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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