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Text of the page (random words):
e represents the theoretical distribution used to generate the points with the gray histogram depicting the actual distribution four means of two numbers a and b constructed as chords on a semicircle the arithmetic geometric and harmonic means are sometimes referred to as the pythagorean means 3 this type of construction highlights the ordering of the values of the different means the most commonly used definition of the average is the arithmetic mean 4 also known as arithmetic average i e the sum divided by the count so the average of the list of numbers 2 3 4 7 9 is generally considered to be 2 3 4 7 9 5 25 5 5 the arithmetic mean of a set of numbers x 1 x 2 x n is typically denoted using an overhead bar x displaystyle bar x note 1 if the numbers are from observing a sample of a larger group the arithmetic mean is termed the sample mean x displaystyle bar x to distinguish it from the group mean or expected value of the underlying distribution denoted μ displaystyle mu or μ x displaystyle mu _ x note 2 5 however other meanings are sometimes used depending on the context which can lead to confusion for instance in teaching average sometimes refers to the three ms mean median and mode 6 7 8 9 10 11 12 the median defined as the value in the center after sorting the group is usually used as the average in situations where the data is skewed or has outliers in order to focus on the main part of the group rather than the long tail for example the average personal income is usually given as the median income so that it represents the majority of the population rather than being overly influenced by the much higher incomes of the few rich people 13 the harmonic mean defined as the reciprocal of the mean of the reciprocals is used in a variety of situations involving rates or ratios such as computing the average speed from multiple measurements taken over the same distance 14 indeed unlike an arithmetic mean or median of speeds a harmonic mean of speeds will give the value of the constant speed that would cause one to travel the same distance in the same amount of time the mode represents the most common value found in the group it can be used when the data is categorical rather than numeric 15 when the frequency of each value is relevant such as where a histogram bar chart or probability density function is being referenced 16 or to find a value that represents the majority of the group 17 other statistics that can be used as an average include the mid range the quadratic mean or the geometric mean but they are rarely referred to as the average central tendency edit for the graph network concept see centrality in statistics a central tendency or measure of central tendency is a central or typical value for a probability distribution 18 informally measures of central tendency are often called averages the term central tendency dates from the late 1920s 19 the most common measures of central tendency are the arithmetic mean the median and the mode a middle tendency can be calculated for either a finite set of values or for a theoretical distribution such as the normal distribution occasionally authors use central tendency to denote the tendency of quantitative data to cluster around some central value 19 20 the central tendency of a distribution is typically contrasted with its dispersion or variability dispersion and central tendency are the often characterized properties of distributions analysis may judge whether data has a strong or a weak central tendency based on its dispersion general properties edit all averages of a collection are somewhere within its bounding box and so for real numbers between its maximum and minimum therefore if a collection consists entirely of the same value any average of it is that value 21 most averages a are monotonic i e moving a member of it in one direction causes the average to move in the same direction or equivalently if two collections of numbers a and b have the same number of elements and they can be arranged such that each entry in a the corresponding entry in b then the average of a the average of b all commonly used averages are linearly homogeneous i e multiplying every value by the same scale factor multiplies the average by that same scale factor most averages b remain identical when the list of items is permuted i e the ordering does not matter statistical location edit see also average statistical location comparison of the arithmetic mean median and mode of two skewed log normal distributions geometric visualization of the mode median and mean of an arbitrary probability density function 22 in descriptive statistics the mean may be confused with the median mode or mid range as any of these may colloquially be called an average more formally a measure of central tendency the mean of a set of observations is the arithmetic average of the values however for skewed distributions the mean is not necessarily the same as the middle value median or the most likely value mode for example mean income is typically skewed upwards by a small number of people with very large incomes so that the majority have an income lower than the mean by contrast the median income is the level at which half the population is below and half is above the mode income is the most likely income and favors the larger number of people with lower incomes while the median and mode are often more intuitive measures for such skewed data many skewed distributions are in fact best described by their mean including the exponential and poisson distributions mean of a probability distribution edit main article expected value see also population mean the mean of a probability distribution is the long run arithmetic average value of a random variable having that distribution if the random variable is denoted by x displaystyle x then the mean is also known as the expected value of x displaystyle x denoted e x displaystyle e x for a discrete probability distribution the mean is given by x p x displaystyle textstyle sum xp x where the sum is taken over all possible values of the random variable and p x displaystyle p x is the probability mass function for a continuous distribution the mean is x f x d x displaystyle textstyle int _ infty infty xf x dx where f x displaystyle f x is the probability density function 23 in all cases including those in which the distribution is neither discrete nor continuous the mean is the lebesgue integral of the random variable with respect to its probability measure the mean need not exist or be finite for some probability distributions the mean is infinite or while for others the mean is undefined relationships between the mean median and mode edit main article nonparametric skew relationships between the mean median and mode for unimodal distributions the following bounds are known and are sharp 24 θ μ σ 3 displaystyle frac theta mu sigma leq sqrt 3 ν μ σ 0 6 displaystyle frac nu mu sigma leq sqrt 0 6 θ ν σ 3 displaystyle frac theta nu sigma leq sqrt 3 where μ is the mean ν is the median θ is the mode and σ is the standard deviation for every distribution 25 26 ν μ σ 1 displaystyle frac nu mu sigma leq 1 possible averages edit see also mean other means and central tendency solutions to variational problems name equation or description as solution to optimization problem arithmetic mean am x 1 n i 1 n x i 1 n x 1 x n displaystyle bar x frac 1 n sum _ i 1 n x_ i frac 1 n x_ 1 cdots x_ n for example the arithmetic mean of five values 4 36 45 50 75 is 4 36 45 50 75 5 210 5 42 displaystyle frac 4 36 45 50 75 5 frac 210 5 42 argmin x r i 1 n x x i 2 displaystyle underset x in mathbb r operatorname argmin sum _ i 1 n x x_ i 2 median 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 argmin x r i 1 n x x i displaystyle underset x in mathbb r operatorname argmin sum _ i 1 n x x_ i geometric median a rotation invariant extension of the median for points in r d displaystyle mathbb r d argmin x r d i 1 n x x i 2 displaystyle underset vec x in mathbb r d operatorname argmin sum _ i 1 n vec x vec x _ i _ 2 tukey median another rotation invariant extension of the median for points in r d displaystyle mathbb r d a point that maximizes the tukey depth argmax x r d min u r d i 1 n 1 if x i x u 0 0 otherwise 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 mode the most frequent value in the data set argmax x r i 1 n 1 if x x i 0 if x x i 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 geometric mean gm i 1 n x i n x 1 x 2 x n n displaystyle sqrt n prod _ i 1 n x_ i sqrt n x_ 1 cdot x_ 2 dotsb x_ n an average that is useful for sets of positive numbers interpreted according to their product and not their sum 1 for example the geometric mean of five values 4 36 45 50 75 is 4 36 45 50 75 1 5 24 300 000 5 30 displaystyle 4 times 36 times 45 times 50 times 75 frac 1 5 sqrt 5 24 300 000 30 argmin x r 0 i 1 n ln x ln x i 2 if x i 0 i 1 n 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 harmonic mean hm n 1 x 1 1 x 2 1 x n displaystyle frac n frac 1 x_ 1 frac 1 x_ 2 cdots frac 1 x_ n for example the harmonic mean of the five values 4 36 45 50 75 is 5 1 4 1 36 1 45 1 50 1 75 5 1 3 15 displaystyle frac 5 tfrac 1 4 tfrac 1 36 tfrac 1 45 tfrac 1 50 tfrac 1 75 frac 5 tfrac 1 3 15 argmin x r 0 i 1 n 1 x 1 x i 2 displaystyle underset x in mathbb r _ neq 0 operatorname argmin sum _ i 1 n left frac 1 x frac 1 x_ i right 2 contraharmonic mean x 1 2 x 2 2 x n 2 x 1 x 2 x n displaystyle frac x_ 1 2 x_ 2 2 cdots x_ n 2 x_ 1 x_ 2 cdots x_ n argmin x r i 1 n x i x x i 2 displaystyle underset x in mathbb r operatorname argmin sum _ i 1 n x_ i x x_ i 2 lehmer mean i 1 n x i p i 1 n x i p 1 displaystyle frac sum _ i 1 n x_ i p sum _ i 1 n x_ i p 1 argmin x r i 1 n x i p 1 x x i 2 displaystyle underset x in mathbb r operatorname argmin sum _ i 1 n x_ i p 1 x x_ i 2 quadratic mean or rms 1 n i 1 n x i 2 1 n x 1 2 x 2 2 x n 2 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 argmin x r 0 i 1 n x 2 x i 2 2 displaystyle underset x in mathbb r _ geq 0 operatorname argmin sum _ i 1 n x 2 x_ i 2 2 cubic mean 1 n i 1 n x i 3 3 1 n x 1 3 x 2 3 x n 3 3 displaystyle sqrt 3 frac 1 n sum _ i 1 n x_ i 3 sqrt 3 frac 1 n left x_ 1 3 x_ 2 3 cdots x_ n 3 right argmin x r 0 i 1 n x 3 x i 3 2 if x i 0 i 1 n 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 generalized mean 1 n i 1 n x i p p displaystyle sqrt p frac 1 n cdot sum _ i 1 n x_ i p argmin x r 0 i 1 n x p x i p 2 if x i 0 i 1 n 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 quasi arithmetic mean f 1 1 n k 1 n f x k displaystyle f 1 left frac 1 n sum _ k 1 n f x_ k right argmin x dom f i 1 n f x f x i 2 if f displaystyle underset x in operatorname dom f operatorname argmin sum _ i 1 n f x f x_ i 2 qquad text if f is monotonic weighted mean i 1 n w i x i i 1 n w i w 1 x 1 w 2 x 2 w n x n w 1 w 2 w n 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 argmin x r i 1 n w i x x i 2 displaystyle underset x in mathbb r operatorname argmin sum _ i 1 n w_ i x x_ i 2 truncated mean the arithmetic mean of data values after a certain number or proportion of the highest and lowest data values have been discarded interquartile mean a special case of the truncated mean using the interquartile range a special case of the inter quantile truncated mean which operates on quantiles often deciles or percentiles that are equidistant but on opposite sides of the median midrange 1 2 max x min x displaystyle frac 1 2 left max x min x right argmin x r max i 1 n x x i displaystyle underset x in mathbb r operatorname argmin underset i in 1 dots n operatorname max x x_ i winsorized mean similar to the truncated mean but rather than deleting the extreme values they are set equal to the largest and smallest values that remain medoid a representative object of a set x displaystyle mathcal x of objects with minimal sum of dissimilarities to all the objects in the set according to some dissimilarity function d displaystyle d argmin y x i 1 n d y x i displaystyle underset y in mathcal x operatorname argmin sum _ i 1 n d y x_ i even though perhaps not an average the τ displaystyle tau th quantile another summary statistic that generalizes the median can similarly be expressed as a solution to the optimization problem argmin x r i 1 n max 1 τ x i x τ x x i argmin x r i 1 n x x i 1 2 τ x displaystyle underset x in mathbb r operatorname argmin sum _ i 1 n max big 1 tau x_ i x tau x x_ i big underset x in mathbb r operatorname argmin sum _ i 1 n big x x_ i 1 2 tau x big which aims to minimize the total tilted absolute value loss or quantile loss or pinball loss other more sophisticated averages are trimmore sophistiean trimedian and normalized mean with their generalizations 27 in a more general fashion one can create their own average metric using the generalized f mean y f 1 1 n f x 1 f x 2 f x n displaystyle y f 1 left frac 1 n left f x_ 1 f x_ 2 cdots f x_ n right right where f is any invertible function the harmonic mean is an example of this using f x 1 x and the geometric mean is another using f x log x however this method for generating means is not general enough to capture all averages a more general method 28 failed verification for defining an average takes any function g x 1 x 2 x n of a list of arguments that is continuous strictly increasing in each argument and symmetric invariant under permutation of the arguments the average y is then the value that when replacing each member of the list results in the same function value g y y y g x 1 x 2 x n 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 g x 1 x 2 x n x 1 x 2 x n provides the arithmetic mean the function g x 1 x 2 x n x 1 x 2 x n where the list elements are positive numbers provides the geometric mean the function g x 1 x 2 x n x 1 1 x 2 1 x n 1 1 where the list elements are positive numbers provides the harmonic mean 28 pythagorean means edit main article 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 29 because of their importan...