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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.

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enter image description here

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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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  • $\begingroup$ 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. =) $\endgroup$
    – user1275515
    Commented Apr 18, 2014 at 14:52
  • $\begingroup$ 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. $\endgroup$
    – bdeonovic
    Commented Apr 18, 2014 at 14:57
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    $\begingroup$ that's another way of saying that the distribution shown in the plot is symmetric. $\endgroup$
    – user603
    Commented Apr 18, 2014 at 14:58
  • $\begingroup$ Thank you so much everyone. The explanation had been lucid, I now understand the concept. Appreciate it lots. $\endgroup$
    – user1275515
    Commented Apr 19, 2014 at 4:13

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