The Standard Normal Distribution

Statisticians like to communicate in a way they can both understand. The use of scaling is important in that we can talk about scores as they are represented in the general population. Usually, the general population can be represented by a normal distribution, which appears as a bell shaped curve, like this:

As you can see, the curve is distributed into intervals that are symmetrical on each side. This is where standardization becomes important, and where using standard deviation to explain variation of scores is essential.

So how do we use standard deviation? Let’s say for a normal distribution of SAT scores for students across the US we have a mean of 1200 and a standard deviation of 150. A score that is 2 standard deviations above the mean will be 1500. A score that is 2 SD’s below the mean will be 900.

Standard deviation measures the level of spread of an average distribution of scores. Here we have different spreads represented by different SD’s.

We can see that for larger values for SD, given the same n, there tends be a wider distribution, meaning that the individual differences from the mean score tend to be more widely distributed.