Final answer:
The z-score increases when the standard deviation decreases because the difference between the value x and the mean becomes relatively larger when the standard deviation is smaller.
Step-by-step explanation:
The z-score increases when the standard deviation decreases because the z-score is calculated by dividing the difference between the value x and the mean, μ, by the standard deviation, σ. When the standard deviation is smaller, the difference between x and μ becomes relatively larger, resulting in a larger z-score. This can be seen in the formula for calculating the z-score: z = (x - μ) / σ.
For example, let's consider two sets of data with the same mean but different standard deviations. In the first set, the standard deviation is relatively large, and in the second set, the standard deviation is relatively small. If we have a value x that is the same distance from the mean in both sets, the difference between x and μ will be greater in the second set with the smaller standard deviation, resulting in a larger z-score.