Why did they divide by 4? Next let another random variable Y have. the p.m.f. g(

Question

Why did they divide by 4?

Next let another random variable Y have. the p.m.f. g(y) = 1/3, y = -2, 0.2. Its mean is also zero, and it is easy to show that Var(Y) = 8/3. so the standard deviation of Y is sigmay = 2 2/3. Here the standard deviation of Y is twice that of the standard deviation of X, reflecting the fact that the probability of Y is spread out twice as much as that of X. Rolling a fair six-sided die five independent times could result in the following sample of n = 5 observations: x1 = 3, x2 = 1, x3 = 2, x4 = 6, x5 = 3. In this case x = 3 + 1 + 2 + 6 + 3/5 = 3 and s2 = (3 - 3)2 + (1 - 3)2 + (2 - 3)2 + (6 - 3)2 + (3 - 3)2/4 = 14/4 = 3.5. It follows that s = 14/4 = 1.87. Roughly, s can be thought of as the average distance that the x-values are away from the sample mean x . Clearly this is no exact because in this example the distances from x = 3 are 0, 2, 1, 3, 0.

Explanation / Answer

In the formula for sample standard deviation we always divide by n-1, not n. Here n=5, so n-1 = 4. Dividing by n-1 makes the sample SD an unbiased estimator of the population SD.

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