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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?

2 Answers 2

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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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    I added that to the answer above,thanks.2012-02-29