Final answer:
The z-score percentiles in the provided list were obtained by using rank-based z-scores that consider the rank of each observation relative to the entire dataset. Percentiles are calculated, and then the corresponding z-scores are found using a z-table or calculator functions like invNorm.
Step-by-step explanation:
To construct a normal probability plot and determine the z percentiles for each observation, we typically use the formula z = (X - μ) / σ, where X is the sample observation, μ is the mean, and σ is the standard deviation. However, the values provided do not seem to match this method, suggesting a different approach, most likely involving the use of rank-based z-scores.
In creating a normal probability plot with rank-based z-scores, you first must rank the observations from smallest to largest. Then for each observation, you calculate the cumulative probability using the formula (i - 0.5) / n, where i is the rank of the data point and n is the sample size. This probability value corresponds to the percentile of the observation. You must then use the standard normal distribution to find the z-score that corresponds to this cumulative probability.
Tools like a z-table, calculator functions like invNorm, or statistical software can find the z-score corresponding to a given percentile. As stated in your provided reference, invNorm(0.975,0,1) can be used for a cumulative probability of 0.975, which gives you the 97.5th percentile z-score.
It seems your provided z-score percentiles were calculated with a method that takes rank into account, which explains why the standard z-score formula did not match.