What is the Z-score for capturing 99.7% of the data in a normal distribution?

To determine the Z-score that captures 99.7% of the data in a normal distribution, it is essential to understand the properties of the standard normal distribution and the empirical rule. The empirical rule states that approximately 68% of the data falls within one standard deviation of the mean, about 95% within two standard deviations, and around 99.7% within three standard deviations.

In a standard normal distribution, the Z-score that corresponds to 99.7% of the data includes all values within three standard deviations from the mean. The Z-score for this scenario is approximately 3.00.

When looking for the Z-score, it is important to recognize that the area under the curve (AUC) from the mean to the Z-score for a one-tailed scenario is crucial. Since 99.7% of the distribution is captured within three standard deviations, the Z-score at this point must reflect that coverage accurately.

Therefore, the Z-score that captures 99.7% of the data in a normal distribution is indeed 3.00, representing that almost all observations fall within this range of standard deviations from the mean. This understanding is essential for quantifying risk and assessing probabilities in various financial contexts using a