In a standard normal distribution, what z-score corresponds to the 97.5th percentile?

In a standard normal distribution, what z-score corresponds to the 97.5th percentile?

• A. 1.64
• B. 2.05
• C. 1.28
• D. 1.96

Answer: (D)

In a standard normal distribution, a percentile tells you the z-score where that fraction of the data lies to the left. The 97.5th percentile means 97.5% of the area is to the left, so the left-tail probability is 0.975. The z-score that gives Φ(z) = 0.975 is about 1.96, because the standard normal CDF at 1.96 is roughly 0.975. This value also matches the upper boundary of the common 95% central interval, -1.96 to 1.96, which helps you see why it’s used as a reference point. Other z-scores correspond to other percentiles, but 1.96 is the one that yields 97.5% to the left.

In a standard normal distribution, a percentile tells you the z-score where that fraction of the data lies to the left. The 97.5th percentile means 97.5% of the area is to the left, so the left-tail probability is 0.975. The z-score that gives Φ(z) = 0.975 is about 1.96, because the standard normal CDF at 1.96 is roughly 0.975. This value also matches the upper boundary of the common 95% central interval, -1.96 to 1.96, which helps you see why it’s used as a reference point. Other z-scores correspond to other percentiles, but 1.96 is the one that yields 97.5% to the left.