As far as i know both variance and standard deviation is used for determining avarage deviating from the mean for each value in data set. So , if we can calculate it by using variance , why did we need to take square root of it to invent standard deviation ?
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1$\begingroup$ The SD is on the same scale as the original variable. This makes it a lot easier to interpret it $\endgroup$– stefgehrigCommented Jan 7, 2022 at 18:51
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$\begingroup$ A specific detail is that in a normal distribution the inflexions of the density function are 1 SD away from the mean. This shouldn't be thought a major rationale as variance and SD arise quite naturally in problems with quite different distributions such binomial, Poisson and exponential. The SD also has a general geometric interpretation (think Pythagoras' theorem in a suitable space). (Spotting which textbooks don't show this correctly on diagrams of normal distributions is a good diagnostic of books to be avoided.) $\endgroup$– Nick CoxCommented Jan 7, 2022 at 19:02
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$\begingroup$ @NickCox can you give the name of a textbook which teaches these concepts clearly $\endgroup$– Not a Salmon FishCommented Jan 7, 2022 at 19:12
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1$\begingroup$ See stats.stackexchange.com/questions/48347 for some suggestions. $\endgroup$– whuber ♦Commented Jan 7, 2022 at 19:36
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1$\begingroup$ I agree with the recommendation of Freedman, Pisani and Purves by @whuber in the thread he links to. $\endgroup$– Nick CoxCommented Jan 7, 2022 at 23:26
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it's actually the other way: variance was invented to measure dispersion.
when we characterize values generated by chance, we can easily two main characterizations of them: a center point and the dispersion. if you think of central point then a few ideas can come to mind: the point of highest concentration of values, the mid point in terms probabilities and a probability weighted sum. so scientists came up with mathematical representations of these concepts: the peak of the density of values, median and mean.
When it comes to dispersion, we also can think of similar concepts. For instance to measure the distance between points to left and right from the median at equal probabilities, i.e. interquartile distance. we could also get the average of absolute value of the distance from the median or mean. we could, at last, take the square root of the square distance from the mean and obtain its probability weighted average - the standard deviation. the latter two approaches are quite similar, as you can see easily - they get the measure of dispersion, of course, in the same units of measure as the variable itself.
