6.11: z Scores and Area Under the Curve
z scores are the standardized values obtained after converting a normal distribution into a standard normal distribution. A z score is measured in units of the standard deviation. The z score tells you how many standard deviations the value x is above (to the right of) or below (to the left of) the mean, μ. Values of x that are larger than the mean have positive z scores, and values of x that are smaller than the mean have negative z scores. If x equals the mean, then x has a z score of zero. The z score allows us to compare data that are normally distributed but scaled differently.
A standardized graph can help determine the probability function. The area under the density curve between two points corresponds to the probability that the variable falls between those two values. The area under the curve is always 1. One can also find the area for a particular z score by referring to the z score table, which shows the cumulative areas under the standard normal distribution from the left side of the curve.
This text is adapted from Openstax, Introductory Statistics, Section 6.1
