Q.6.114 - Elementary Statistics

Question

Express the quartiles, $Q_1,Q_2$, and $Q_3$, of a normally distributed variable in terms of its mean, $\mu$, and standard deviation, $\sigma$.

Step-by-Step Solution

Answer

The required quartile,

$Q_1=\mu -0.67\sigma$

$Q_2=\mu$

$Q_3=\mu +0.67\sigma$

Step 1: Given Information

Consider normally distributed variable.

Step 2: Explanation

In a standard normal distribution,

The quartile $Q_1$ is the area under the curve that falls below $25\%$ of data.

The area to the left of the $z$ score $0.25$

$=0.67$

$\frac{x-\mu }{\sigma }=-0.67$

Hence, $Q_1=\mu -0.67\sigma$

Step 3: Explanation

The quartile $Q_2$ is the area under the curve that falls below $25\%$ of data.

The area to the left of the $z$ score $0.50$.

$=0.0$

$\frac{x-\mu }{\sigma }=0$

$x=\mu$

Hence, $Q_2=\mu$

The quartile $Q_3$ is the area under the curve that falls below $75\%$ of data.

The area to the left of the $z$ score $0.75$

$=0.67$

$\frac{x-\mu }{\sigma }=0.67$

$x=\mu +0.67\sigma$

Hence, $Q_3=\mu +0.67\sigma$