Q. 6.87 - Elementary Statistics

Question

Explain why the percentage of all possible observations of a normally distributed variable that lie within two standard deviations to either side of the mean equals the area under the standard normal curve between $-2$ and $2$.

Step-by-Step Solution

Answer

The area under the standard normal curve for the range $-2$ to $2$ is then the percentage of observations within the range.

Step 1: Given information

For a normally distributed random variable, the proportion of all possible values within the interval of two standard deviations from either side of the mean is equal to the area between $-2$ and $2$ under the standard normal curve.

Step 2: Explanation

The random variable $X$ have $z-$score which is given by

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

Where $\mu$ is the mean and

$\sigma$ is the standard deviation

Let the interval limits be

$a=\mu -2\sigma$

$b=\mu +2\sigma$

For a,

$z=\frac{(\mu -2\sigma )-\mu }{\sigma }$

    $=-2.$

For b, 

$z=\frac{(\mu +2\sigma )-\mu }{\sigma }$

    $=2$

Hence the values that lie at two standard deviations from the mean have the $z-$ scores $-2$ or $2.$