So far, you have worked exclusively with data values that are a whole number of standard deviations from the mean. Not all data values correspond so nicely to the standard deviation, so it is useful to be able to determine the number of standard deviations any data value is from the mean. The z-score formula allows you to do this; you may have discovered a similar formula in the Warm Up.

In this formula, z is the z-score, x is the data value of interest, ยต is the mean, and ฯƒ is the standard deviation.

The x - ยต gives the distance between the mean and the data value and dividing that difference by the size of one standard deviation gives the number of standard deviations, the z-score, that the data value, x, is from the mean.

Example 2

Determine the z-scores of 101, 116, and 157 for a normal distribution with a mean of 119 and a standard deviation of 17. Sketch where each value would lie on the normal curve.

Example 3

Determine the data value for a normally distributed set of data that has a z-score of 2.1, where the mean is 65 and the standard deviation is 4.3.