Misplaced Pages

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60-444: For a single given sample, the relative atomic mass of a given element is the weighted arithmetic mean of the masses of the individual atoms (including all its isotopes ) that are present in the sample. This quantity can vary significantly between samples because the sample's origin (and therefore its radioactive history or diffusion history) may have produced combinations of isotopic abundances in varying ratios. For example, due to

120-720: A different probability distribution with known variance σ i 2 {\displaystyle \sigma _{i}^{2}} , all having the same mean, one possible choice for the weights is given by the reciprocal of variance: The weighted mean in this case is: and the standard error of the weighted mean (with inverse-variance weights) is: Note this reduces to σ x ¯ 2 = σ 0 2 / n {\displaystyle \sigma _{\bar {x}}^{2}=\sigma _{0}^{2}/n} when all σ i = σ 0 {\displaystyle \sigma _{i}=\sigma _{0}} . It

180-404: A different mixture of stable carbon-12 and carbon-13 isotopes, a sample of elemental carbon from volcanic methane will have a different relative atomic mass than one collected from plant or animal tissues. The more common, and more specific quantity known as standard atomic weight ( A r,standard ) is an application of the relative atomic mass values obtained from many different samples. It

240-473: A different radioactive history and so different isotopic composition. To reflect this natural variability, the IUPAC made the decision in 2010 to list the standard relative atomic masses of 10 elements as an interval rather than a fixed number. Weighted arithmetic mean The weighted arithmetic mean is similar to an ordinary arithmetic mean (the most common type of average ), except that instead of each of

300-448: A few counterintuitive properties, as captured for instance in Simpson's paradox . Given two school classes   —   one with 20 students, one with 30 students   —   and test grades in each class as follows: The mean for the morning class is 80 and the mean of the afternoon class is 90. The unweighted mean of the two means is 85. However, this does not account for

360-614: A linear combination is called a convex combination . Using the previous example, we would get the following weights: Then, apply the weights like this: Formally, the weighted mean of a non-empty finite tuple of data ( x 1 , x 2 , … , x n ) {\displaystyle \left(x_{1},x_{2},\dots ,x_{n}\right)} , with corresponding non-negative weights ( w 1 , w 2 , … , w n ) {\displaystyle \left(w_{1},w_{2},\dots ,w_{n}\right)}

420-554: A sample, is denoted as P ( I i = 1 ∣ Some sample of size  n ) = π i {\displaystyle P(I_{i}=1\mid {\text{Some sample of size }}n)=\pi _{i}} , and the one-draw probability of selection is P ( I i = 1 | one sample draw ) = p i ≈ π i n {\displaystyle P(I_{i}=1|{\text{one sample draw}})=p_{i}\approx {\frac {\pi _{i}}{n}}} (If N

480-407: A tick mark if multiplying by the indicator function. I.e.: y ˇ i ′ = I i y ˇ i = I i y i π i {\displaystyle {\check {y}}'_{i}=I_{i}{\check {y}}_{i}={\frac {I_{i}y_{i}}{\pi _{i}}}} In this design based perspective,

540-573: Is which expands to: Therefore, data elements with a high weight contribute more to the weighted mean than do elements with a low weight. The weights may not be negative in order for the equation to work . Some may be zero, but not all of them (since division by zero is not allowed). The formulas are simplified when the weights are normalized such that they sum up to 1, i.e., ∑ i = 1 n w i ′ = 1 {\textstyle \sum \limits _{i=1}^{n}{w_{i}'}=1} . For such normalized weights,

600-451: Is a nuclide . It is a stable isotope of oxygen , with 8 neutrons and 8 protons in its nucleus , and when not ionized, 8 electrons orbiting the nucleus. Oxygen-16 has a mass of 15.994 914 619 56  u . It is the most abundant isotope of oxygen and accounts for 99.757% of oxygen's natural abundance . The relative and absolute abundances of oxygen-16 are high because it is a principal product of stellar evolution and because it

660-583: Is a primordial isotope , meaning it can be made by stars that were initially made exclusively of hydrogen . Most oxygen-16 is synthesized at the end of the helium fusion process in stars; the triple-alpha process creates carbon-12, which captures an additional helium-4 to make oxygen-16. The neon-burning process also makes it. Oxygen-16 is doubly magic . Solid samples (organic and inorganic) for oxygen-16 studies are usually stored in silver cups and measured with pyrolysis and mass spectrometry . Researchers need to avoid improper or prolonged storage of

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720-457: Is a special case of the general formula in previous section, The equations above can be combined to obtain: The significance of this choice is that this weighted mean is the maximum likelihood estimator of the mean of the probability distributions under the assumption that they are independent and normally distributed with the same mean. The weighted sample mean, x ¯ {\displaystyle {\bar {x}}} ,

780-491: Is as follows: The estimation of the uncertainty is complicated, especially as the sample distribution is not necessarily symmetrical: the IUPAC standard relative atomic masses are quoted with estimated symmetrical uncertainties, and the value for silicon is 28.0855(3). The relative standard uncertainty in this value is 10 or 10 ppm. Apart from this uncertainty by measurement, some elements have variation over sources. That is, different sources (ocean water, rocks) have

840-524: Is called a Ratio estimator and it is approximately unbiased for R . In this case, the variability of the ratio depends on the variability of the random variables both in the numerator and the denominator - as well as their correlation. Since there is no closed analytical form to compute this variance, various methods are used for approximate estimation. Primarily Taylor series first-order linearization, asymptotics, and bootstrap/jackknife. The Taylor linearization method could lead to under-estimation of

900-561: Is considered constant, and the variability comes from the selection procedure. This in contrast to "model based" approaches in which the randomness is often described in the y values. The survey sampling procedure yields a series of Bernoulli indicator values ( I i {\displaystyle I_{i}} ) that get 1 if some observation i is in the sample and 0 if it was not selected. This can occur with fixed sample size, or varied sample size sampling (e.g.: Poisson sampling ). The probability of some element to be chosen, given

960-478: Is exemplified for silicon , whose relative atomic mass is especially important in metrology . Silicon exists in nature as a mixture of three isotopes: Si, Si and Si. The atomic masses of these nuclides are known to a precision of one part in 14 billion for Si and about one part in one billion for the others. However, the range of natural abundance for the isotopes is such that the standard abundance can only be given to about ±0.001% (see table). The calculation

1020-424: Is fixed, and the randomness comes from it being included in the sample or not ( I i {\displaystyle I_{i}} ), we often talk about the multiplication of the two, which is a random variable. To avoid confusion in the following section, let's call this term: y i ′ = y i I i {\displaystyle y'_{i}=y_{i}I_{i}} . With

1080-413: Is itself a random variable. Its expected value and standard deviation are related to the expected values and standard deviations of the observations, as follows. For simplicity, we assume normalized weights (weights summing to one). If the observations have expected values E ( x i ) = μ i , {\displaystyle E(x_{i})={\mu _{i}},} then

1140-486: Is known we can estimate the population mean using Y ¯ ^ known  N = Y ^ p w r N ≈ ∑ i = 1 n w i y i ′ N {\displaystyle {\hat {\bar {Y}}}_{{\text{known }}N}={\frac {{\hat {Y}}_{pwr}}{N}}\approx {\frac {\sum _{i=1}^{n}w_{i}y'_{i}}{N}}} . If

1200-410: Is rather limited due to the strong assumption about the y observations. This has led to the development of alternative, more general, estimators. From a model based perspective, we are interested in estimating the variance of the weighted mean when the different y i {\displaystyle y_{i}} are not i.i.d random variables. An alternative perspective for this problem

1260-406: Is sometimes interpreted as the expected range of the relative atomic mass values for the atoms of a given element from all terrestrial sources, with the various sources being taken from Earth . "Atomic weight" is often loosely and incorrectly used as a synonym for standard atomic weight (incorrectly because standard atomic weights are not from a single sample). Standard atomic weight is nevertheless

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1320-565: Is that of some arbitrary sampling design of the data in which units are selected with unequal probabilities (with replacement). In Survey methodology , the population mean, of some quantity of interest y , is calculated by taking an estimation of the total of y over all elements in the population ( Y or sometimes T ) and dividing it by the population size – either known ( N {\displaystyle N} ) or estimated ( N ^ {\displaystyle {\hat {N}}} ). In this context, each value of y

1380-539: Is the ratio of the mass of a single atom to the atomic mass constant ( m u = 1 Da ). This ratio is dimensionless. Modern relative atomic masses (a term specific to a given element sample) are calculated from measured values of atomic mass (for each nuclide ) and isotopic composition of a sample. Highly accurate atomic masses are available for virtually all non-radioactive nuclides, but isotopic compositions are both harder to measure to high precision and more subject to variation between samples. For this reason,

1440-681: Is the probability of selecting both i and j. And Δ ˇ i j = 1 − π i π j π i j {\displaystyle {\check {\Delta }}_{ij}=1-{\frac {\pi _{i}\pi _{j}}{\pi _{ij}}}} , and for i=j: Δ ˇ i i = 1 − π i π i π i = 1 − π i {\displaystyle {\check {\Delta }}_{ii}=1-{\frac {\pi _{i}\pi _{i}}{\pi _{i}}}=1-\pi _{i}} . If

1500-444: Is the quotient of the two weights, which makes the value dimensionless (having no unit). This quotient also explains the word relative : the sample mass value is considered relative to that of carbon-12. It is a synonym for atomic weight, though it is not to be confused with relative isotopic mass . Relative atomic mass is also frequently used as a synonym for standard atomic weight and these quantities may have overlapping values if

1560-428: Is therefore a more general term that can more broadly refer to samples taken from non-terrestrial environments or highly specific terrestrial environments which may differ substantially from Earth-average or reflect different degrees of certainty (e.g., in number of significant figures ) than those reflected in standard atomic weights. The prevailing IUPAC definitions (as taken from the " Gold Book ") are: and Here

1620-481: Is very large and each p i {\displaystyle p_{i}} is very small). For the following derivation we'll assume that the probability of selecting each element is fully represented by these probabilities. I.e.: selecting some element will not influence the probability of drawing another element (this doesn't apply for things such as cluster sampling design). Since each element ( y i {\displaystyle y_{i}} )

1680-1074: The π {\displaystyle \pi } -estimator. This estimator can be itself estimated using the pwr -estimator (i.e.: p {\displaystyle p} -expanded with replacement estimator, or "probability with replacement" estimator). With the above notation, it is: Y ^ p w r = 1 n ∑ i = 1 n y i ′ p i = ∑ i = 1 n y i ′ n p i ≈ ∑ i = 1 n y i ′ π i = ∑ i = 1 n w i y i ′ {\displaystyle {\hat {Y}}_{pwr}={\frac {1}{n}}\sum _{i=1}^{n}{\frac {y'_{i}}{p_{i}}}=\sum _{i=1}^{n}{\frac {y'_{i}}{np_{i}}}\approx \sum _{i=1}^{n}{\frac {y'_{i}}{\pi _{i}}}=\sum _{i=1}^{n}w_{i}y'_{i}} . The estimated variance of

1740-415: The oxygen-16 relative isotopic mass or else the oxygen relative atomic mass (i.e., atomic weight) for reference. See the article on the history of the modern unified atomic mass unit for the resolution of these problems. The IUPAC commission CIAAW maintains an expectation-interval value for relative atomic mass (or atomic weight) on Earth named standard atomic weight. Standard atomic weight requires

1800-735: The pwr -estimator is given by: Var ⁡ ( Y ^ p w r ) = n n − 1 ∑ i = 1 n ( w i y i − w y ¯ ) 2 {\displaystyle \operatorname {Var} ({\hat {Y}}_{pwr})={\frac {n}{n-1}}\sum _{i=1}^{n}\left(w_{i}y_{i}-{\overline {wy}}\right)^{2}} where w y ¯ = ∑ i = 1 n w i y i n {\displaystyle {\overline {wy}}=\sum _{i=1}^{n}{\frac {w_{i}y_{i}}{n}}} . The above formula

1860-443: The sampling design is one that results in a fixed sample size n (such as in pps sampling ), then the variance of this estimator is: The general formula can be developed like this: The population total is denoted as Y = ∑ i = 1 N y i {\displaystyle Y=\sum _{i=1}^{N}y_{i}} and it may be estimated by the (unbiased) Horvitz–Thompson estimator , also called

Relative atomic mass - Misplaced Pages Continue

1920-469: The "unified atomic mass unit" refers to 1/12 of the mass of an atom of C in its ground state . The IUPAC definition of relative atomic mass is: The definition deliberately specifies " An atomic weight ...", as an element will have different relative atomic masses depending on the source. For example, boron from Turkey has a lower relative atomic mass than boron from California , because of its different isotopic composition . Nevertheless, given

1980-475: The Earthly sources vary systematically). Atomic mass ( m a ) is the mass of a single atom. It defines the mass of a specific isotope, which is an input value for the determination of the relative atomic mass. An example for three silicon isotopes is given below. A convenient unit of mass for atomic mass is the dalton (Da), which is also called the unified atomic mass unit (u). The relative isotopic mass

2040-2535: The above notation, the parameter we care about is the ratio of the sums of y i {\displaystyle y_{i}} s, and 1s. I.e.: R = Y ¯ = ∑ i = 1 N y i π i ∑ i = 1 N 1 π i = ∑ i = 1 N y ˇ i ∑ i = 1 N 1 ˇ i = ∑ i = 1 N w i y i ∑ i = 1 N w i {\displaystyle R={\bar {Y}}={\frac {\sum _{i=1}^{N}{\frac {y_{i}}{\pi _{i}}}}{\sum _{i=1}^{N}{\frac {1}{\pi _{i}}}}}={\frac {\sum _{i=1}^{N}{\check {y}}_{i}}{\sum _{i=1}^{N}{\check {1}}_{i}}}={\frac {\sum _{i=1}^{N}w_{i}y_{i}}{\sum _{i=1}^{N}w_{i}}}} . We can estimate it using our sample with: R ^ = Y ¯ ^ = ∑ i = 1 N I i y i π i ∑ i = 1 N I i 1 π i = ∑ i = 1 N y ˇ i ′ ∑ i = 1 N 1 ˇ i ′ = ∑ i = 1 N w i y i ′ ∑ i = 1 N w i 1 i ′ = ∑ i = 1 n w i y i ′ ∑ i = 1 n w i 1 i ′ = y ¯ w {\displaystyle {\hat {R}}={\hat {\bar {Y}}}={\frac {\sum _{i=1}^{N}I_{i}{\frac {y_{i}}{\pi _{i}}}}{\sum _{i=1}^{N}I_{i}{\frac {1}{\pi _{i}}}}}={\frac {\sum _{i=1}^{N}{\check {y}}'_{i}}{\sum _{i=1}^{N}{\check {1}}'_{i}}}={\frac {\sum _{i=1}^{N}w_{i}y'_{i}}{\sum _{i=1}^{N}w_{i}1'_{i}}}={\frac {\sum _{i=1}^{n}w_{i}y'_{i}}{\sum _{i=1}^{n}w_{i}1'_{i}}}={\bar {y}}_{w}} . As we moved from using N to using n, we actually know that all

2100-418: The class means by the number of students in each class. The larger class is given more "weight": Thus, the weighted mean makes it possible to find the mean average student grade without knowing each student's score. Only the class means and the number of students in each class are needed. Since only the relative weights are relevant, any weighted mean can be expressed using coefficients that sum to one. Such

2160-415: The cost and difficulty of isotope analysis , it is common practice to instead substitute the tabulated values of standard atomic weights , which are ubiquitous in chemical laboratories and which are revised biennially by the IUPAC's Commission on Isotopic Abundances and Atomic Weights (CIAAW). Older (pre-1961) historical relative scales based on the atomic mass unit (symbol: a.m.u. or amu ) used either

2220-522: The data elements are independent and identically distributed random variables with variance σ 2 {\displaystyle \sigma ^{2}} , the standard error of the weighted mean , σ x ¯ {\displaystyle \sigma _{\bar {x}}} , can be shown via uncertainty propagation to be: For the weighted mean of a list of data for which each element x i {\displaystyle x_{i}} potentially comes from

2280-438: The data points contributing equally to the final average, some data points contribute more than others. The notion of weighted mean plays a role in descriptive statistics and also occurs in a more general form in several other areas of mathematics. If all the weights are equal, then the weighted mean is the same as the arithmetic mean . While weighted means generally behave in a similar fashion to arithmetic means, they do have

2340-505: The difference in number of students in each class (20 versus 30); hence the value of 85 does not reflect the average student grade (independent of class). The average student grade can be obtained by averaging all the grades, without regard to classes (add all the grades up and divide by the total number of students): x ¯ = 4300 50 = 86. {\displaystyle {\bar {x}}={\frac {4300}{50}}=86.} Or, this can be accomplished by weighting

2400-410: The expectation of the weighted sample mean will be that value, E ( x ¯ ) = μ . {\displaystyle E({\bar {x}})=\mu .} When treating the weights as constants, and having a sample of n observations from uncorrelated random variables , all with the same variance and expectation (as is the case for i.i.d random variables), then

2460-610: The following expectancy: E [ y i ′ ] = y i E [ I i ] = y i π i {\displaystyle E[y'_{i}]=y_{i}E[I_{i}]=y_{i}\pi _{i}} ; and variance: V [ y i ′ ] = y i 2 V [ I i ] = y i 2 π i ( 1 − π i ) {\displaystyle V[y'_{i}]=y_{i}^{2}V[I_{i}]=y_{i}^{2}\pi _{i}(1-\pi _{i})} . When each element of

Relative atomic mass - Misplaced Pages Continue

2520-1470: The formula from above. An alternative term, for when the sampling has a random sample size (as in Poisson sampling ), is presented in Sarndal et al. (1992) as: Var ⁡ ( Y ¯ ^ pwr (known  N ) ) = 1 N 2 ∑ i = 1 n ∑ j = 1 n ( Δ ˇ i j y ˇ i y ˇ j ) {\displaystyle \operatorname {Var} ({\hat {\bar {Y}}}_{{\text{pwr (known }}N{\text{)}}})={\frac {1}{N^{2}}}\sum _{i=1}^{n}\sum _{j=1}^{n}\left({\check {\Delta }}_{ij}{\check {y}}_{i}{\check {y}}_{j}\right)} With y ˇ i = y i π i {\displaystyle {\check {y}}_{i}={\frac {y_{i}}{\pi _{i}}}} . Also, C ( I i , I j ) = π i j − π i π j = Δ i j {\displaystyle C(I_{i},I_{j})=\pi _{ij}-\pi _{i}\pi _{j}=\Delta _{ij}} where π i j {\displaystyle \pi _{ij}}

2580-441: The indicator variables get 1, so we could simply write: y ¯ w = ∑ i = 1 n w i y i ∑ i = 1 n w i {\displaystyle {\bar {y}}_{w}={\frac {\sum _{i=1}^{n}w_{i}y_{i}}{\sum _{i=1}^{n}w_{i}}}} . This will be the estimand for specific values of y and w, but

2640-501: The most widely published variant of relative atomic mass. Additionally, the continued use of the term "atomic weight" (for any element) as opposed to "relative atomic mass" has attracted considerable controversy since at least the 1960s, mainly due to the technical difference between weight and mass in physics. Still, both terms are officially sanctioned by the IUPAC . The term "relative atomic mass" now seems to be replacing "atomic weight" as

2700-408: The population mean as a ratio of an estimated population total ( Y ^ {\displaystyle {\hat {Y}}} ) with a known population size ( N {\displaystyle N} ), and the variance was estimated in that context. Another common case is that the population size itself ( N {\displaystyle N} ) is unknown and is estimated using

2760-414: The preferred term, although the term " standard atomic weight" (as opposed to the more correct " standard relative atomic mass") continues to be used. Relative atomic mass is determined by the average atomic mass, or the weighted mean of the atomic masses of all the atoms of a particular chemical element found in a particular sample, which is then compared to the atomic mass of carbon-12. This comparison

2820-432: The relative atomic mass used is that for an element from Earth under defined conditions. However, relative atomic mass (atomic weight) is still technically distinct from standard atomic weight because of its application only to the atoms obtained from a single sample; it is also not restricted to terrestrial samples, whereas standard atomic weight averages multiple samples but only from terrestrial sources. Relative atomic mass

2880-504: The relative atomic masses of the 22 mononuclidic elements (which are the same as the isotopic masses for each of the single naturally occurring nuclides of these elements) are known to especially high accuracy. For example, there is an uncertainty of only one part in 38 million for the relative atomic mass of fluorine , a precision which is greater than the current best value for the Avogadro constant (one part in 20 million). The calculation

2940-482: The same estimator, since multiplying w i {\displaystyle w_{i}} by some factor would lead to the same estimator. It also means that if we scale the sum of weights to be equal to a known-from-before population size N , the variance calculation would look the same. When all weights are equal to one another, this formula is reduced to the standard unbiased variance estimator. Oxygen-16 Oxygen-16 (symbol: O or 8 O )

3000-826: The sample (i.e.: N ^ {\displaystyle {\hat {N}}} ). The estimation of N {\displaystyle N} can be described as the sum of weights. So when w i = 1 π i {\displaystyle w_{i}={\frac {1}{\pi _{i}}}} we get N ^ = ∑ i = 1 n w i I i = ∑ i = 1 n I i π i = ∑ i = 1 n 1 ˇ i ′ {\displaystyle {\hat {N}}=\sum _{i=1}^{n}w_{i}I_{i}=\sum _{i=1}^{n}{\frac {I_{i}}{\pi _{i}}}=\sum _{i=1}^{n}{\check {1}}'_{i}} . With

3060-641: The sample is inflated by the inverse of its selection probability, it is termed the π {\displaystyle \pi } -expanded y values, i.e.: y ˇ i = y i π i {\displaystyle {\check {y}}_{i}={\frac {y_{i}}{\pi _{i}}}} . A related quantity is p {\displaystyle p} -expanded y values: y i p i = n y ˇ i {\displaystyle {\frac {y_{i}}{p_{i}}}=n{\check {y}}_{i}} . As above, we can add

SECTION 50

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3120-2159: The selection probability are uncorrelated (i.e.: ∀ i ≠ j : C ( I i , I j ) = 0 {\displaystyle \forall i\neq j:C(I_{i},I_{j})=0} ), and when assuming the probability of each element is very small, then: We assume that ( 1 − π i ) ≈ 1 {\displaystyle (1-\pi _{i})\approx 1} and that Var ⁡ ( Y ^ pwr (known  N ) ) = 1 N 2 ∑ i = 1 n ∑ j = 1 n ( Δ ˇ i j y ˇ i y ˇ j ) = 1 N 2 ∑ i = 1 n ( Δ ˇ i i y ˇ i y ˇ i ) = 1 N 2 ∑ i = 1 n ( ( 1 − π i ) y i π i y i π i ) = 1 N 2 ∑ i = 1 n ( w i y i ) 2 {\displaystyle {\begin{aligned}\operatorname {Var} ({\hat {Y}}_{{\text{pwr (known }}N{\text{)}}})&={\frac {1}{N^{2}}}\sum _{i=1}^{n}\sum _{j=1}^{n}\left({\check {\Delta }}_{ij}{\check {y}}_{i}{\check {y}}_{j}\right)\\&={\frac {1}{N^{2}}}\sum _{i=1}^{n}\left({\check {\Delta }}_{ii}{\check {y}}_{i}{\check {y}}_{i}\right)\\&={\frac {1}{N^{2}}}\sum _{i=1}^{n}\left((1-\pi _{i}){\frac {y_{i}}{\pi _{i}}}{\frac {y_{i}}{\pi _{i}}}\right)\\&={\frac {1}{N^{2}}}\sum _{i=1}^{n}\left(w_{i}y_{i}\right)^{2}\end{aligned}}} The previous section dealt with estimating

3180-454: The sources be terrestrial, natural, and stable with regard to radioactivity. Also, there are requirements for the research process. For 84 stable elements, CIAAW has determined this standard atomic weight. These values are widely published and referred to loosely as 'the' atomic weight of elements for real-life substances like pharmaceuticals and commercial trade. Also, CIAAW has published abridged (rounded) values and simplified values (for when

3240-449: The statistical properties comes when including the indicator variable y ¯ w = ∑ i = 1 n w i y i ′ ∑ i = 1 n w i 1 i ′ {\displaystyle {\bar {y}}_{w}={\frac {\sum _{i=1}^{n}w_{i}y'_{i}}{\sum _{i=1}^{n}w_{i}1'_{i}}}} . This

3300-722: The variance for small sample sizes in general, but that depends on the complexity of the statistic. For the weighted mean, the approximate variance is supposed to be relatively accurate even for medium sample sizes. For when the sampling has a random sample size (as in Poisson sampling ), it is as follows: If π i ≈ p i n {\displaystyle \pi _{i}\approx p_{i}n} , then either using w i = 1 π i {\displaystyle w_{i}={\frac {1}{\pi _{i}}}} or w i = 1 p i {\displaystyle w_{i}={\frac {1}{p_{i}}}} would give

3360-959: The variance of the weighted mean can be estimated as the multiplication of the unweighted variance by Kish's design effect (see proof ): With σ ^ y 2 = ∑ i = 1 n ( y i − y ¯ ) 2 n − 1 {\displaystyle {\hat {\sigma }}_{y}^{2}={\frac {\sum _{i=1}^{n}(y_{i}-{\bar {y}})^{2}}{n-1}}} , w ¯ = ∑ i = 1 n w i n {\displaystyle {\bar {w}}={\frac {\sum _{i=1}^{n}w_{i}}{n}}} , and w 2 ¯ = ∑ i = 1 n w i 2 n {\displaystyle {\overline {w^{2}}}={\frac {\sum _{i=1}^{n}w_{i}^{2}}{n}}} However, this estimation

3420-404: The weighted mean is equivalently: One can always normalize the weights by making the following transformation on the original weights: The ordinary mean 1 n ∑ i = 1 n x i {\textstyle {\frac {1}{n}}\sum \limits _{i=1}^{n}{x_{i}}} is a special case of the weighted mean where all data have equal weights. If

3480-427: The weighted sample mean has expectation E ( x ¯ ) = ∑ i = 1 n w i ′ μ i . {\displaystyle E({\bar {x}})=\sum _{i=1}^{n}{w_{i}'\mu _{i}}.} In particular, if the means are equal, μ i = μ {\displaystyle \mu _{i}=\mu } , then

3540-416: The weights, used in the numerator of the weighted mean, are obtained from taking the inverse of the selection probability (i.e.: the inflation factor). I.e.: w i = 1 π i ≈ 1 n × p i {\displaystyle w_{i}={\frac {1}{\pi _{i}}}\approx {\frac {1}{n\times p_{i}}}} . If the population size N

3600-2418: Was taken from Sarndal et al. (1992) (also presented in Cochran 1977), but was written differently. The left side is how the variance was written and the right side is how we've developed the weighted version: Var ⁡ ( Y ^ pwr ) = 1 n 1 n − 1 ∑ i = 1 n ( y i p i − Y ^ p w r ) 2 = 1 n 1 n − 1 ∑ i = 1 n ( n n y i p i − n n ∑ i = 1 n w i y i ) 2 = 1 n 1 n − 1 ∑ i = 1 n ( n y i π i − n ∑ i = 1 n w i y i n ) 2 = n 2 n 1 n − 1 ∑ i = 1 n ( w i y i − w y ¯ ) 2 = n n − 1 ∑ i = 1 n ( w i y i − w y ¯ ) 2 {\displaystyle {\begin{aligned}\operatorname {Var} ({\hat {Y}}_{\text{pwr}})&={\frac {1}{n}}{\frac {1}{n-1}}\sum _{i=1}^{n}\left({\frac {y_{i}}{p_{i}}}-{\hat {Y}}_{pwr}\right)^{2}\\&={\frac {1}{n}}{\frac {1}{n-1}}\sum _{i=1}^{n}\left({\frac {n}{n}}{\frac {y_{i}}{p_{i}}}-{\frac {n}{n}}\sum _{i=1}^{n}w_{i}y_{i}\right)^{2}={\frac {1}{n}}{\frac {1}{n-1}}\sum _{i=1}^{n}\left(n{\frac {y_{i}}{\pi _{i}}}-n{\frac {\sum _{i=1}^{n}w_{i}y_{i}}{n}}\right)^{2}\\&={\frac {n^{2}}{n}}{\frac {1}{n-1}}\sum _{i=1}^{n}\left(w_{i}y_{i}-{\overline {wy}}\right)^{2}\\&={\frac {n}{n-1}}\sum _{i=1}^{n}\left(w_{i}y_{i}-{\overline {wy}}\right)^{2}\end{aligned}}} And we got to

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