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

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.

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).