Wikipedia / Weighted arithmetic mean
Wikipedia / 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 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 a few counterintuitive properties, as captured for instance in Simpson's paradox.
Examples
Basic example
Given two school with 20 students, one with 30 test grades in each class as follows:
Morning class = {62, 67, 71, 74, 76, 77, 78, 79, 79, 80, 80, 81, 81, 82, 83, 84, 86, 89, 93, 98}
Afternoon class = {81, 82, 83, 84, 85, 86, 87, 87, 88, 88, 89, 89, 89, 90, 90, 90, 90, 91, 91, 91, 92, 92, 93, 93, 94, 95, 96, 97, 98, 99}
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 the class means and the number of students in each class are needed.
Convex combination example
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:
Mathematical definition
Formally, the weighted mean of a non-empty finite tuple, with corresponding non-negative weights 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).
. For such normalized weights, the weighted mean is equivalently:
.
One can always normalize the weights by making the following transformation on the original weights:
.
The ordinary mean is a special case of the weighted mean where all data have equal weights.
If the data elements are independent and identically distributed random variables, the standard error of the weighted mean, can be shown via uncertainty propagation to be:
Variance-defined weights
potentially comes from a different probability distribution with known variance, 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:
. It 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.
Statistical properties
Expectancy
, 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).
Variance
Simple i.i.d. case
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) can be estimated as the multiplication of the unweighted variance by Kish's design effect (see proof):
However, this estimation is rather limited due to the strong assumption about the x observations. This has led to the development of alternative, more general, estimators. For example, using the residuals and the a posteriori variance of unit weight:
and:
Other alternatives follow below.
Survey sampling perspective
From a model based are not i.i.d random variables. An alternative perspective for this problem 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). In this context, each value of y 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) 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 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).
.
-expanded y-expanded y. As above, we can add a tick mark if multiplying by the indicator function
In this design based.
Variance of the weighted sum (pwr-estimator for totals)
If the population size N.
If 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:
An alternative term, for when the sampling has a random sample size (as in Poisson sampling), is presented in Sarndal et al. (1992) as:
is the probability of selecting both i and j..
), and when assuming the probability of each element is very small, then:
Variance of the weighted mean (-estimator for ratio-mean)
. This will be the estimand.
This 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 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:
.
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.
We have (at least) two versions of variance for the weighted mean: one with known and one with unknown population size estimation. There is no uniformly better approach, but the literature presents several arguments to prefer using the population estimation version (even when the population size is known). For example: if all y values are constant, the estimator with unknown population size will give the correct result, while the one with known population size will have some variability. Also, when the sample size itself is random (e.g.: in Poisson sampling), the version with unknown population mean is considered more stable. Lastly, if the proportion of sampling is negatively correlated with the values (i.e.: smaller chance to sample an observation that is large), then the un-known population size version slightly compensates for that.
For the trivial case in which all the weights are equal to 1, the above formula is just like the regular formula for the variance of the mean (but notice that it uses the maximum likelihood estimator for the variance instead of the unbiased variance. I.e.: dividing it by n instead of (n-1)).
Bootstrapping validation
It has been shown, by Gatz et al. (1995), that in comparison to bootstrapping methods, the following (variance estimation of ratio-mean using Taylor series linearization) is a reasonable estimation for the square of the standard error of the mean (when used in the context of measuring chemical constituents):
. Further simplification leads to
Gatz et al. mention that the above formulation was published by Endlich et al. (1988) when treating the weighted mean as a combination of a weighted total estimator divided by an estimator of the population size, based on the formulation published by Cochran (1977), as an approximation to the ratio mean. However, Endlich et al. didn't seem to publish this derivation in their paper (even though they mention they used it), and Cochran's book includes a slightly different formulation. Still, it's almost identical to the formulations described in previous sections.
Replication-based estimators
Because there is no closed analytical form for the variance of the weighted mean, it was proposed in the literature to rely on replication methods such as the Jackknife and Bootstrapping.
Other notes
, the variance of the weighted sample mean is
whose square root can be called the standard error of the weighted mean (general case).
, the weighted sample mean will have variance
, i.e., it degenerates into the standard error of the mean, squared.
.
Related concepts
Mediant
is equal to their mediant.
Weighted sample variance
Typically when a mean is calculated it is important to know the variance and standard deviation is used, the variance of the weighted sample is different from the variance of the unweighted sample.
The biased weighted sample variance is defined similarly to the normal biased:
for normalized weights. If the weights are frequency weights (and thus are random variables), it can be shown for iid Gaussian observations.
For small samples, it is customary to use an unbiased estimator for the population variance. In normal unweighted samples, the N in the denominator (corresponding to the sample size) is changed to N−1 (see Bessel's correction). In the weighted setting, there are actually two different unbiased estimators, one for the case of frequency weights and another for the case of reliability weights.
Frequency weights
If the weights are frequency weights (where a weight equals the number of occurrences), then the unbiased estimator is:
, and we get the same result either way.
are normalized to 1, then the correct expression after Bessel's correction becomes
has a different meaning other than frequency weight.
The estimator can be unbiased only if the weights are not standardized nor normalized, these processes changing the data's mean and variance and thus leading to a loss of the base rate (the population count, which is a requirement for Bessel's correction).
Reliability weights
If the weights are instead reliability weights, taking expectations we have,
is the effective sample size, ensuring that the expected value of the estimated variance equals the actual variance of the sampling distribution. The final unbiased estimate of sample variance is:
. The degrees of freedom of this weighted, unbiased sample variance vary accordingly from N−1 down to0. The standard deviation is simply the square root of the variance above.
As a side note, other approaches have been described to compute the weighted sample variance.
Weighted sample covariance
(each set of single observations on each of the K.
Then the weighted mean is given by
And the weighted covariance matrix is given by:
Similarly to weighted sample variance, there are two different unbiased estimators depending on the type of the weights.
Frequency weights
If the weights are frequency weights, the unbiased, with Bessel's correction, is given by:
This estimator can be unbiased only if the weights are not standardized nor normalized, these processes changing the data's mean and variance and thus leading to a loss of the base rate (the population count, which is a requirement for Bessel's correction).
Reliability weights
In the case of reliability weights, the weights are normalized:
:
Then the weighted mean can be simplified to
and the unbiased is:
The reasoning here is the same as in the previous section.
and this reduces to:
, then the weighted mean and covariance reduce to the unweighted sample mean and covariance above.
Vector-valued estimates
The above generalizes easily to the case of taking the mean of vector-valued estimates. For example, estimates of position on a plane may have less certainty in one direction than another. As in the scalar case, the weighted mean of multiple estimates can provide a maximum likelihood by the covariance matrix and the arithmetic inverse by the matrix inverse (both denoted in the same way, via superscripts); the weight matrix then reads:
(where the order of the matrix–vector product is not commutative
For example, consider the weighted mean of the point [1 0] with high variance in the second component and [0 1] with high variance in the first component. Then
then the weighted mean is:
which makes sense: the [1 0] estimate is "compliant" in the second component and the [0 1] estimate is compliant in the first component, so the weighted mean is nearly [1 1].
Accounting for correlations
is the covariance matrix is a design matrix equal to a vector of ones). The Gauss–Markov theorem states that the estimate of the mean having minimum variance is given by:
and
where:
Decreasing strength of interactions
Consider the time series of an independent variable.
Exponentially decreasing weights
normalized weights by
is simply
.
such that the tail area is sufficiently small.
Weighted averages of functions
The concept of weighted average can be extended to functions. Weighted averages of functions play an important role in the systems of weighted differential and integral calculus.
Correcting for over- or under-dispersion
is too large. The correction that must be made is
is the reduced chi-squared:
can be called the standard error of the weighted mean (variance weights, scale corrected).
, which again reduces to the standard error of the mean, formulated in terms of the sample standard deviation (squared),
See also
Notes
References
Further reading
External links