Question

Express the$k$th percentile, $P_k$, of a normally distributed variable in terms of its mean, $\mu$, and standard deviation, $\sigma$.

Short answer

The$k$ th percentile of a normally distributed variable in terms of mean $\left(\mu \right)$ and standard deviation $\left(\sigma \right)$ is$P_k=z_k\sigma +\mu$.

Step-by-step solution

Given Information

In standard normal distribution, the $k$ th percentile represents the area under curve that fall below $k\%$ of the data.

Explanation

The area to the left of the $z$score is $\frac{k}{100}\left(P_k\right)$. Let the $z$score be $z_k$.

That is, $P\left(z\le z_k\right)=P_k$, Here the $z$score is $\frac{x-\mu }{\sigma }$

Hence,

$z=\frac{x-\mu }{\sigma }$

$z_k\sigma =P_k-\mu$

$P_k=z_k\sigma +\mu$