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58-492: 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 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

116-430: A convex space , not only a vector space. The arithmetic mean has several properties that make it interesting, especially as a measure of central tendency. These include: The arithmetic mean may be contrasted with the median . The median is defined such that no more than half the values are larger, and no more than half are smaller than it. If elements in the data increase arithmetically when placed in some order, then

174-479: A data set X {\displaystyle X} is denoted as X ¯ {\displaystyle {\overline {X}}} ). The arithmetic mean can be similarly defined for vectors in multiple dimensions, not only scalar values; this is often referred to as a centroid . More generally, because the arithmetic mean is a convex combination (meaning its coefficients sum to 1 {\displaystyle 1} ), it can be defined on

232-720: 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

290-416: A result of 180 ° . This is incorrect for two reasons: In general application, such an oversight will lead to the average value artificially moving towards the middle of the numerical range. A solution to this problem is to use the optimization formulation (that is, define the mean as the central point: the point about which one has the lowest dispersion) and redefine the difference as a modular distance (i.e.,

348-554: 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

406-411: A situation with n {\displaystyle n} numbers being averaged). If a numerical property, and any sample of data from it, can take on any value from a continuous range instead of, for example, just integers, then the probability of a number falling into some range of possible values can be described by integrating a continuous probability distribution across this range, even when

464-407: 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,

522-427: Is 3 + 5 2 = 4 {\displaystyle {\frac {3+5}{2}}=4} , or equivalently 3 ⋅ 1 2 + 5 ⋅ 1 2 = 4 {\displaystyle 3\cdot {\frac {1}{2}}+5\cdot {\frac {1}{2}}=4} . In contrast, a weighted mean in which the first number receives, for example, twice as much weight as the second (perhaps because it

580-572: 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,

638-457: 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}}} ,

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696-525: Is assumed to appear twice as often in the general population from which these numbers were sampled) would be calculated as 3 ⋅ 2 3 + 5 ⋅ 1 3 = 11 3 {\displaystyle 3\cdot {\frac {2}{3}}+5\cdot {\frac {1}{3}}={\frac {11}{3}}} . Here the weights, which necessarily sum to one, are 2 3 {\displaystyle {\frac {2}{3}}} and 1 3 {\displaystyle {\frac {1}{3}}} ,

754-520: 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

812-561: 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

870-424: 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

928-413: 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

986-486: 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

1044-410: 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

1102-563: 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

1160-677: 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

1218-479: 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}} )

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1276-1072: 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

1334-593: The arithmetic mean ( / ˌ æ r ɪ θ ˈ m ɛ t ɪ k / arr-ith- MET -ik ), arithmetic average , or just the mean or average (when the context is clear) is the sum of a collection of numbers divided by the count of numbers in the collection. The collection is often a set of results from an experiment , an observational study , or a survey . The term "arithmetic mean" is preferred in some mathematics and statistics contexts because it helps distinguish it from other types of means, such as geometric and harmonic . In addition to mathematics and statistics,

1392-407: The distribution of income for which a few people's incomes are substantially higher than most people's, the arithmetic mean may not coincide with one's notion of "middle". In that case, robust statistics, such as the median , may provide a better description of central tendency. The arithmetic mean of a set of observed data is equal to the sum of the numerical values of each observation, divided by

1450-733: 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

1508-443: 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

1566-469: The 1980s, the median income in the United States has increased more slowly than the arithmetic average of income. A weighted average, or weighted mean, is an average in which some data points count more heavily than others in that they are given more weight in the calculation. For example, the arithmetic mean of 3 {\displaystyle 3} and 5 {\displaystyle 5}

1624-2535: 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

1682-452: The arithmetic mean is frequently used in economics , anthropology , history , and almost every academic field to some extent. For example, per capita income is the arithmetic average income of a nation's population. While the arithmetic mean is often used to report central tendencies , it is not a robust statistic : it is greatly influenced by outliers (values much larger or smaller than most others). For skewed distributions , such as

1740-475: The arithmetic mean is: If the data set is a statistical population (i.e., consists of every possible observation and not just a subset of them), then the mean of that population is called the population mean and denoted by the Greek letter μ {\displaystyle \mu } . If the data set is a statistical sample (a subset of the population), it is called the sample mean (which for

1798-809: 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 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)}

Weighted arithmetic mean - Misplaced Pages Continue

1856-526: The data Amount of total numbers within the data {\displaystyle {\frac {\text{Total of all numbers within the data}}{\text{Amount of total numbers within the data}}}} For example, if the monthly salaries of 10 {\displaystyle 10} employees are { 2500 , 2700 , 2400 , 2300 , 2550 , 2650 , 2750 , 2450 , 2600 , 2400 } {\displaystyle \{2500,2700,2400,2300,2550,2650,2750,2450,2600,2400\}} , then

1914-522: 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

1972-453: The distance on the circle: so the modular distance between 1° and 359° is 2°, not 358°). The arithmetic mean is often denoted by a bar ( vinculum or macron ), as in x ¯ {\displaystyle {\bar {x}}} . Some software ( text processors , web browsers ) may not display the "x̄" symbol correctly. For example, the HTML symbol "x̄" combines two codes —

2030-410: 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

2088-610: 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

2146-410: The former being twice the latter. The arithmetic mean (sometimes called the "unweighted average" or "equally weighted average") can be interpreted as a special case of a weighted average in which all weights are equal to the same number ( 1 2 {\displaystyle {\frac {1}{2}}} in the above example and 1 n {\displaystyle {\frac {1}{n}}} in

2204-1478: 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}}

2262-467: 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 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

2320-441: 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

2378-466: The median and arithmetic average are equal. For example, consider the data sample { 1 , 2 , 3 , 4 } {\displaystyle \{1,2,3,4\}} . The mean is 2.5 {\displaystyle 2.5} , as is the median. However, when we consider a sample that cannot be arranged to increase arithmetically, such as { 1 , 2 , 4 , 8 , 16 } {\displaystyle \{1,2,4,8,16\}} ,

Weighted arithmetic mean - Misplaced Pages Continue

2436-407: The median and arithmetic average can differ significantly. In this case, the arithmetic average is 6.2 {\displaystyle 6.2} , while the median is 4 {\displaystyle 4} . The average value can vary considerably from most values in the sample and can be larger or smaller than most. There are applications of this phenomenon in many fields. For example, since

2494-407: 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 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

2552-402: The naive probability for a sample number taking one certain value from infinitely many is zero. In this context, the analog of a weighted average, in which there are infinitely many possibilities for the precise value of the variable in each range, is called the mean of the probability distribution . The most widely encountered probability distribution is called the normal distribution ; it has

2610-408: 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

2668-407: The property that all measures of its central tendency, including not just the mean but also the median mentioned above and the mode (the three Ms ), are equal. This equality does not hold for other probability distributions, as illustrated for the log-normal distribution here. Particular care is needed when using cyclic data, such as phases or angles . Taking the arithmetic mean of 1° and 359° yields

2726-476: 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. Arithmetic mean In mathematics and statistics ,

2784-826: 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

2842-639: 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

2900-2159: 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

2958-447: 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

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3016-424: The total number of observations. Symbolically, for a data set consisting of the values x 1 , … , x n {\displaystyle x_{1},\dots ,x_{n}} , the arithmetic mean is defined by the formula: (For an explanation of the summation operator, see summation .) In simpler terms, the formula for the arithmetic mean is: Total of all numbers within

3074-718: 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

3132-959: 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

3190-404: 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

3248-427: 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

3306-416: 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

3364-2419: 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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