n.

平均、平均値

adj.

平均する、平均の

関: even、mean、mean value

WordNet

(sports) the ratio of successful performances to opportunities
achieve or reach on average; "He averaged a C"
a statistic describing the location of a distribution; "it set the norm for American homes" (同)norm
an intermediate scale value regarded as normal or usual; "he is about average in height"; "the snowfall this month is below average"
amount to or come to an average, without loss or gain; "The number of hours I work per work averages out to 40" (同)average out
compute the average of (同)average out
lacking exceptional quality or ability; "a novel of average merit"; "only a fair performance of the sonata"; "in fair health"; "the caliber of the students has gone from mediocre to above average"; "the performance was middling at best" (同)fair, mediocre, middling
around the middle of a scale of evaluation; "an orange of average size"; "intermediate capacity"; "medium bombers" (同)intermediate, medium
approximating the statistical norm or average or expected value; "the average income in New England is below that of the nation"; "of average height for his age"; "the mean annual rainfall" (同)mean
lacking special distinction, rank, or status; commonly encountered; "average people"; "the ordinary (or common) man in the street" (同)ordinary
used as an intensive especially to indicate something unexpected; "even an idiot knows that"; "declined even to consider the idea"; "I dont have even a dollar!"
become even or more even; "even out the surface" (同)even out
make even or more even (同)even out
equal in degree or extent or amount; or equally matched or balanced; "even amounts of butter and sugar"; "on even terms"; "it was a fifty-fifty (or even) split"; "had a fifty-fifty (or even) chance"; "an even fight" (同)fifty-fifty
occurring at fixed intervals; "a regular beat"; "the even rhythm of his breathing" (同)regular
symmetrically arranged; "even features"; "regular features"; "a regular polygon" (同)regular
to a greater degree or extent; used with comparisons; "looked sick and felt even worse"; "an even (or still) more interesting problem"; "still another problem must be solved"; "a yet sadder tale" (同)yet, still
being level or straight or regular and without variation as e.g. in shape or texture; or being in the same plane or at the same height as something else (i.e. even with); "an even application of varnish"; "an even floor"; "the road was not very even"; "the picture is even with the window"
divisible by two
in spite of; notwithstanding; "even when he is sick, he works"; "even with his head start she caught up with him"
to the full extent; "loyal even unto death"
have a specified degree of importance; "My ex-husband means nothing to me"; "Happiness means everything"
an average of n numbers computed by adding some function of the numbers and dividing by some function of n (同)mean_value
mean or intend to express or convey; "You never understand what I mean!"; "what do his words intend?" (同)intend
denote or connote; "`maison means `house in French"; "An example sentence would show what this word means" (同)intend, signify, stand_for
(used of persons or behavior) characterized by or indicative of lack of generosity; "a mean person"; "he left a miserly tip" (同)mingy, miserly, tight
destine or designate for a certain purpose; "These flowers were meant for you"
excellent; "famous for a mean backhand"
the state of being that is average; indicates normality but with connotations of mediocrity
ordinariness as a consequence of being average and not outstanding (同)mediocrity
a measure of a students academic achievement at a college or university; calculated by dividing the total number of grade points received by the total number attempted (同)GPA
total cost for all units bought (or produced) divided by the number of units

PrepTutorEJDIC

『平均』;並み,標準 / 平均値 / 『平均の』;並みの,標準の / 〈数〉'を'『平均する』 / 平均して…'を'する(受動態にできない) / 平均して…となる
『平らな』,平たんな / 『同じ高さの』,同一平面の,平行の / (運動・動作・品質などが)『均一の』,規則的な,一様の,むらのない / (数・量などが)『同じの』,等しい,釣り合のとれた互角の / 『偶数の』 / 公平な,公正な(fair) / 平静な,穏やかな,落ち着いた(calm) / 貸借にない,清算済みの / 端数のない,ちょうどの,きっかりの / 《意味を強めて,不審や意外の念を含み》…『でさえも』,までも / 《比較級の前に用いて》『なおいっそう』,さらに(still,yet) / 《形容詞の前または後の用いて》(…と)すら言える;《古》《時を表す副詞の前に用いて》ちょうど / …‘を'平らにする,ならす / …‘を'釣り合わせる,平均させる《+『up』+『名』,+『名』+『up』》 / …‘の'変動をなくする,‘を'安定させる《+『out』+『名』,+『名』+『out』》
夕べ,晩
〈人が〉…‘を'『意図する』,‘の'つもりである;…‘を'言おうとある;示そうとする / 〈単語・句・文章などが〉…‘を'『意味を持つ』,‘を'『意味する』 / (人に対して)…‘を'当てる,向ける《+『名』+『for』+『名』》 / 《受動態で》〈人・物〉‘を'(…)向きにする / 〈物事が〉…‘を'示す,表す / (結果として)〈物事が〉…‘を'もたらす,生じさせる / 《『mean』+『for』+『名』+『to』 do》《米話》(…が…するような)つもりでいる
《名詞の前にのみ用いて》(品質・価値・才能などが)『劣った』,つまらない / (人・行為などが)『卑劣な』,さもしい / 《古》(身分・地位などが)卑しい / 《まれ》(身なり・建物などが)みずぼらしい,見るもあわれな / 《米話》(動物が)御しにくい;(一般に)扱いにくい
中庸,中道 / (数学で) / 平均[値];(特に)算術平均,相加平均 / (比例式の)中項,内項 / (両極端の)中間の,中央に位置する / (大きさ・品質・程度などが)平均的な,平均の

Wikipedia preview

出典(authority):フリー百科事典『ウィキペディア（Wikipedia）』「2013/05/24 16:17:15」(JST)

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[Wiki en表示]

In colloquial language average usually means the sum of a list of numbers divided by the size of the list, in other words the arithmetic mean. However it can alternatively mean the median, the mode, or some other central or typical value. In statistics, these are all known as measures of central tendency. The concept of an average can be extended in various ways in mathematics, but in those contexts it is usually referred to as a mean (for example the mean of a function)

1 Calculation
- 1.1 Pythagorean means
  - 1.1.1 Arithmetic mean
  - 1.1.2 Geometric mean
  - 1.1.3 Harmonic mean
  - 1.1.4 Inequality concerning AM, GM, and HM
- 1.2 Mode
- 1.3 Median
2 Summary of Types
3 Miscellaneous types
- 3.1 Average Percentage Return and CAGR
4 In data streams
5 Etymology
6 See also
7 Notes
8 References
9 External links

Calculation [edit]

Pythagorean means [edit]

The three most common averages are the Pythagorean means – the arithmetic mean, the geometric mean, and the harmonic mean.

Arithmetic mean [edit]

Main article: Arithmetic mean

If n numbers are given, each number denoted by a_i, where i = 1, ..., n, the arithmetic mean is the [sum] of the a_i's divided by n or

The arithmetic mean, often simply called the mean, of two numbers, such as 2 and 8, is obtained by finding a value A such that 2 + 8 = A + A. One may find that A = (2 + 8)/2 = 5. Switching the order of 2 and 8 to read 8 and 2 does not change the resulting value obtained for A. The mean 5 is not less than the minimum 2 nor greater than the maximum 8. If we increase the number of terms in the list for which we want an average, we get, for example, that the arithmetic mean of 2, 8, and 11 is found by solving for the value of A in the equation 2 + 8 + 11 = A + A + A. One finds that A = (2 + 8 + 11)/3 = 7.

Geometric mean [edit]

The geometric mean of n non-negative numbers is obtained by multiplying them all together and then taking the nth root. In algebraic terms, the geometric mean of a₁, a₂, ..., a_n is defined as

Geometric mean can be thought of as the antilog of the arithmetic mean of the logs of the numbers.

Example: Geometric mean of 2 and 8 is

Harmonic mean [edit]

Harmonic mean for a non-empty collection of numbers a₁, a₂, ..., a_n, all different from 0, is defined as the reciprocal of the arithmetic mean of the reciprocals of the a_i's:

One example where it is useful is calculating the average speed for a number of fixed-distance trips. For example, if the speed for going from point A to B was 60 km/h, and the speed for returning from B to A was 40 km/h, then the average speed is given by

Inequality concerning AM, GM, and HM [edit]

A well known inequality concerning arithmetic, geometric, and harmonic means for any set of positive numbers is

It is easy to remember noting that the alphabetical order of the letters A, G, and H is preserved in the inequality. See Inequality of arithmetic and geometric means.

Mode [edit]

Comparison of arithmetic mean, median and mode of two log-normal distributions with different skewness.

Main article: Mode (statistics)

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. The mode is not necessarily uniquely defined; for example, the list (1, 2, 2, 3, 3, 5) has the two modes 2 and 3. The mode is more meaningful and potentially useful if there are many numbers in the list, and the frequency of the numbers progresses smoothly (e.g., if out of a group of 1000 people, 30 people weigh 61 kg, 32 weigh 62 kg, 29 weigh 63 kg, and all the other possible weights occur less frequently, then 62 kg is the mode).

The mode has the advantage that it can be used with non-numerical data (e.g., red cars are most frequent), while other averages cannot.

Median [edit]

Main article: Median

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.)

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 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.

Summary of Types [edit]

The table of mathematical symbols explains the symbols used below.

Name	Equation or description
Arithmetic mean
Median	The middle value that separates the higher half from the lower half of the data set
Geometric median	A rotation invariant extension of the median for points in Rⁿ
Mode	The most frequent value in the data set
Geometric mean
Harmonic mean
Quadratic mean (or RMS)	$\sqrt{\frac{1}{n} \sum_{i=1}^{n} x_i^2} = <p>\sqrt {\frac{x_1^2 + x_2^2 + \cdots + x_n^2}{n}} </p>$
Generalized mean
Weighted mean
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
Midrange
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
Annualization

Miscellaneous types [edit]

Other more sophisticated averages are: trimean, trimedian, and normalized mean, with their generalizations.^[1]

One can create one's own average metric using the generalized f-mean:

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^[2] for defining an average takes any function g(x₁, x₂, ..., 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(x₁, x₂, ..., 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(x₁, x₂, ..., x_n) = x₁+x₂+ ··· + x_n provides the arithmetic mean. The function g(x₁, x₂, ..., x_n) = x₁x₂···x_n (where the list elements are positive numbers) provides the geometric mean. The function g(x₁, x₂, ..., x_n) = −(x₁⁻¹+x₂⁻¹+ ··· + x_n⁻¹) (where the list elements are positive numbers) provides the harmonic mean.^[2]

Average Percentage Return and CAGR [edit]

Main article: Compound annual growth rate

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 the return in the second year is +60%, then the average percentage return or CAGR, R, can be obtained by solving the equation: (1 − 10%) × (1 + 60%) = (1 − 0.1) × (1 + 0.6) = (1 + R) × (1 + R). The value of R 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. Note that 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%.

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 return, R, that is the solution of the following equation: (1 − 0.23)^0.5 × (1 + 0.13)^2.5 = (1 + R)^0.5+2.5, giving an average percentage return R of 0.0600 or 6.00%.

In data streams [edit]

The concept of an average can be applied to a stream of data as well as a bounded set, the goal being to find a value that recent data clusters about in some way. The stream may be distributed in time, as in samples taken by some data acquisition system from which we want to remove noise, or in space, as in pixels in an image from which we want to extract some property. An easy-to-understand and widely used application of average to a stream is the simple moving average in which we compute the arithmetic mean of the most recent N data items in the stream. To advance one position in the stream, we add 1/N times the new data item and subtract 1/N times the data item N places back in the stream.

Update rule for a window of size upon seeing new element :

Etymology [edit]

"Few words have received more etymological investigation." ^[3] In the 16th century average meant a customs duty, or the like, and was used in the Mediterranean area. It came to mean the cost of damage sustained at sea. From that came an "average adjuster" who decided how to apportion a loss between the owners and insurers of a ship and cargo.

Marine damage is either particular average, which is borne only by the owner of the damaged property, or general average, 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".

The root is found in Arabic as awar, in Italian as avaria, in French as avarie and in Dutch as averij. It is unclear in which language the word first appeared.

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 Domesday Book (1085).

Notes [edit]

^ Merigo, Jose M.; Cananovas, Montserrat (2009). "The Generalized Hybrid Averaging Operator and its Application in Decision Making". Journal of Quantitative Methods for Economics and Business Administration 9: 69–84. ISSN 1886-516X.
^ ^a ^b John Bibby (1974). “Axiomatisations of the average and a further generalisation of monotonic sequences”. Glasgow Mathematical Journal, vol. 15, pp. 63–65.
^ Oxford English Dictionary

References [edit]

Hardy, G.H.; Littlewood, J.E.; Pólya, G. (1988). Inequalities (2nd ed.). Cambridge University Press. ISBN 978-0-521-35880-4

External links [edit]

Look up average in Wiktionary, the free dictionary.

Median as a weighted arithmetic mean of all Sample Observations
Calculations and comparison between arithmetic and geometric mean of two values

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