Which statement about the effect of changing the standard deviation on Z-score for a fixed difference is true?

• A. Increasing the SD yields a higher Z-score.
• B. The Z-score is independent of SD.
• C. The Z-score decreases when the SD decreases.
• D. Decreasing the SD yields a higher Z-score.

Answer: (D)

Z-score tells you how many standard deviations a value is from the mean. If you keep the difference between the value and the mean fixed, changing the spread of the distribution (the standard deviation) changes that Z-score inversely.

If the variability gets smaller, the same difference becomes more significant in terms of standard deviations, so the Z-score goes up. If the variability increases, the same difference becomes less significant, so the Z-score goes down. For a concrete sense, with a fixed difference of 6 from the mean, Z would be larger when the SD is 2 than when the SD is 3 (3 vs 2), and even larger if the SD drops to 1.

Thus, decreasing the standard deviation yields a higher Z-score.