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If the normal distribution curve is symmetrical about the vertical line then the

mean = mode = median

This would mean that:

Prob( X < mean) = Prob( X > mean) = .....  

What is the missing part of this concept?

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2 Answers

up vote 1 down vote accepted

Generally for continuous probability distributions that are symmetric about their median, if they have a mean at all, then the probability of being greater than the mean and the probability of being less than the mean are both equal to $1/2$.

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Since

Prob(X < mean) + Prob (X = mean) + Prob (X > mean)=1

and Prob(X = mean)=0 (continuous distribution)

You get

Prob(X < mean) = Prob (X > mean)=0.5

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You might need something like: " for a continuous distribution $\Pr(X = \text{mean})=0$". –  Henry Feb 29 '12 at 15:15
    
I added that to the answer above,thanks. –  chemeng Feb 29 '12 at 15:18
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