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Is there a formal difference between weighted average and weighted mean?

I get corrected to the latter if I type in the former in wikipedia, and then there is a lot of stuff about the name "average" so I'm not sure about anything anymore.

As a small bonus question: Logically, what is the field which introduces weighted averages? To associate it with measure theory seems a little over the top.

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Average is the same as mean. – Thomas May 17 '12 at 16:54
@Gigili: Is this related to the "bonus"? I'll accept them once I understand them. I work a 40 hour job, so that might take a while, but I'll do it eventually. – NikolajK May 17 '12 at 17:08
Well, instead of setting bonus on your question, you could accept some of the good answers you've got to your previous questions! – Gigili May 17 '12 at 17:10
up vote 2 down vote accepted

As far as I know average and mean informally are interchangeable terms.

As far as weighing, the classical Greeks already were aware of not only arithmetic, geometric and harmonic means but possibly as many as 10 different types.

As discussed in Graziani and Veronese in "How to compute a mean? The Chisini approach and its applications" Am Stat 2009:

"(Oscar) Chisini in 1929 pointed out that in a practical context a mean should simplify the problem under investigation (by replacing several observations by a single value) so that the overall evaluation of the problem itself remains unchanged. Therefore, the main issue is the specification of the invariance requirement, being a function of the observation, that we want to remain unchanged while replacing the observations by their mean"

The authors also write:

"The approach has a double advantage. First by discouraging any automatic procedure it makes students understand the substance of the problem for which a mean is required. Second, it does not require a preliminary (and necessariy imcomplete) list of different mean formulas."

Examples of invariance requirements and resulting means listed in Table 1 include weighed arithmetic, weighed quadratic, weighed harmonic, weighed geometric, weighed power, weighed exponential.

The paper works through several easily followed practical applications including Mean Traveling Speed, Mean Interest Rate, Mean Exchange Rate and others.

Finally, the authors note that Chisini Mean does not directly address important statistics like mode and median, but generalization by A. Herzel in 1961 (A paper I've been searching for) recasts the invariance constraint as an optimization problem to handle these.

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I think the statistics profession deserves credit for weighted average. It is a young profession with significant development only beginning at around 1890. In survey sampling ( a branch of statistics) stratifed random samples are often taken and if the sample proportion in the strata are not the same as in the population you have to take an unequally weighted average to get an unbiased estimate of the population mean. This is where it got a lot of use even though it may not be the first example. – Michael Chernick May 18 '12 at 22:13
@MichaelChernick, but which weighing? The Chisini mean approach is close to the spirit of physics by placing the focus on the invariants (symmetries) of the problem. Bias, unlike variance, is not an intrinsic property of the data. – alancalvitti May 19 '12 at 1:39
Bias depends on the population distribution. If a sample is selected in a non random way it can be biased with respect to the target population. Bias and variance are both properties that data can have when viewed as a sample from a target population. I am not claiming statisticians were the first but we did give it emphasis. I don't know what you mean by your question "which weighting?" I am talking about any weighting that is unequal. In the case of survey sampling take two strata for simplicity. Suppose in the population there are twice as many people in strata 1 than in strata 2. – Michael Chernick May 19 '12 at 2:08
Now we can get measurements for the height of every individual in the population. We sample equally from the strata. Strata 1 has mean 65 inches and strata 2 has mean 73 inches. we select 25 people at random from strata 1 and 25 from strata 2. Z1 is the sample mean for strata 1 and Z2 for strata 2. E(Z1)=65 and E(Z2)=73 The population mean is [2(65)+73]/3. But an equally weighted mean would be the pooled mewn and it expected value would be [65+73]/2. The population mean is 66.67. But the expected value for the sample is 69. – Michael Chernick May 19 '12 at 2:38
Therefore the bias is 1.33. But the weighted average (2 Z1a + Z2a)/3 has expected value 66.67 where Z1a is the sample average for group 1 and Z2a is the sample average for strata 2 sample average. – Michael Chernick May 19 '12 at 2:39

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