# Zach’s facts

Zach’s facts have been extracted from the book to remind you of the key concepts you and Zach have learned in each chapter.

Zach's Facts 6.1 z-scores

• Scores are sometimes expressed in a standard form known as z-scores.

• To transform a score into a z-score you subtract from it the mean of all scores and divide the result by the standard deviation of all scores.

• The sign of the z-score tells us whether the original score was above or below the mean; the value of the z-score tells us how far the score was from the mean in standard deviation units. Therefore, a z-score tells us about the location of a score within the distribution.

Zach's Facts 6.2 z-score distributions

• If you convert an entire distribution of scores to z-scores:
• The shape of the distribution is unchanged.

• The transformed distribution will have a mean of 0.

• The transformed distribution will have a standard deviation of 1.
• Two distributions of scores that have been converted to z-scores are directly comparable – irrespective of their original units of measurement - because the conversion changes the units of measurement to standard deviations.

• Once you have converted an entire distribution of scores to z-scores, you can further transform the scores to have any mean or standard deviation you want by multiplying each score by the desired standard deviation and then adding the desired value of the mean.