![]() ![]() ![]() Q1) The Standard Deviation is the "mean of mean". Note: for more detailed information on the advantages of the related Absolute Mean Deviation, see. Instead, a measure of variance that shows the spreading of the data from each other should be used rather than the standard one, which shows the order-2 spreading from the mean. For non-Gaussian data, this def of variance is erroneous. But non-Gaussian distributions also frequently arise (such as when making many measurements with a ruler). In summary, the definition of variance given here by Khan only applies to Gaussian distributions, which frequently arise in nature and in human behavior. For general data, the mean is not defined, and other, more robust statistical measures must be found. In fact, the mean itself only works when there IS a mean,which is when the data is normally distributed. As soon as you have data derived from two or more normal distributions, or a gamma distribution, or a Poisson distribution, or anything else, using abs val works better. The point is that the use of the mean and squaring in the definition of variance works great only for normal (Gaussian) distributions. We might say, a least-squares fit, since one of the motivations is fitting 2nd-order polynomials to the error data. No, the real reason is historical: Gauss used the square variance definition to introduce his concept of normal distributions, where it is a perfect and natural fit. In fact, it is an n-dimensional problem, where n is the number of data measurements. There is not much justification for this, as variance is not obviously an unconstrained 2-D distance problem. Next, some people like Euclidean distance over Manhattan (rectangular) distance. But computer numerical analysis can handle discontinuities, so calculations using the abs val definition should be easy using an advanced computer calculator program. ![]() For example, the first derivative of the abs val func has a discontinuity at zero. This is largely irrelevant, since we have computers to aid us. I've done a quick Web search on this question, and I believe I understand this better.įirst, almost all the the reasons given have to do with ease of computation. The Codomain is actually part of the definition of the function.Īnd The Range is the set of values that actually do come out.There are many questioners here (including myself) wondering why squaring is used in the definition of variance instead of the more sensible absolute value. The Codomain is the set of values that could possibly come out. The Codomain and Range are both on the output side, but are subtly different. Or if we are studying whole numbers, the domain is assumed to be whole numbers.īut in more advanced work we need to be more careful! Codomain vs Range.Usually it is assumed to be something like "all numbers that will work".Yes, but in simpler mathematics we never notice this, because the domain is assumed: So, the domain is an essential part of the function. In this case the range of g(x) also includes 0.Īlso they will have different properties.įor example f(x) always gives a unique answer, but g(x) can give the same answer with two different inputs (such as g(-2)=4, and also g(2)=4) Example: a simple function like f(x) = x 2 can have the domain (what goes in) of just the counting numbers Įven though both functions take the input and square it, they have a different set of inputs, and so give a different set of outputs.
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