Standard Normal Distribution

Topics: Probability - Distribution Function - Normal Distribution

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(definition)

The standard normal distribution, commonly denoted $Z$, is a normal distribution such that $\mu =0$ and ${\sigma }^2=1$.

In other words, if $X$ a random variable is such that $X\sim \mathcal{N}(0,1)$, we say that it follows a standard normal distribution, and commonly denote it $Z$.

Standardisation §

The values of $Z$’s distribution function are well known and can be easily found in tables. We can use these tables to facilitate our calculations with any other normal distribution $X\sim \mathcal{N}(\mu ,{\sigma }^2)$, since we can standardise $X$ as follows:

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

Properties §

Let $Z$ be a random variable with a standard normal distribution. Then:

• The mean, the median and the mode of $Z$ are all equal to $\mu$.
• $68\%$ of the observations fall in the interval $[\mu -\sigma ,\mu +\sigma ]$.
• $95\%$ of the observations fall in the interval $[\mu -2\sigma ,\mu +2\sigma ]$.
• $99.7\%$ of the observations fall in the interval $[\mu -3\sigma ,\mu +3\sigma ]$.