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$2$.

Short answer

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

Step-by-step solution

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.

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$2.$