Section 58 Standard Normal Distribution: Properties

Normal Distribution

Standard Normal Distribution

Every normal distribution is symmetric about its mean.

Therefore, for the standard normal distribution:

\[\large Prob(z < -1.96) = Prob(z > +1.96) = 0.025 \]

In other words, we can also say:

\[\large Prob(-1.96 < z < +1.96) = 0.95 \]

This is an important result, partly because it generalizes to the following, for any normal distribution:

When: \[\large X \sim Normal(\mu, \sigma^2)\]

\[\large Prob(\mu - 1.96\sigma < X < \mu + 1.96\sigma) = 0.95 \]

The probability that X lies within 1.96 standard deviations of its mean is 0.95.