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assume a probability distribution with a mean of %10 and standard deviation of %1.5. In wanting to solve the probability being lower than %5, the normal distribution is written down and integrated as follows:

$\int_{-∞}^{0.05} e^{(-\frac{1}{2} ((x - 0.1)/0.015)^2)}/(\sqrt{2 π} 0.015) dx = 0.00042906$

which gives %0.04

if you calculate it through the z score(assuming a sample size of 10) it would be:

Z=(0.05-0.10)/(0.015/sqrt(10))=-10.5409

looking at a z score table would then give a probability of 2.798*10^-26

why do the two approaches give different answeres?

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    $\begingroup$ Where does the sqrt(10) come from? Clearly, $$z\equiv \frac{x-\mu}{\sigma}=-10/3$$, and $N(z)=0.0042906$ $\endgroup$
    – Kermittfrog
    Aug 23, 2022 at 9:20
  • $\begingroup$ "standard deviation of %1.5" is unusually small. Are you sure of this value? I think 15% (or more) would be a more usual value for stock returns. No surprise then that the probability you found is quite low. Any outcome outside of $\mu \pm 2 \sigma$ is rare. $\endgroup$
    – nbbo2
    Aug 23, 2022 at 9:57

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Seems like you calculated the z-score incorrectly. There’s no need for the sqrt(10) in the denominator since we just want the standard deviation there.

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