3/17/2024 0 Comments Math range definitionThe arithmetic mean of this data set right here, it is But if you are going to goįurther in statistics, I just want to make thatĬlarification. Samples of it, and you're going to try to estimate With, as you see, the population measuresįancy words. This is the entire population of our data. Statistics, you're going to understand the differenceīetween a population and a sample. Now let's calculate theĪrithmetic mean for both of these data sets. Let's say I have negativeġ0, 0, 10, 20 and 30. Video is to expand that a little bit to understand I have a longer, more detailed answer here (I really wish the KA links were shorter):Ībout different ways to represent the central tendency And even better, the sampling distribution of the sample mean converges to the Normal distribution, so a lot of methods can be built on top of the Normal distribution, and as long as the sample mean is the starting point, everything should work out. The method of least squares also results in the sample mean - a very intuitive and common measure of central tendency - being the "best" measure of center. The Normal distribution goes hand-in-hand with the notion of squaring deviations, and scientists centuries ago noticed that the Normal distribution worked quite well to model their astronomical data. Though it's not entirely the only reason. Using squares (or the method of "least squares") certainly does often make derivations easier. But remember, by squaring the differences, we get a wider spread between the differences, which is called higher weighting. Thanks to Lura Ercolano for clearing my misconception about using absolute value to get the variance. Q2) I think we could use the absolute value,but for the official definition, you have to square the differences. Therefore, the difference between Variance and the Standard Difference is that the Variance is "The average of the squared differences from the Mean" and the Standard Deviation is it's square-root. And of course, you will see the same when you have endured the boring process of calculating the Variance and then the Standard Deviation. You can see clearly that the data-points are grouped closely together more in the first set than the second set of data-points. The smaller the Standard Deviation, the closely grouped the data point are. Standard Deviation is the measure of how far a typical value in the set is from the average. Basically, it is the square-root of the Variance (the mean of the differences between the data points and the average). 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. There are many questioners here (including myself) wondering why squaring is used in the definition of variance instead of the more sensible absolute value.
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