
* are the
coefficient of variation and the mean of the distribution respectively..
The geometric (''G''), arithmetic and harmonic means of the distribution are related by
[Rossman LA (1990) Design stream flows based on harmonic means. J Hydr Eng ASCE 116(7) 946–950]
:
Pareto distribution
The harmonic mean of type 1
Pareto distribution is
[Johnson NL, Kotz S, Balakrishnan N (1994) Continuous univariate distributions Vol 1. Wiley Series in Probability and Statistics.]
:
where ''k'' is the scale parameter and ''α'' is the shape parameter.
Statistics
For a random sample, the harmonic mean is calculated as above. Both the
mean
There are several kinds of mean in mathematics, especially in statistics. Each mean serves to summarize a given group of data, often to better understand the overall value (magnitude and sign) of a given data set.
For a data set, the '' ari ...
and the
variance
In probability theory and statistics, variance is the expectation of the squared deviation of a random variable from its population mean or sample mean. Variance is a measure of dispersion, meaning it is a measure of how far a set of numbe ...
may be
infinite
Infinite may refer to:
Mathematics
* Infinite set, a set that is not a finite set
*Infinity, an abstract concept describing something without any limit
Music
*Infinite (group), a South Korean boy band
*''Infinite'' (EP), debut EP of American m ...
(if it includes at least one term of the form 1/0).
Sample distributions of mean and variance
The mean of the sample ''m'' is asymptotically distributed normally with variance ''s''
2.
:
The variance of the mean itself is
[Zelen M (1972) Length-biased sampling and biomedical problems. In: Biometric Society Meeting, Dallas, Texas]
:
where ''m'' is the arithmetic mean of the reciprocals, ''x'' are the variates, ''n'' is the population size and ''E'' is the expectation operator.
Delta method
Assuming that the variance is not infinite and that the
central limit theorem
In probability theory, the central limit theorem (CLT) establishes that, in many situations, when independent random variables are summed up, their properly normalized sum tends toward a normal distribution even if the original variables themsel ...
applies to the sample then using the
delta method
In statistics, the delta method is a result concerning the approximate probability distribution for a function of an asymptotically normal statistical estimator from knowledge of the limiting variance of that estimator.
History
The delta meth ...
, the variance is
:
where ''H'' is the harmonic mean, ''m'' is the arithmetic mean of the reciprocals
:
''s''
2 is the variance of the reciprocals of the data
:
and ''n'' is the number of data points in the sample.
Jackknife method
A
jackknife method of estimating the variance is possible if the mean is known.
[Lam FC (1985) Estimate of variance for harmonic mean half lives. J Pharm Sci 74(2) 229-231] This method is the usual 'delete 1' rather than the 'delete m' version.
This method first requires the computation of the mean of the sample (''m'')
:
where ''x'' are the sample values.
A series of value ''w
i'' is then computed where
:
The mean (''h'') of the ''w''
i is then taken:
:
The variance of the mean is
:
Significance testing and
confidence intervals for the mean can then be estimated with the
t test
A ''t''-test is any statistical hypothesis testing, statistical hypothesis test in which the test statistic follows a Student's t-distribution, Student's ''t''-distribution under the null hypothesis. It is most commonly applied when the test stati ...
.
Size biased sampling
Assume a random variate has a distribution ''f''( ''x'' ). Assume also that the likelihood of a variate being chosen is proportional to its value. This is known as length based or size biased sampling.
Let ''μ'' be the mean of the population. Then the
probability density function
In probability theory, a probability density function (PDF), or density of a continuous random variable, is a function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) ca ...
''f''*( ''x'' ) of the size biased population is
:
The expectation of this length biased distribution E
*( ''x'' ) is
: