For example, if you were to regress $Y$ on $X$ and find that the standard error of the $X$ coefficient is $10$, what does the $10$ mean in layman's language?
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What about "the quadratic mean of the error" ? – Peter Jun 16 '19 at 12:28
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See https://math.stackexchange.com/questions/700160/intuition-behind-variance-forumla/700231#700231 – Michael Hoppe Jun 16 '19 at 19:00
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It also can be described as the "root-mean-square" deviation - that is the square root of the mean of the squares of the deviations of the values from the average. For a normal distribution, approximately 2/3 of the data points will lie within this distance from the average (actually, a little greater than 2/3). So it gives you an idea of how widely spread out your data is. The greater the std deviation is, the greater influence factors you are not accounting for are having on your data. – Paul Sinclair Jun 17 '19 at 01:39