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I know that SD depends on change of scale and not on change of origin.

But what about Variance? And why?

Fred
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  • Variance in that sense behaves like standard deviation, the variance of $aX+b$ is $a^2$ times the variance of $X$. It is a consequence of the fact that the variance of $Y$ is defined as $E(Y-E(Y))^2$. – André Nicolas Apr 03 '15 at 06:19
  • @AndréNicolas This means that Variance too like SD is independent of change of origin but dependent on change of scale. The difference in the dependence is SD is 'a' times then Var is a^2 times? – Fred Apr 03 '15 at 06:35
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    Yes, that's the difference. Another way of thinking about it is that the standard deviation is the square root of the variance. I think of the variance as being more fundamental. – André Nicolas Apr 03 '15 at 06:38
  • @AndréNicolas Yes. Variance=SD^2. Thanks a lot. – Fred Apr 03 '15 at 06:39
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    You are welcome. – André Nicolas Apr 03 '15 at 06:39

1 Answers1

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Here is a numerical example with fake data: 100 observations $X_i$ simulated from a normal distribution with population mean 50 and population SD 3. Now it is easy to give a numerical illustration of the formulas in the comments.

The sample mean is $\bar X = 49.82$; the sample variance is $S_X^2 = 8.05$; and the standard deviation is $S_X = 2.84.$ (BTW: As is typical with 100 such observations, $\bar X$ is pretty close to the population mean, and $S_X$ has a somewhat larger percentage error. Population means are easier to estimate with accuracy than are population standard deviations.)

Now I add 10 to all $n = 100$ observations: $Y_i = X_i + 10.$ Then the sample mean is $\bar Y = 59.82$; the sample variance is $S_Y^2 = 8.05$; and the standard deviation is $S_Y = 2.84.$

Finally, I multiply each of the 100 original 100 observations by 10: $W_i = 10X_i.$ The results are now as follows: the sample mean is $\bar W = 498.2,$ sample variance is $S_W^2 = 805$; and the
standard deviation is $S_Y = 28.4.$

Note: All numbers have been rounded to a reasonable number of significant digits for readability.

BruceET
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