One of the most fundamental properties of the normal probability curve is the area property. Area under normal curve is used to determine the probability of
occurence of a given value, the value is first standardized so that it
can be converted to a Z score.

The area under the normal probability curve between the coordinates at X = $\mu$ - $\sigma$ and X = $\mu$ + $\sigma$ is 0.6826.

Area under the normal probability curve between the ordinates at X = $\mu$ - 2$\sigma$ and X = $\mu$ +

2$\sigma$ is 0.9544 and the area under the normal probability curve between the ordinates at X = $\mu$

- 3$\sigma$ and X = $\mu$ + 3$\sigma$ is 0.9973. this range covers almost the entire area which is 1.

The standard normal variate corresponding to X is Z =

$\frac{X - \mu}{\sigma}$

When X = $\mu$ + $\sigma$, Z =

$\frac{\mu + \sigma - \mu}{\sigma}$ = 1;

When X = $mu$ - $\sigma$, Z =

$\frac{\mu - \sigma - \mu}{\sigma}$ = - 1;

When X = $\mu$ + 2 $\sigma$, Z =

$\frac{\mu + 2\sigma - \mu}{\sigma}$ = 2;

When X = $mu$ - 2$\sigma$, Z =

$\frac{\mu - 2\sigma - \mu}{\sigma}$ = - 2;

When X = $\mu$ + 3 $\sigma$, Z =

$\frac{\mu + 3\sigma - \mu}{\sigma}$ = 3;

When X = $mu$ - 3 $\sigma$, Z =

$\frac{\mu - 3 \sigma - \mu}{\sigma}$ = - 3;

Therefore the area under the standard normal probability curve

**1)** Between the ordinates at Z = $\pm$ 1 is 0.6826.

**2)** Between the ordinates at Z = $\pm$ 2 is 0.9544.

3) Between the ordinates at Z = $\pm$ 3 is 0.9973.