How to deal with Z-score greater than 3?

  • In a standard normal distribution how do I deal with a $Z$ value greater than 3?

    I know that z-score ranges form -3 to 3

    Consider this one ...

    mean = 70, standard deviation = 4

    I need to find $P(65 < X < 85)$.

    Transforming to standard normal gives $P(-1.25 < Z < 3.75)$

    How to deal with the $3.75$?

    Edit *Actually it's not greater than 3, $z < 3.75$. I meant smaller than, sorry. Should I assume it's just 0.4990 or what?

    z scores range from $-\infty$ to $+\infty$.

    ok i know that but i'm talkin about the standard .. which has table with values from -3 from 3 , i know it's up to infinity but what i want is to find values bigger than 3 which is not listed in the table .. i know there is another exponential formula but it's not for the standard dist.

    The $p$-value of $|z|$ statistics that are $\ge 3$ are all *small*. They never get big again... just smaller and smaller. So if your table of $p$-values for $z$ statistics stops at three, it probably says the $p$-value is 0.000 or something like that (really meaning < 0.001), and you can simply use that to conduct your tests.

  • Sergio

    Sergio Correct answer

    6 years ago

    Let me repeat (and correct) what I've said in my comment ad reply to your edit.

    You have to transfer from $X$ to $Z$ in order to use a z-score table. Since a z-score table contains a small finite subset of values, you often must settle for an approximation. So you could also settle for $P(Z<-3)\approx 0$ and $P(Z< 3)\approx 1$ (NB: $P(Z>3)\approx 1$ was a typo, sorry.)

    As to $P(-1.25<Z<3.75)$, I'll use this z-score table: $$P(-1.25<Z<3.75)=P(Z<3.75)-P(Z<-1.25)\approx 1-0.1056=0.8944$$

License under CC-BY-SA with attribution

Content dated before 6/26/2020 9:53 AM