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52-420: (Redirected from Halfwidth ) Half width may refer to Full width at half maximum Halfwidth and fullwidth forms Half-width kana Topics referred to by the same term [REDACTED] This disambiguation page lists articles associated with the title Half-width . If an internal link led you here, you may wish to change the link to point directly to

104-560: A continuous real-valued random variable X with probability density function p ( x ) is σ = ∫ X ( x − μ ) 2 p ( x ) d x ,  where  μ = ∫ X x p ( x ) d x , {\displaystyle \sigma ={\sqrt {\int _{\mathbf {X} }(x-\mu )^{2}\,p(x)\,\mathrm {d} x}},{\text{ where }}\mu =\int _{\mathbf {X} }x\,p(x)\,\mathrm {d} x,} and where

156-614: A built-in bias. See the discussion on Bessel's correction further down below. or, by using summation notation, σ = 1 N ∑ i = 1 N ( x i − μ ) 2 ,  where  μ = 1 N ∑ i = 1 N x i . {\displaystyle \sigma ={\sqrt {{\frac {1}{N}}\sum _{i=1}^{N}(x_{i}-\mu )^{2}}},{\text{ where }}\mu ={\frac {1}{N}}\sum _{i=1}^{N}x_{i}.} If, instead of having equal probabilities,

208-623: A class of eight students (that is, a statistical population ) are the following eight values: 2 ,   4 ,   4 ,   4 ,   5 ,   5 ,   7 ,   9. {\displaystyle 2,\ 4,\ 4,\ 4,\ 5,\ 5,\ 7,\ 9.} These eight data points have the mean (average) of 5: μ = 2 + 4 + 4 + 4 + 5 + 5 + 7 + 9 8 = 40 8 = 5. {\displaystyle \mu ={\frac {2+4+4+4+5+5+7+9}{8}}={\frac {40}{8}}=5.} First, calculate

260-724: A correction factor to produce an unbiased estimate. For the normal distribution, an unbiased estimator is given by ⁠ s / c 4 ⁠ , where the correction factor (which depends on N ) is given in terms of the Gamma function , and equals: c 4 ( N ) = 2 N − 1 Γ ( N 2 ) Γ ( N − 1 2 ) . {\displaystyle c_{4}(N)\,=\,{\sqrt {\frac {2}{N-1}}}\,\,\,{\frac {\Gamma \left({\frac {N}{2}}\right)}{\Gamma \left({\frac {N-1}{2}}\right)}}.} This arises because

312-798: A finite data set x 1 , x 2 , ..., x N , with each value having the same probability, the standard deviation is σ = 1 N [ ( x 1 − μ ) 2 + ( x 2 − μ ) 2 + ⋯ + ( x N − μ ) 2 ] ,  where  μ = 1 N ( x 1 + ⋯ + x N ) , {\displaystyle \sigma ={\sqrt {{\frac {1}{N}}\left[(x_{1}-\mu )^{2}+(x_{2}-\mu )^{2}+\cdots +(x_{N}-\mu )^{2}\right]}},{\text{ where }}\mu ={\frac {1}{N}}(x_{1}+\cdots +x_{N}),} Note: The above expression has

364-441: A height within 6 inches of the mean ( 63–75 inches ) – two standard deviations. If the standard deviation were zero, then all men would share an identical height of 69 inches. Three standard deviations account for 99.73% of the sample population being studied, assuming the distribution is normal or bell-shaped (see the 68–95–99.7 rule , or the empirical rule, for more information). Let μ be

416-407: A mean for each sample. The mean's standard error turns out to equal the population standard deviation divided by the square root of the sample size, and is estimated by using the sample standard deviation divided by the square root of the sample size. For example, a poll's standard error (what is reported as the margin of error of the poll), is the expected standard deviation of the estimated mean if

468-435: A modified quantity that is an unbiased estimate of the population standard deviation (the standard deviation of the entire population). Suppose that the entire population of interest is eight students in a particular class. For a finite set of numbers, the population standard deviation is found by taking the square root of the average of the squared deviations of the values subtracted from their average value. The marks of

520-779: Is defined as σ ≡ E ⁡ [ ( X − μ ) 2 ] = ∫ − ∞ + ∞ ( x − μ ) 2 f ( x ) d x , {\displaystyle \sigma \equiv {\sqrt {\operatorname {E} \left[(X-\mu )^{2}\right]}}={\sqrt {\int _{-\infty }^{+\infty }(x-\mu )^{2}f(x)\,\mathrm {d} x}},} which can be shown to equal E ⁡ [ X 2 ] − ( E ⁡ [ X ] ) 2 . {\textstyle {\sqrt {\operatorname {E} \left[X^{2}\right]-(\operatorname {E} [X])^{2}}}.} Using words,

572-429: Is a very technically involved problem. Most often, the standard deviation is estimated using the corrected sample standard deviation (using N  − 1), defined below, and this is often referred to as the "sample standard deviation", without qualifiers. However, other estimators are better in other respects: the uncorrected estimator (using N ) yields lower mean squared error, while using N  − 1.5 (for

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624-448: Is an unbiased estimator for the population variance, s is still a biased estimator for the population standard deviation, though markedly less biased than the uncorrected sample standard deviation. This estimator is commonly used and generally known simply as the "sample standard deviation". The bias may still be large for small samples ( N less than 10). As sample size increases, the amount of bias decreases. We obtain more information and

676-540: Is applied to antenna beam width , it is called half-power beam width . If the considered function is the density of a normal distribution of the form f ( x ) = 1 σ 2 π exp ⁡ [ − ( x − x 0 ) 2 2 σ 2 ] {\displaystyle f(x)={\frac {1}{\sigma {\sqrt {2\pi }}}}\exp \left[-{\frac {(x-x_{0})^{2}}{2\sigma ^{2}}}\right]} where σ

728-829: Is invariant under translations. The area within this FWHM is approximately 76% of the total area under the function. In spectroscopy half the width at half maximum (here γ ), HWHM, is in common use. For example, a Lorentzian/Cauchy distribution of height ⁠ 1 / πγ ⁠ can be defined by f ( x ) = 1 π γ [ 1 + ( x − x 0 γ ) 2 ]  and  F W H M = 2 γ . {\displaystyle f(x)={\frac {1}{\pi \gamma \left[1+\left({\frac {x-x_{0}}{\gamma }}\right)^{2}\right]}}\quad {\text{ and }}\quad \mathrm {FWHM} =2\gamma .} Another important distribution function, related to solitons in optics ,

780-403: Is known as Bessel's correction . Roughly, the reason for it is that the formula for the sample variance relies on computing differences of observations from the sample mean, and the sample mean itself was constructed to be as close as possible to the observations, so just dividing by n would underestimate the variability. If the population of interest is approximately normally distributed,

832-420: Is suited for all but the smallest samples or highest precision: for N = 3 the bias is equal to 1.3%, and for N = 9 the bias is already less than 0.1%. A more accurate approximation is to replace N − 1.5 above with N − 1.5 + ⁠ 1 / 8( N − 1) ⁠ . For other distributions, the correct formula depends on the distribution, but a rule of thumb is to use the further refinement of

884-412: Is that, unlike the variance, it is expressed in the same unit as the data. The standard deviation of a population or sample and the standard error of a statistic (e.g., of the sample mean) are quite different, but related. The sample mean's standard error is the standard deviation of the set of means that would be found by drawing an infinite number of repeated samples from the population and computing

936-709: Is the hyperbolic secant : f ( x ) = sech ⁡ ( x X ) . {\displaystyle f(x)=\operatorname {sech} \left({\frac {x}{X}}\right).} Any translating element was omitted, since it does not affect the FWHM. For this impulse we have: F W H M = 2 arcsch ⁡ ( 1 2 ) X = 2 ln ⁡ ( 2 + 3 ) X ≈ 2.634 X {\displaystyle \mathrm {FWHM} =2\operatorname {arcsch} \left({\tfrac {1}{2}}\right)X=2\ln(2+{\sqrt {3}})\;X\approx 2.634\;X} where arcsch

988-403: Is the inverse hyperbolic secant . This applied mathematics –related article is a stub . You can help Misplaced Pages by expanding it . Standard deviation In statistics , the standard deviation is a measure of the amount of variation of the values of a variable about its mean . A low standard deviation indicates that the values tend to be close to the mean (also called

1040-425: Is the standard deviation and x 0 is the expected value , then the relationship between FWHM and the standard deviation is F W H M = 2 2 ln ⁡ 2 σ ≈ 2.355 σ . {\displaystyle \mathrm {FWHM} =2{\sqrt {2\ln 2}}\;\sigma \approx 2.355\;\sigma .} The FWHM does not depend on the expected value x 0 ; it

1092-492: Is the mean of these values: σ 2 = 9 + 1 + 1 + 1 + 0 + 0 + 4 + 16 8 = 32 8 = 4. {\displaystyle \sigma ^{2}={\frac {9+1+1+1+0+0+4+16}{8}}={\frac {32}{8}}=4.} and the population standard deviation is equal to the square root of the variance: σ = 4 = 2. {\displaystyle \sigma ={\sqrt {4}}=2.} This formula

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1144-524: Is unbiased if the variance exists and the sample values are drawn independently with replacement. N  − 1 corresponds to the number of degrees of freedom in the vector of deviations from the mean, ( x 1 − x ¯ , … , x n − x ¯ ) . {\displaystyle \textstyle (x_{1}-{\bar {x}},\;\dots ,\;x_{n}-{\bar {x}}).} Taking square roots reintroduces bias (because

1196-537: Is valid only if the eight values with which we began form the complete population. If the values instead were a random sample drawn from some large parent population (for example, they were 8 students randomly and independently chosen from a class of 2 million), then one divides by 7 (which is n − 1) instead of 8 (which is n ) in the denominator of the last formula, and the result is s = 32 / 7 ≈ 2.1. {\textstyle s={\sqrt {32/7}}\approx 2.1.} In that case,

1248-867: The confidence interval or CI. To show how a larger sample will make the confidence interval narrower, consider the following examples: A small population of N = 2 has only one degree of freedom for estimating the standard deviation. The result is that a 95% CI of the SD runs from 0.45 × SD to 31.9 × SD; the factors here are as follows : Pr ( q α 2 < k s 2 σ 2 < q 1 − α 2 ) = 1 − α , {\displaystyle \Pr \left(q_{\frac {\alpha }{2}}<k{\frac {s^{2}}{\sigma ^{2}}}<q_{1-{\frac {\alpha }{2}}}\right)=1-\alpha ,} where q p {\displaystyle q_{p}}

1300-411: The expected value (the average) of random variable X with density f ( x ) : μ ≡ E ⁡ [ X ] = ∫ − ∞ + ∞ x f ( x ) d x {\displaystyle \mu \equiv \operatorname {E} [X]=\int _{-\infty }^{+\infty }xf(x)\,\mathrm {d} x} The standard deviation σ of X

1352-418: The expected value ) of the set, while a high standard deviation indicates that the values are spread out over a wider range. The standard deviation is commonly used in the determination of what constitutes an outlier and what does not. Standard deviation may be abbreviated SD or Std Dev , and is most commonly represented in mathematical texts and equations by the lowercase Greek letter σ (sigma), for

1404-519: The spectral width of sources used for optical communications and the resolution of spectrometers . The convention of "width" meaning "half maximum" is also widely used in signal processing to define bandwidth as "width of frequency range where less than half the signal's power is attenuated", i.e., the power is at least half the maximum. In signal processing terms, this is at most −3  dB of attenuation, called half-power point or, more specifically, half-power bandwidth . When half-power point

1456-599: The standard deviation of the sample (considered as the entire population), and is defined as follows: s N = 1 N ∑ i = 1 N ( x i − x ¯ ) 2 , {\displaystyle s_{N}={\sqrt {{\frac {1}{N}}\sum _{i=1}^{N}\left(x_{i}-{\bar {x}}\right)^{2}}},} where { x 1 , x 2 , … , x N } {\displaystyle \{x_{1},\,x_{2},\,\ldots ,\,x_{N}\}} are

1508-497: The approximation: σ ^ = 1 N − 1.5 − 1 4 γ 2 ∑ i = 1 N ( x i − x ¯ ) 2 , {\displaystyle {\hat {\sigma }}={\sqrt {{\frac {1}{N-1.5-{\frac {1}{4}}\gamma _{2}}}\sum _{i=1}^{N}\left(x_{i}-{\bar {x}}\right)^{2}}},} where γ 2 denotes

1560-411: The bias is below 1%. Thus for very large sample sizes, the uncorrected sample standard deviation is generally acceptable. This estimator also has a uniformly smaller mean squared error than the corrected sample standard deviation. If the biased sample variance (the second central moment of the sample, which is a downward-biased estimate of the population variance) is used to compute an estimate of

1612-1155: The deviations of each data point from the mean, and square the result of each: ( 2 − 5 ) 2 = ( − 3 ) 2 = 9 ( 5 − 5 ) 2 = 0 2 = 0 ( 4 − 5 ) 2 = ( − 1 ) 2 = 1 ( 5 − 5 ) 2 = 0 2 = 0 ( 4 − 5 ) 2 = ( − 1 ) 2 = 1 ( 7 − 5 ) 2 = 2 2 = 4 ( 4 − 5 ) 2 = ( − 1 ) 2 = 1 ( 9 − 5 ) 2 = 4 2 = 16. {\displaystyle {\begin{array}{lll}(2-5)^{2}=(-3)^{2}=9&&(5-5)^{2}=0^{2}=0\\(4-5)^{2}=(-1)^{2}=1&&(5-5)^{2}=0^{2}=0\\(4-5)^{2}=(-1)^{2}=1&&(7-5)^{2}=2^{2}=4\\(4-5)^{2}=(-1)^{2}=1&&(9-5)^{2}=4^{2}=16.\\\end{array}}} The variance

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1664-399: The difference between 1 N {\displaystyle {\frac {1}{N}}} and 1 N − 1 {\displaystyle {\frac {1}{N-1}}} becomes smaller. For unbiased estimation of standard deviation , there is no formula that works across all distributions, unlike for mean and variance. Instead, s is used as a basis, and is scaled by

1716-493: The estimator (or the value of the estimator, namely the estimate) is called a sample standard deviation, and is denoted by s (possibly with modifiers). Unlike in the case of estimating the population mean of a normal distribution, for which the sample mean is a simple estimator with many desirable properties ( unbiased , efficient , maximum likelihood), there is no single estimator for the standard deviation with all these properties, and unbiased estimation of standard deviation

1768-454: The findings). By convention, only effects more than two standard errors away from a null expectation are considered "statistically significant" , a safeguard against spurious conclusion that is really due to random sampling error. When only a sample of data from a population is available, the term standard deviation of the sample or sample standard deviation can refer to either the above-mentioned quantity as applied to those data, or to

1820-636: The integrals are definite integrals taken for x ranging over the set of possible values of the random variable  X . In the case of a parametric family of distributions , the standard deviation can be expressed in terms of the parameters. For example, in the case of the log-normal distribution with parameters μ and σ , the standard deviation is ( e σ 2 − 1 ) e 2 μ + σ 2 . {\displaystyle {\sqrt {\left(e^{\sigma ^{2}}-1\right)e^{2\mu +\sigma ^{2}}}}.} One can find

1872-423: The intended article. Retrieved from " https://en.wikipedia.org/w/index.php?title=Half-width&oldid=1145778088 " Category : Disambiguation pages Hidden categories: Short description is different from Wikidata All article disambiguation pages All disambiguation pages Full width at half maximum FWHM is applied to such phenomena as the duration of pulse waveforms and

1924-423: The normal distribution) almost completely eliminates bias. The formula for the population standard deviation (of a finite population) can be applied to the sample, using the size of the sample as the size of the population (though the actual population size from which the sample is drawn may be much larger). This estimator, denoted by s N , is known as the uncorrected sample standard deviation , or sometimes

1976-418: The observed values of the sample items, and x ¯ {\displaystyle {\bar {x}}} is the mean value of these observations, while the denominator  N stands for the size of the sample: this is the square root of the sample variance, which is the average of the squared deviations about the sample mean. This is a consistent estimator (it converges in probability to

2028-417: The population excess kurtosis . The excess kurtosis may be either known beforehand for certain distributions, or estimated from the data. The standard deviation we obtain by sampling a distribution is itself not absolutely accurate, both for mathematical reasons (explained here by the confidence interval) and for practical reasons of measurement (measurement error). The mathematical effect can be described by

2080-466: The population standard deviation, or the Latin letter s , for the sample standard deviation. The standard deviation of a random variable , sample , statistical population , data set , or probability distribution is the square root of its variance. It is algebraically simpler, though in practice less robust , than the average absolute deviation . A useful property of the standard deviation

2132-421: The population value as the number of samples goes to infinity), and is the maximum-likelihood estimate when the population is normally distributed. However, this is a biased estimator , as the estimates are generally too low. The bias decreases as sample size grows, dropping off as 1/ N , and thus is most significant for small or moderate sample sizes; for N > 75 {\displaystyle N>75}

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2184-456: The population's standard deviation, the result is s N = 1 N ∑ i = 1 N ( x i − x ¯ ) 2 . {\displaystyle s_{N}={\sqrt {{\frac {1}{N}}\sum _{i=1}^{N}\left(x_{i}-{\bar {x}}\right)^{2}}}.} Here taking the square root introduces further downward bias, by Jensen's inequality , due to

2236-431: The result of the original formula would be called the sample standard deviation and denoted by s {\textstyle s} instead of σ . {\displaystyle \sigma .} Dividing by n − 1 {\textstyle n-1} rather than by n {\textstyle n} gives an unbiased estimate of the variance of the larger parent population. This

2288-417: The same poll were to be conducted multiple times. Thus, the standard error estimates the standard deviation of an estimate, which itself measures how much the estimate depends on the particular sample that was taken from the population. In science , it is common to report both the standard deviation of the data (as a summary statistic) and the standard error of the estimate (as a measure of potential error in

2340-735: The sampling distribution of the sample standard deviation follows a (scaled) chi distribution , and the correction factor is the mean of the chi distribution. An approximation can be given by replacing N  − 1 with N  − 1.5 , yielding: σ ^ = 1 N − 1.5 ∑ i = 1 N ( x i − x ¯ ) 2 , {\displaystyle {\hat {\sigma }}={\sqrt {{\frac {1}{N-1.5}}\sum _{i=1}^{N}\left(x_{i}-{\bar {x}}\right)^{2}}},} The error in this approximation decays quadratically (as ⁠ 1 / N ⁠ ), and it

2392-653: The square root is a nonlinear function which does not commute with the expectation, i.e. often E [ X ] ≠ E [ X ] {\textstyle E[{\sqrt {X}}]\neq {\sqrt {E[X]}}} ), yielding the corrected sample standard deviation, denoted by s: s = 1 N − 1 ∑ i = 1 N ( x i − x ¯ ) 2 . {\displaystyle s={\sqrt {{\frac {1}{N-1}}\sum _{i=1}^{N}\left(x_{i}-{\bar {x}}\right)^{2}}}.} As explained above, while s

2444-713: The square root's being a concave function . The bias in the variance is easily corrected, but the bias from the square root is more difficult to correct, and depends on the distribution in question. An unbiased estimator for the variance is given by applying Bessel's correction , using N  − 1 instead of N to yield the unbiased sample variance, denoted s : s 2 = 1 N − 1 ∑ i = 1 N ( x i − x ¯ ) 2 . {\displaystyle s^{2}={\frac {1}{N-1}}\sum _{i=1}^{N}\left(x_{i}-{\bar {x}}\right)^{2}.} This estimator

2496-497: The standard deviation is the square root of the variance of X . The standard deviation of a probability distribution is the same as that of a random variable having that distribution. Not all random variables have a standard deviation. If the distribution has fat tails going out to infinity, the standard deviation might not exist, because the integral might not converge. The normal distribution has tails going out to infinity, but its mean and standard deviation do exist, because

2548-423: The standard deviation of an entire population in cases (such as standardized testing ) where every member of a population is sampled. In cases where that cannot be done, the standard deviation σ is estimated by examining a random sample taken from the population and computing a statistic of the sample, which is used as an estimate of the population standard deviation. Such a statistic is called an estimator , and

2600-560: The standard deviation provides information on the proportion of observations above or below certain values. For example, the average height for adult men in the United States is about 69 inches , with a standard deviation of around 3 inches . This means that most men (about 68%, assuming a normal distribution ) have a height within 3 inches of the mean ( 66–72 inches ) – one standard deviation – and almost all men (about 95%) have

2652-399: The tails diminish quickly enough. The Pareto distribution with parameter α ∈ ( 1 , 2 ] {\displaystyle \alpha \in (1,2]} has a mean, but not a standard deviation (loosely speaking, the standard deviation is infinite). The Cauchy distribution has neither a mean nor a standard deviation. In the case where X takes random values from

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2704-684: The values have different probabilities, let x 1 have probability p 1 , x 2 have probability p 2 , ..., x N have probability p N . In this case, the standard deviation will be σ = ∑ i = 1 N p i ( x i − μ ) 2 ,  where  μ = ∑ i = 1 N p i x i . {\displaystyle \sigma ={\sqrt {\sum _{i=1}^{N}p_{i}(x_{i}-\mu )^{2}}},{\text{ where }}\mu =\sum _{i=1}^{N}p_{i}x_{i}.} The standard deviation of

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