Which of the following is NOT a characteristic of z-scores?

Which of the following is NOT a characteristic of z-scores?

• A. A z-score of 0 is equal to the mean.
• B. The standard deviation and variance of standard scores always equal 1.00.
• C. The larger the absolute value of a z-score, the less frequently that raw score occurs.
• D. The standard deviation of a set of standard scores can be zero.

Answer: (D)

Standardizing data to z-scores puts different measures on a common scale: the transformed values have a mean of 0 and a spread of one standard deviation. A z-score shows how far a value is from the mean in units of standard deviations. This makes the statements about z-scores that the mean is zero and the spread is one standard deviation true. The absolute value interpretation also fits: larger absolute z-scores indicate values further from the mean and thus rarer in a normal-like distribution. The claim that the standard deviation of a set of standard scores can be zero contradicts the whole idea of standardization. By construction, the z-scores have a fixed spread of 1 (and hence a variance of 1) for any valid data set; a zero standard deviation would imply no variability, which isn’t possible once you’ve standardized.

Standardizing data to z-scores puts different measures on a common scale: the transformed values have a mean of 0 and a spread of one standard deviation. A z-score shows how far a value is from the mean in units of standard deviations.

This makes the statements about z-scores that the mean is zero and the spread is one standard deviation true. The absolute value interpretation also fits: larger absolute z-scores indicate values further from the mean and thus rarer in a normal-like distribution.

The claim that the standard deviation of a set of standard scores can be zero contradicts the whole idea of standardization. By construction, the z-scores have a fixed spread of 1 (and hence a variance of 1) for any valid data set; a zero standard deviation would imply no variability, which isn’t possible once you’ve standardized.