How to Find a Z-Score from a Percentile

To find a z-score from a percentile, you essentially need to reverse the process of finding a percentile from a z-score. This involves using a standard normal distribution table (also known as a z-table) or a statistical calculator that can perform inverse normal distribution calculations.

Steps to Find the Z-Score:

1. Understand the Percentile: A percentile represents the percentage of values in a distribution that are below a specific value. For example, the 80th percentile means 80% of the values are below that point.

2. Use a Standard Normal Distribution Table (Z-Table) or Calculator:

   • Z-Table: Find the area in the z-table that corresponds to your percentile. Remember that z-tables usually show the area to the left of the z-score, which directly corresponds to the percentile. Look for the closest value in the table to your percentile (expressed as a decimal, e.g., 80th percentile = 0.80). Once you find the closest area, read the corresponding z-score from the row and column headings of the table.
   • Statistical Calculator/Software: Most statistical calculators or software packages have a function that calculates the inverse normal distribution (also sometimes called the quantile function). You input the percentile as a decimal (e.g., 0.80 for the 80th percentile), and the function returns the corresponding z-score. This is usually denoted as invNorm or similar.

3. Interpret the Z-Score: The z-score tells you how many standard deviations the value corresponding to that percentile is away from the mean. A positive z-score indicates the value is above the mean, while a negative z-score indicates it is below the mean.

Example:

Let's say you want to find the z-score corresponding to the 90th percentile.

1. Percentile: 90th percentile = 0.90

2. Using a Z-Table: Look for 0.90 (or the closest value) inside the z-table. You might find a value close to 0.8997, which corresponds to a z-score of approximately 1.28.

3. Using a Calculator: Use the invNorm function on your calculator, inputting 0.90. The result will be approximately 1.28.

Therefore, the z-score corresponding to the 90th percentile is approximately 1.28. This means a value at the 90th percentile is 1.28 standard deviations above the mean.

Z-Score Formula (for reference, but not directly used to find z from percentile):

While the question specifically asks how to find the z-score from the percentile (using tables/calculators), it's good to remember the basic z-score formula:

Z = (x - μ) / σ

Where:

• Z = Z-score
• x = Raw score
• μ = Population mean
• σ = Population standard deviation

This formula is used to find the z-score if you know the raw score (x), the mean (μ), and the standard deviation (σ). But when starting with a percentile, the z-table/calculator method described above is what you need.