std. dev. = sqrt (1/N E (x_i – mean)^2)

(std. dev.)^2 = (1/N E (x_i – mean)^2)

11.4^2 = (1/40 E (x_i – mean)^2)

129.96 = (1/40 E (x_i – mean)^2)

(129.96)(40) = E (x_i – mean)^2)

5198.4 = E (x_i – mean)^2

So now you need 40 “differences squared” that will add up to a GRAND total of 5198.4… so let’s take the worst scattered case scenario…. score of 0 (that will give you the greatest difference from 4.2, right?)…. and we’ll take 40 of those differences (sounds like I’m ordering at McDonald’s… LOL) …. so…

(0 – 4.2)^2 (40) = 705.6 …. and you need a total of 5198.4 … and that was just assuming that all the scores were 0!!!! … which wouldn’t give you a 4.2 average I know because I’m just trying to illustrate a point here… but when you have a LARGE standard deviation… that means that your scores are “scattered” and not clustered around the mean… but I was assuming the worst case of 0 scores…. and all the differences squared… don’t even come CLOSE to the “standard deviation squared times 40” that I needed…. so with just a few 0’s… and then those close to 0 (meaning smaller differences), well, we just can’t expect those to add up any closer than what we found with scores of 0… if anything, our total will be LESS than what we figured using scores of 0… less than the 705.6 that we found… and so farther from the 5198.4 that we need…

might help to read the link I’m including… then you’ll see what I’m talking about…

http://en.wikipedia.org/wiki/Standard_deviation

Well… I hope that this helps and makes sense to you…

The quiz has five possible points. The mean calculates that the average score was 4.2 out of those 5–all well and good.

But by the Empirical rule, 68% of the students scored within one standard deviation of the mean. Using the professor’s values, the majority of the students got between 15.6 out of 5, and -7.2 out of 5–clearly nonsense. (And the other 32% must have scored beyond those values in one direction or another…)

I hope that all made sense.