Standard deviation

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In statistics, the standard deviation ( SD , also represented by the Greek letter sigma ? or Latin letter s) is the measure used to measure the number of variations or dispersion of a set of data values. The low standard deviation indicates that the data points tend to be close to the average (also called the expected value) of the set, while the high standard deviation indicates that the data points are spread over a wider range of values.

Standard deviations from random variables, population statistics, data sets, or probability distributions are the square roots of the variations. Algebra is simpler, though in practice less powerful, than the average absolute aberration. A useful property of the standard deviation is that, unlike variance, it is expressed in units equal to the data.

In addition to expressing population variability, standard deviations are generally used to measure confidence in statistical inference. For example, the margin of error in the voting data is determined by calculating the expected standard deviation in the results if the same poll is done many times. The derivation of this standard deviation is often called the "standard error of mean" when referring to the mean. This is calculated as the standard deviation of all means to be calculated from the population if the number of unlimited samples is taken and the mean for each sample is calculated.

It is important to note that the standard deviation of the population and the standard error of statistics derived from that population (such as the mean) are quite different but related (as opposed to the square root of the observed number). The reported error margin of a poll is calculated from the average standard error (or as an alternative to the standard population deviation product and the inverse of the square root of sample size, which is the same thing) and is usually about twice the standard deviation - half the width of the interval 95 percent confidence.

In science, many researchers report the standard deviation of experimental data, and only the effects that fall further than two standard deviations far from what are expected to be considered statistically significant - normal random errors or variations in measurements in this way are distinguished from possible original effects or associations. Standard deviations are also important in finance, where standard deviations on investment returns are a measure of investment volatility.

When only sample data from the population is available, the term sample deviation standard or sample deviation standard may refer to the quantity mentioned above as applied to the modified data or quantity which is an estimate not biased from the population standard deviation (standard deviation of the entire population).

Video Standard deviation

Contoh dasar

Contoh deviasi standar tingkat metabolisme Northern Fulmars

Logan gives the following example. Furness and Bryant measured the rate of resting metabolism for 8 male and 6 female breeding Northern fulmars. The table shows the Furness data set.

The graph shows the metabolic rate for men and women. With visual inspection, it appears that the variability of the metabolic rate is greater for males than for females.

Standar deviasi sampel dari tingkat metabolisme untuk fulmars wanita dihitung sebagai berikut. Rumus untuk standar deviasi sampel adalah

${\ displaystyle s = {\ sqrt {\ frac {\ sum _ {i = 1} ^ {N} (x_ {i} - {\ overline {x}}) ^ {2}} {N-1}}}.}  ÂÂ$

di mana ${\ displaystyle \ scriptstyle \ {x_ {1}, \, x_ {2}, \, \ ldots, \, x_ {N} \}}  ÂÂ$ adalah nilai yang teramati dari item sampel, ${\ displaystyle \ scriptstyle {\ overline {x}}}  ÂÂ$ adalah nilai rata-rata dari observasi ini, dan N adalah jumlah observasi dalam sampel.

In the sample standard deviation formula, for this example, the numerator is the sum of the squared deviations of each animal's metabolic rate from the average metabolic rate. The table below shows the calculation of the number of squared deviations for the female fulmars. For women, the sum of squared deviations is 886047.09, as shown in the table.

Penyebut dalam rumus standar deviasi sampel adalah N - 1, di mana N adalah jumlah hewan. Dalam contoh ini, ada N = 6 perempuan, jadi penyebutnya adalah 6 - 1 = 5. Standar deviasi sampel untuk fulmars perempuan adalah

${\ displaystyle s = {\ sqrt {\ frac {\ sum _ {i = 1} ^ {N} (x_ {i} - {\ overline {x}}) ^ {2}} {N-1}}} = {\ sqrt {\ frac {886047.09} {5}}} = 420.96.}  ÂÂ$

For male fulmars, similar calculations provide a standard deviation sample of 894.37, about twice as large as the standard deviation for women. The graph shows data on metabolic rate, means (red dots), and standard deviation (red line) for women and men.

The use of the standard deviation of the sample implies that these 14 fulms are samples of the larger fulma population. If these 14 fulmars consist of the entire population (perhaps the last 14 surviving fulmars), then than the standard deviation of the sample, the calculation will use the standard deviation of the population. In the population standard deviation formula, the denominator is N rather than N - 1. Very rare measurements can be made for the entire population, so, by default, software package statistics calculate the standard deviation of the sample. Similarly, journal articles report the standard deviation of samples unless otherwise specified.

Population standard deviations of eight students

Suppose the entire population of interest is eight students in a particular class. For a limited set of numbers, the population standard deviation is found by taking the square root of the mean deviation of the squares from the values ​​of their average value. The class marks of eight students (ie, population statistics) are the following eight values:

${\ displaystyle 2, \ 4, \ 4, \ 4, \ 5, \ 5, \ 7, \ 9.}$

Delapan titik data ini memiliki rata-rata (rata-rata) 5:

${\ displaystyle \ mu = {\ frac {2 4 4 4 5 5 7 9} {8}} = 5.}  ÂÂ$

Variansnya adalah mean dari nilai-nilai ini:

${\ displaystyle \ sigma ^ {2} = {\ frac {9 1 1 1 0 0 4 16} {8}} = 4.}  ÂÂ$

dan populasi standar deviasi sama dengan akar kuadrat dari varians:

${\ displaystyle \ sigma = {\ sqrt {4}} = 2.}  ÂÂ$

This formula only applies if the eight values ​​we start from the complete population. If the opposite values ​​are random samples drawn from some large populations of parents (eg, they are 8 randomly and independently marked from the 2 millionth class), then one is often divided by 7Ã, (the n - 1) than 8Ã, (which n ) in the denominator of the last formula. In this case the result of the original formula will be called the standard deviation sample . Dividing by n Ã,-1 rather than by n providing an unbiased estimate of the larger parent population variance. This is known as Bessel correction .

Average high standard deviation for adult men

If the population of interest is approximately normally distributed, the standard deviation provides information about the proportion of observations above or below certain values. For example, the average height for adult men in the United States is about 70 inches (177.8 cm), with a standard deviation of about 3 inches (7.62 cm). This means that most men (about 68%, assuming normal distribution) have a height in 3Ã,¼ (7.62Ã, cm) of the average (67-73Ã, inches (170.18-185.42 cm)) - one standard deviation - and almost all men (about 95%) have a height in 6 inches (15.24 cm) of average (64-76 inches (162.56-193.04 cm)) - two standard deviations. If the standard deviation is zero, then all men will be exactly 70 inches (177.8 cm) tall. If the standard deviation is 20 inches (50.8 cm), then the man will have more variable heights, with a typical range of about 50-90 inches (127-228.6 cm). Three standard deviations accounted for 99.7% of the sample population under study, assuming a normal distribution (bell-shaped). (See rules 68-95-99.7, or empirical rules , for more information.)

Maps Standard deviation

Definition of population value

Biarkan X menjadi variabel acak dengan nilai rata-rata ? :

${\ displaystyle \ operatorname {E} [X] = \ mu. \, \!}  ÂÂ$

(derived using the properties of the expected value).

In other words, the standard deviation ? (sigma) is the square root of the variance X ; that is, the square root of the mean value ( X Ã, - ? ) ² .

The standard deviation of a probability distribution (univariate) is equal to a random variable having that distribution. Not all random variables have a standard deviation, since the expected value does not need to exist. For example, the standard deviation of a random variable following the Cauchy distribution is undefined because the expected value is ? is undefined.

Discrete random variable

Dalam kasus di mana X mengambil nilai acak dari kumpulan data yang terbatas x ₁ , x ₂ ,..., x _N , dengan setiap nilai memiliki probabilitas yang sama, standar deviasi adalah

${\ displaystyle \ sigma = {\ sqrt {{\ frac {1} {N}} \ kiri [(x_ {1} - \ mu) ^ {2} ( x_ {2} - \ mu) ^ {2} \ cdots (x_ {N} - \ mu) ^ {2} \ right]}}, {\ rm {\ \ where \ \}} \ mu = { \ frac {1} {N}} (x_ {1} \ cdots x_ {N}),}  ÂÂ$

atau, menggunakan notasi penjumlahan,

${\ displaystyle \ sigma = {\ sqrt {{\ frac {1} {N}} \ jumlah _ {i = 1} ^ {N} (x_ {i} - \ mu) ^ {2}}}, {\ rm {\ \ where \ \}} \ mu = {\ frac {1} {N}} \ jumlah _ {i = 1} ^ {N} x_ {i}.}  ÂÂ$

Jika, daripada memiliki probabilitas yang sama, nilai-nilai memiliki probabilitas yang berbeda, biarkan x ₁ memiliki probabilitas p ₁ , x ₂ memiliki probabilitas p ₂ ,..., x _N memiliki probabilitas p _N . Dalam hal ini, standar deviasinya akan

${\ displaystyle \ sigma = {\ sqrt {\ sum _ {i = 1} ^ {N} p_ {i} (x_ {i} - \ mu) ^ {2 }}}, {\ rm {\ \ where \ \}} \ mu = \ sum _ {i = 1} ^ {N} p_ {i} x_ {i}.}  ÂÂ$

Variabel acak kontinyu

Simpangan baku dari variabel acak bernilai real kontinyu X dengan fungsi kepadatan probabilitas p ( x ) adalah

${\ displaystyle \ sigma = {\ sqrt {\ int_ {\ mathbf {X}} (x- \ mu) ^ {2} \, p (x) \, {\ rm {d}} x}}, {\ rm {\ \ di mana \ \}} \ mu = \ int_ {\ mathbf {X}} x \, p (x) \, {\ rm {d} } x,}  ÂÂ$

and where the integral is a specific integral taken for x ranges above the set of possible values ​​of the random variable X .

Dalam kasus keluarga distribusi parametrik, standar deviasi dapat dinyatakan dalam parameter. Misalnya, dalam kasus distribusi log normal dengan parameter ? dan ? ² , standar deviasinya adalah

${\ displaystyle {\ sqrt {(e ^ {\ sigma ^ {2}} - 1) e ^ {2 \ mu \ sigma ^ {2}}}}. }  ÂÂ$

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Estimasi

One can find standard deviations of the entire population in cases (such as standard testing) in which each member of the population is sampled. In cases where that can not be done, the standard deviation ? is estimated by examining random samples taken from the population and calculating sample statistics, which are used as an estimate of population standard deviation. Such statistics are called estimators, and estimators (or estimator values, ie estimates) are called standard deviations of samples , and are denoted by (possibly with modifiers). However, unlike in the case of estimating the average population, the sample mean is a simple estimator with many desired properties (unbiased, efficient, maximum probability), no single estimator for standard deviation with all these properties, and no bias. standard deviation estimates are a very technical issue involved. Most often, the standard deviation is estimated using the corrected sample deviation standard (using N Ã,-1), defined below, and this is often referred to as the "standard deviation sample" without qualification. However, other predictions are better in other ways: the uncorrected estimator (using N ) yields a lower average squared error while using N Ã,-1, 5 (for normal distribution) almost completely eliminates the bias.

Uncorrected sample deviation standard

The formula for the standard deviation of the population (limited population) can be applied to the sample, using the sample size as the population size (although the actual population size from which the sample is drawn may be much greater). This estimate, denoted by s _N , is known as the uncorrected standard deviation sample , or sometimes < b>> the standard deviation of the sample (regarded as the entire population), and defined as follows:

_{S                ÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂ... N                          =                                   ÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂ...     Â 1      ÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂ,        ÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂ,                ÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂ,               ?                    Â         me               =     Â 1        ÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂ,                        ÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂ,        ÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂ,        ÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂ/           (      Â              x                    Â         me        ÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂ,          Â     ÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÃ, -                ÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂ...                x                Ã,¯       ÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂ,                ÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂ,             )                             2        ÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂ,         Â  <Â>                           ,               {\ displaystyle s_ {N} = {\ sqrt {{\ frac {1} {N}} \ sum _ {i = 1} ^ {N} (x_ {i} - {\ overline {x}}) ^ {2}}},}  Â}

di mana $            {\displaystyle                   \{          {}_{}}$

Source of the article : Wikipedia