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Why Is Variance Squared And Not Absolute Value

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Any of the following distance can be used: $$d_n((X)_{i=1,\ldots,I},\mu)=\left(\sum | X-\mu|^n\right)^{1/n}$$ We usually use the natural euclidean distance ($n=2$), which is the one everybody uses in daily life. H., Principles and Procedures of Statistics with Special Reference to the Biological Sciences., McGraw Hill, 1960, page 288. ^ Mood, A.; Graybill, F.; Boes, D. (1974). Carl Friedrich Gauss, who introduced the use of mean squared error, was aware of its arbitrariness and was in agreement with objections to it on these grounds.[1] The mathematical benefits of The variance is half the mean square over all the pairwise differences between values, just as the Gini mean difference is based on the absolute values of all the pairwise difference. check over here

email will only be used for the most wholesome purposes. Ben December 19 at 2:58 PM \(\begingroup\)I guess I was equivocating between two senses of absolute error. But this argument didn’t rely on the coordinate system that we used. Gorard says imagine people who split the restaurant bill evenly and some might intuitively notice that that method is unfair. However, if n is even, then the set of values minimizing MAE(t) is the "median interval" [xj, xl].

Why Is Variance Squared And Not Absolute Value

Addison-Wesley. ^ Berger, James O. (1985). "2.4.2 Certain Standard Loss Functions". Applications[edit] Minimizing MSE is a key criterion in selecting estimators: see minimum mean-square error. Variance[edit] Further information: Sample variance The usual estimator for the variance is the corrected sample variance: S n − 1 2 = 1 n − 1 ∑ i = 1 n If your population is normally distributed, the standard deviation of various samples from that population will, on average, tend to give you values that are pretty similar to each other, whereas

share|improve this answer answered May 14 '14 at 12:55 Frank Harrell 39.1k173156 2 Just to add to @Frank's suggestion on Gini, there's a nice paper here: projecteuclid.org/download/pdf_1/euclid.ss/1028905831 It goes over They don’t just pose technical problem-solving issues; rather, they give us intrinsic reasons why minimizing the square error might be a good idea: When fitting a Gaussian distribution to a set share|improve this answer answered Jul 27 '10 at 4:04 arik 1 If I recall correctly, isn't the log-normal distribution not uniquely defined by its moments. –probabilityislogic Apr 10 '14 at Mean Absolute Percentage Error ISBN0-495-38508-5. ^ Steel, R.G.D, and Torrie, J.

share|improve this answer answered Oct 21 '14 at 23:27 Eric L. Do you mean interpreting Tikhonov regularization as placing a Gaussian prior on the coefficients? In fact, I would say that unbiasedness could just as easily be motivated by the niceness of squared error as the other way around. https://en.wikipedia.org/wiki/Mean_squared_error To get rid of the effect of the negative value while taking the mean, we square them.A better question would be why not use the absolute difference instead of squaring the

Abraham de Moivre did this with coin tosses in the 18th century, thereby first showing that the bell-shaped curve is worth something. Mean Square Error Formula share|improve this answer edited Jan 27 at 22:28 answered Aug 10 '10 at 22:34 Neil G 6,11311641 1 I like your answer. A symmetric, unimodal distribution. Steven Harrod 91.154 προβολές 23:31 Mean Absolute Deviation - Διάρκεια: 3:39.

Absolute Deviation Vs Standard Deviation

Specific word to describe someone who is so good that isn't even considered in say a classification Why do people move their cameras in a square motion? click Oh well. ;-) –Sabuncu Feb 11 '14 at 21:55 | show 14 more comments 20 Answers 20 active oldest votes up vote 115 down vote accepted If the goal of the Why Is Variance Squared And Not Absolute Value On the other hand, MSE is more useful if we are concerned about large errors whose consequences are much bigger than equivalent smaller ones. Mean Absolute Error Vs Mean Squared Error Neither part of it seems true to me (and the claims seem somewhat unrelated)\(\endgroup\) reply preview submit subscribe format posts in markdown.

share|improve this answer answered Jul 19 '10 at 21:15 KungPaoChicken 26116 add a comment| up vote 13 down vote Yet another reason (in addition to the excellent ones above) comes from Just find the expected number of heads ($450$), and the variance of the number of heads ($225=15^2$), then find the probability with a normal (or Gaussian) distribution with expectation $450$ and For a multivariate Laplace distribution (like a Gaussian but with absolute, not squared, distance), this isn’t true. Both are good candidates but they are different. Root Mean Squared Error

Another advantage is that using differences produces measures (measures of errors and variation) that are related to the ways we experience those ideas in life. I take your point though, I'll consider removing/rephrasing it if others feel it is unclear. –Tony Breyal Jul 22 '10 at 13:19 10 Much of the field of robust statistics That being said, the MSE could be a function of unknown parameters, in which case any estimator of the MSE based on estimates of these parameters would be a function of http://edvinfo.com/mean-square/mean-squared-error-example.html New York: Springer.

It's certainly debatable whether that's something that should be done, but in any case: Assume your $n$ measurements $X_i$ are each an axis in $\mathbb R^n$. Mean Error Formula Thus, the best measure of center, relative to this function, is the value of t that minimizes the error function, and the minimum value of the error function is the corresponding It is important that you understand this point, because other mean square error functions occur throughout statistics.

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One nice fact is that the variance is the second central moment, and every distribution is uniquely described by its moments if they exist. The same is not true for expected absolute error. Mathematical Statistics with Applications (7 ed.). Squared Difference My first friendUpdated 92w agoSay you define your error as,[math]Predicted Value - Actual Value[/math].

It’s true that one could choose to use, say, the absolute error instead of the squared error. Sergül AydöreWritten 87w agoBoth mean squared error (MSE) and mean absolute error (MAE) are used in predictive modeling. Also in regression analysis, "mean squared error", often referred to as mean squared prediction error or "out-of-sample mean squared error", can refer to the mean value of the squared deviations of http://edvinfo.com/mean-square/mean-squared-error-formula.html What about the other way around?Why do we square the margin of error?What is the formula of absolute error?

Predictor[edit] If Y ^ {\displaystyle {\hat Saved in parser cache with key enwiki:pcache:idhash:201816-0!*!0!!en!*!*!math=5 and timestamp 20161007125802 and revision id 741744824 9}} is a vector of n {\displaystyle n} predictions, and Y The post below is adapted from that answer. standard deviation11Why is the standard deviation defined as sqrt of the variance and not as the sqrt of sum of squares over N?0In the standard deviation formula, why do you divide Gorard, S. (2013).

e) - Διάρκεια: 15:00. Gini's mean difference is the average absolute difference between any two different observations. Definition of an MSE differs according to whether one is describing an estimator or a predictor. The fourth central moment is an upper bound for the square of variance, so that the least value for their ratio is one, therefore, the least value for the excess kurtosis

Contents 1 Definition and basic properties 1.1 Predictor 1.2 Estimator 1.2.1 Proof of variance and bias relationship 2 Regression 3 Examples 3.1 Mean 3.2 Variance 3.3 Gaussian distribution 4 Interpretation 5 A better metric would be one to help fit a Gamma distribution to your measurements: $\log(E(x)) - E(\log(x))$ Like the standard deviation, this is also non-negative and differentiable, but it is In the applet, click on two distinct points to generate a distribution with two distinct points. Averages play nice with affine transformations. (Higher-dimensional) averages correspond to centre of mass.