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I am doing some self study on statistics and noticed that in the notes that I was using the $|Z|$ expression as attached in the photos below. I am confused with the "$|$" that is being used. The only logical conclusion that I was able to draw was that $P(|Z|\le 2)$ represents $P(-2 < Z < 2)$.

I would appreciate some advice on what the "$|$" symbol means.

enter image description here

enter image description here

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Thanks for the link Glen. –  user1275515 Apr 19 at 4:51

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up vote 6 down vote accepted

It's the absolute value. So |-5| = 5, and |5| = 5.

$$ P( |Z| \leq 2) = P( Z \leq 2\text{ and}-Z \leq 2) = P(Z \leq 2\text{ and }Z \geq -2) = P(-2 \leq Z \leq 2) = P(Z \geq -2) - P(Z \geq 2) = 1 - H(2) - H(2)$$

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Thanks Benjamin for the speedy reply. I am curious also why when we have the | symbol in place, we are looking at a two-tail in our calculation? Thank you so much once again. =) –  user1275515 Apr 18 at 14:52
    
By defining the p-value using absolute values you don't need 2 defnitinos for 2-tails and 1-tail tests, but rather 1 equation will suffice. –  Benjamin Apr 18 at 14:57
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that's another way of saying that the distribution shown in the plot is symmetric. –  user603 Apr 18 at 14:58
    
Thank you so much everyone. The explanation had been lucid, I now understand the concept. Appreciate it lots. –  user1275515 Apr 19 at 4:13

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