Properties of the Standard Normal Distribution


The normal distribution probability is specific type of continuous probability distribution. A normal distribution variable can take random values on the whole real line, and the probability that the variable belongs to any certain interval is obtained by using its density function. For the non-technical readers, a density is a function that allows to compute probabilities via integration on appropriate ranges, but for most practical applications, we can use software to skip the mathematical details. The main properties of a normally distributed variable are:

  • It is bell-shaped, where most of the area of curve is concentrated around the mean, with rapidly decaying tails.

  • It has two parameters that determine its shape. Those parameters are the population mean and population standard deviation.

  • It is symmetric with respect to its mean.

  • The mean, median and mode of the distribution coincide

If you need to compute normal distribution probabilities, please go to our normal distribution curve calculator, where you’ll find an online tool that will help with the calculation and it will graph the corresponding area.

A very special case consists of the case of the standard normal distribution. This corresponds to the case of a normal distribution with mean equal to \(\mu\) = 0, and standard deviation equal to \(\sigma\) = 1. The importance of a the standard normal distribution is that with the appropriate transformations (this is, converting normal scores into z-scores), all normal probability calculations can be reduced to calculations with the standard normal distribution.

What are the z-scores? Z-scores are simply values of a standard normal distribution. EVERY other normal distribution can be turned into a standard normal distribution in the following way. Assume that X has a normal distribution with mean \(\mu\) and standard deviation \(\sigma\). Then if we define \(Z = \frac{X – \mu}{\sigma}\) we have that Z has a standard normal distribution.

Now, that is all great, but how do you compute any normal probability using the standard normal distribution? Simple. Think of the following example:

I want to compute \(\Pr(X \le 40)\), where X is a normally distributed variable, with mean \(\mu\) = 35 and a standard deviation of \(\sigma\) = 25. So then I compute the z-score of X = 40:

\[Z = \frac{X – \mu}{\sigma} = \frac{40 – 35}{25} = 0.2\]

and now we make the critical observation that \(\Pr(X \le 40) = Pr(Z \le 0.2)\), and this last probability can be obtained with readily available standard normal distribution tables, or using software such as Excel or others. In fact, using a standard normal distribution table we find that \(\Pr(Z \le 0.2) = 0.5793\). Hence

\[ \Pr(X \le 40) = Pr(Z \le 0.2) = 0.5793\]

If you need to compute normal distribution probabilities, please go to our normal distribution curve calculator




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