Mean

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In mathematics, mean has several different definitions depending on the context.

In probability and statistics, the population averages and the expected value is used synonymously to refer to a measure of the central tendency of either the probability distribution or of the random variable characterized by the distribution. In the case of the discrete probability distribution of the random variable X , the mean is equal to the sum of every possible value weighed by the probability of that value; that is, calculated by taking the product of any possible value x from X and its probability P ( x ), and then adding all these products together same, give ${\ displaystyle \ mu = \ number of xP (x)}$ . The analog formula is valid for the case of continuous probability distributions. Not every probability distribution has a clear meaning; see Cauchy's distribution as an example. In addition, for some unlimited average distributions.

For a data set, mean arithmetic, mathematical expectations, and sometimes mean synonyms are used to refer to the center values ​​of a series of discrete numbers: in particular, the sum of the values ​​divided by the sum of values. The arithmetic mean of a set of numbers x ₁ , x ₂ n is usually denoted by $Â\; ÂÂ\; Â\; Â\; ÂÂ\; Â\; Â\; Â\; Â{\displaystyle Â\; Â\; Â\; Â\; Â\; Â\; Â\; ÂÂ\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; ÂÂ\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\stackrel{}{x}}$



















< Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â {\ displaystyle {\ bar {x}}} Â Â , pronounced " x bar". If the data set is based on a series of observations obtained with a sampling of the statistical population, the arithmetic mean is called sample mean (denoted



















< Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â {\ displaystyle {\ bar {x}}} ) to distinguish them from the mean population (denoted ${\ displaystyle \ mu}$ or ${\ displaystyle \ mu_ {x}}$ ).

For a limited population, the population mean of the property is equal to the arithmetic mean of the given property while considering every member of the population. For example, the average high population is equal to the sum of the height of each individual divided by the total number of individuals. The mean of the sample may be different from the population mean, especially for small samples. The law in large numbers states that the larger the sample size, the more likely that the sample mean will be near the population mean.

Beyond probability and statistics, various other meanings of "mean" are often used in geometry and analysis; examples are given below.

Video Mean

Jenis mean

Pythagoras berarti

Arithmetic mean (AM)

The aritmetik mean (atau hanya "mean") dari sampel ${\ displaystyle x_ {1}, x_ {2}, \ ldots, x_ {n}}  ÂÂ$ , biasanya dilambangkan dengan ${\ displaystyle {\ bar {x}}}  ÂÂ$ , adalah jumlah dari nilai-nilai sampel dibagi dengan jumlah item dalam contoh

${\ displaystyle {\ bar {x}} = {\ frac {1} {n}} \ kiri (\ jumlah _ {i = 1} ^ {n} {x_ {i}} \ right) = {\ frac {x_ {1} x_ {2} \ cdots x_ {n}} {n}}}  ÂÂ$

Sebagai contoh, mean aritmetik dari lima nilai: 4, 36, 45, 50, 75 adalah:

${\ displaystyle {\ frac {4 36 45 50 75} {5}} = {\ frac {210} {5}} = 42.}  ÂÂ$

Geometric mean (GM)

Mean geometrik adalah rata-rata yang berguna untuk set angka positif yang ditafsirkan sesuai dengan produk mereka dan bukan jumlah mereka (seperti halnya dengan mean aritmetik) mis. tingkat pertumbuhan.

${\ displaystyle {\ bar {x}} = \ kiri (\ prod _ {i = 1} ^ {n} {x_ {i}} \ right) ^ {\ tfrac {1} {n}} = \ kiri (x_ {1} x_ {2} \ cdots x_ {n} \ right) ^ {1/n}}  ÂÂ$

Sebagai contoh, rata-rata geometrik dari lima nilai: 4, 36, 45, 50, 75 adalah:

${\ displaystyle (4 \ kali 36 \ kali 45 \ kali 50 \ kali 75) ^ {1/5} = {\ sqrt [{5}] {24 \; 300 \; 000}} = 30.}  ÂÂ$

Harmonic mean (HM)

Mean harmonik adalah rata-rata yang berguna untuk set angka yang didefinisikan dalam kaitannya dengan beberapa unit, misalnya kecepatan (jarak per satuan waktu).

${\ displaystyle {\ bar {x}} = n \ cdot \ left (\ jumlah _ {i = 1} ^ {n} {\ frac {1} {x_ { i}}} \ right) ^ {- 1}}  ÂÂ$

Sebagai contoh, rata-rata harmonik dari lima nilai: 4, 36, 45, 50, 75 adalah

${\ displaystyle {\ frac {5} {{\ tfrac {1} {4}} {\ tfrac {1} {36}} {\ tfrac {1} {45}} {\ tfrac {1} {50}} {\ tfrac {1} {75}}}} = {\ frac {5} {\; {\ tfrac {1} {3}} \; }} = 15.}  ÂÂ$

Hubungan antara AM, GM, dan HM

AM, GM, dan HM memuaskan ketidaksetaraan ini:

${\ displaystyle \ mathrm {AM} \ geq \ mathrm {GM} \ geq \ mathrm {HM} \,}  ÂÂ$

Equality applies if and only if all elements of the given sample are the same.

Location statistics

In descriptive statistics, the mean may be confused with the median, mode or mid-range, since all of these can be called "average" (more formal, central tendency measure). The mean of a set of observations is the arithmetic average of the values; However, for oblique distributions, the mean does not have to be the same as the median, or the most probable value (mode). For example, average income is usually inclined upward by a small number of people with very large incomes, so the majority have lower than average earnings. Conversely, the average income is the rate at which half the population is below and half above. Fashion revenue is the most likely income, and more people are earning less. While medians and modes are often a more intuitive measure for such oblique data, many of the skewed distributions are actually best explained by their means, including the exponential and Poisson distributions.

The average of the probability distribution

Mean dari distribusi probabilitas adalah nilai rata-rata aritmatika jangka panjang dari variabel acak yang memiliki distribusi itu. Dalam konteks ini, ia juga dikenal sebagai nilai yang diharapkan. Untuk distribusi probabilitas diskrit, rata-rata diberikan oleh ${\ displaystyle \ textstyle \ jumlah xP (x)}  ÂÂ$ , di mana jumlah tersebut diambil alih semua nilai yang mungkin dari variabel acak dan ${\ displaystyle P (x)}  ÂÂ$ adalah fungsi massa probabilitas. Untuk distribusi berkelanjutan, mean adalah ${\ displaystyle \ textstyle \ int _ {- \ infty} ^ {\ infty} xf (x) \, dx}  ÂÂ$ , di mana ${\ displaystyle f (x)}  ÂÂ$ adalah fungsi kepadatan probabilitas. Dalam semua kasus, termasuk di mana distribusi tidak diskrit atau kontinu, mean adalah integral Lebesgue dari variabel acak sehubungan dengan ukuran probabilitasnya. Kebutuhan rata-rata tidak ada atau terbatas; untuk beberapa distribusi probabilitas mean tidak terbatas ( ? atau -? ), sementara yang lain tidak memiliki mean.

Alat umum

Kekuatan berarti

Mean umum, juga dikenal sebagai mean kekuasaan atau HÃÆ'¶lder mean, adalah abstraksi dari makna kuadrat, aritmatika, geometrik dan harmonik. Ini didefinisikan untuk satu set n bilangan positif x _i oleh

${\ displaystyle {\ bar {x}} (m) = \ kiri ({\ frac {1} {n}} \ cdot \ sum _ {i = 1} ^ {n} {x_ {i} ^ {m}} \ right) ^ {\ tfrac {1} {m}}}  ÂÂ$

By selecting different values ​​for the m parameter, the following types of tools are obtained:

? -mean

Ini dapat digeneralisasikan lebih lanjut sebagai f-mean umum

${\ displaystyle {\ bar {x}} = f ^ {- 1} \ kiri ({{\ frac {1} {n}} \ cdot \ sum _ {i = 1} ^ {n} {f (x_ {i})}} \ right)}  ÂÂ$

and again a suitable option from which can be reversed ? will give

Weighted arithmetic means

Mean aritmetik terboboti (atau rata-rata tertimbang) digunakan jika seseorang ingin menggabungkan nilai rata-rata dari sampel populasi yang sama dengan ukuran sampel yang berbeda:

${\ displaystyle {\ bar {x}} = {\ frac {\ jumlah _ {i = 1} ^ {n} {w_ {i} \ cdot x_ {i} }} {\ sum _ {i = 1} ^ {n} {w_ {i}}}}.}  ÂÂ$

Bobot ${\ displaystyle w_ {i}}  ÂÂ$ mewakili ukuran sampel yang berbeda. Dalam aplikasi lain mereka mewakili ukuran untuk keandalan pengaruh pada mean dengan nilai masing-masing.

Truncated mean

Sometimes a set of numbers may contain a call, that is, a data value much lower or much higher than another. Often, outliers are wrong data caused by artifacts. In this case, one can use a truncated mean. This involves removing certain parts of the data at the top or the lower end, usually the same number at each end, and then taking the arithmetic average of the remaining data. The amount of the deleted value is indicated as a percentage of the total value.

Interquartile averages

Mean interkuartil adalah contoh spesifik dari mean yang terpotong. Ini hanyalah rata-rata aritmetika setelah menghapus nilai terendah dan tertinggi dari nilai.

${\ displaystyle {\ bar {x}} = {2 \ over n} \ jumlah _ {i = (n/4) 1} ^ {3n/4} { x_ {i}}}  ÂÂ$

assuming the values ​​have been ordered, so it is just a specific example of a weighted average for a particular set of weights.

Means function

Dalam beberapa keadaan, para matematikawan dapat menghitung nilai rata-rata dari kumpulan nilai yang tak terbatas (bahkan tak terhitung). Ini dapat terjadi saat menghitung nilai rata-rata ${\ displaystyle y _ {\ text {ave}}}  ÂÂ$ dari fungsi