Standardising

Normal curves are all the same when measured in standard deviations.
The turning points in every normal curve are 1 standard deviation to either side of the mean.
68% of the area under a normal curve (68% of a normal distributions observations) is within 1 standard deviation of the mean.
95% are withing 2 standard deviations of the mean, and
99.7% are within 3.

We can find the proportion of observations in a normal distribution greater than or less than a value by describing that value in terms of standard deviations from the mean. Doing so allows us to use known properties of normal curves in general, rather than try to find information about a specific one.

To convert a value to standard deviations from the mean we use the formula:

The result (z) is called a standardised value, or a z-score.
z has the normal distribution N(0,1) - this is called the "standard normal distribution"

The area to the left of z under a standard normal curve can be found using a calculator, computer, or table of standard normal probabilities. This area is the proportion of observations in the distribution less than x. 1 - this area is the proportion of observations greater than x (the area under a density curve = 1, so 1 - the area to the left of z = the area to the right of z.)

To find a value given the proportion of observations less than or greater than it, simply do the above in reverse.

Continue to next section: Sampling Distributions