Why $ \Phi (z)=0.8790$ is equal to $F(x)=Φ\left(\frac{(x-μ)}σ\right)= 0.8790$?

Question

I have a data set which consists of measured time in seconds.

Secs= ${3000, 3857, 2400, 3323}.$ Mean $\mu =3145$. Standard deviation $\sigma=609.556$.

I calculated the Standard Normal variable for $3857$. $Z=\frac{(X-μ)}σ=\frac{(3857-3145)}{609.556}=1.1680$. Using the table of Standard Normal Distribution I found $ \Phi (z)=0.8790$.

I also calculated the cdf using: $F(x)=Φ\left(\frac{(x-μ)}σ\right)= \frac{1}{2}\left [1+erf\left(\frac{(x-μ)}{(σ\sqrt{2})}\right)\right]$. $F(3857)=0.8790$

Can somebody explain to me why I get the same result both times, and what each result means? Is it because my dataset is Normally distributed?

Thank you.

Answer 1

Compare $\Phi(z)$ with $z=\displaystyle\frac{(x-μ)}{σ}$ (your first computation), and $\displaystyle F(x)=\Phi \frac{(x-μ)}{σ}$ (your second computation).