Outline

Discuss characteristics of the z-score distribution

Discuss other standardized distributions based on z scores

Work on problems

Procedures for describing an individual score’s location within a distribution, using the mean as a reference point

Notion of z-scores

How scores can be transformed to z-scores

We will examine a procedure for describing an individual score’s location within a distribution.

This procedure is based on the concepts of M and SD

The M will be used as a reference point.

The SD will serve as a yardstick for measuring how much an individual differs from the group average.

E.g. You receive a score of x = 76 on an exam.

How did you do?

If M = 70, you may be in a better position than if M = 85.

The M by itself is not sufficient to tell you the exact location of your score.

A score by itself does not provide much information about its position.

Raw scores are often transformed, or standardized.

z-scores provide a simple procedure for standardizing any distribution.

A z-score takes information about the M and SD and uses it to produce a single value that specifies the location of any raw score within any distribution.

A raw score is transformed into a signed number so that

·        The sign tells whether the score is above (+) or below (-) the M.

·        The number tells the distance between the score and the mean in terms of SD units.

Since the SD is essential in converting scores to z-scores, the amount of variability in a distribution and the relative position of a particular score are interrelated.

Example: Suppose that in Caribou, Maine, the average snowfall per year is m = 110 inches, with s = 30.  In Boston, m = 24 inches, with s = 5.  Last year Caribou had 125 inches of snow, while Boston had 39 inches.

In which city was the winter much worse than average?

z-score formula

z = (x - m) / s

for a sample: (x-M) / s