Useful Properties of the Normal Distribution

in Properties, Normal, z-test

If you're not a statistician it might come as a surprise to learn that many power and sample size formulas can easily be derived using only a minimal amount of basic algebra and knowing just a few useful properties of the Normal distribution.

Properties of Normal Random Variables

  • If $n$ random variables are independent and normally distributed, each with mean $\mu$ and variance $\sigma^2$, then their average is also normally distributed with the same mean $\mu$ and variance $\sigma^2/n$.
    $X_i\overset{iid}\sim N(\mu,\sigma^2)\;,\; i=1,\dots,n \quad\Rightarrow\quad \bar{X}\sim N(\mu,\sigma^2/n)$

  • If a random variable is normally distributed with mean $\mu$ and variance $\sigma^2$, then subtracting $a$ makes the mean $\mu-a$, and dividing by $b$ makes the variance $\sigma^2/b^2$.
    $X\sim N(\mu,\sigma^2) \quad\Rightarrow\quad \frac{\displaystyle X-a}{\displaystyle b}\sim N\left(\mu-a,\frac{\displaystyle\sigma^2}{\displaystyle b^2}\right)$

  • Using the previous property, subtracting the mean and dividing by the standard deviation $\sigma$ give a standard normal random variable.
    $X\sim N(\mu,\sigma^2) \quad\Rightarrow\quad \frac{\displaystyle X-\mu}{\displaystyle \sigma}\sim N(0,1)$

  • If two independent random variables are normally distributed with means $\mu_A$ and $\mu_B$ and variances $\sigma_A^2$ and $\sigma_B^2$, then the difference between them is normally distributed with mean $\mu_A-\mu_B$ and variance $\sigma_A^2+\sigma_B^2$.
    $X_i\overset{indp.}\sim N(\mu_i,\sigma_i^2)\;,\; i=A,B \quad\Rightarrow\quad X_A-X_B\sim N(\mu_A-\mu_B,\sigma_A^2+\sigma_B^2)$

Symmetry Properties

  • Let $z_a$ be the $a^{th}$ quantile of the standard normal distribution. That is, the area under the standard normal curve to the left of $z_a$ is $a$. Then
    $z_a = -z_{1-a}$

  • Let $\Phi$ be the standard normal distribution function. That is, the area under the standard normal curve to the left of $z$ is $\Phi(z)$. Then
    $\Phi(-z)=1-\Phi(z)$



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