1. What is a Z Score?

A Z score (standard score) represents how many standard deviations an element is from the mean. It's a statistical measurement that describes a value's relationship to the mean of a group of values.

2. How Does the Calculator Work?

The calculator uses the inverse normal distribution function:

Where:

• \( z \) — The Z score
• \( \Phi^{-1} \) — The inverse cumulative distribution function
• \( p \) — The percentile (decimal between 0 and 1)

Explanation: The calculator finds the Z score that corresponds to a given percentile in a standard normal distribution.

3. Importance of Z Score Calculation

Details: Z scores are crucial in statistics for comparing results from different normal distributions, identifying outliers, and standardizing scores for comparison across different datasets.

4. Using the Calculator

Tips: Enter the percentile as a decimal between 0.0001 and 0.9999. For example, enter 0.95 for the 95th percentile.

5. Frequently Asked Questions (FAQ)

Q1: What is the relationship between percentiles and Z scores?
A: A Z score tells you how many standard deviations a value is from the mean, while a percentile tells you what percentage of values fall below a certain point in a distribution.

Q2: What are common Z score values?
A: Common values include -1.96 and 1.96 (95% confidence interval), -2.58 and 2.58 (99% confidence interval), and -1.645 and 1.645 (90% confidence interval).

Q3: Can Z scores be negative?
A: Yes, Z scores can be negative, indicating the value is below the mean of the distribution.

Q4: What is the Z score for the median?
A: The median corresponds to the 50th percentile, which has a Z score of 0.

Q5: How accurate is this calculator?
A: This calculator uses a standard approximation method for the inverse normal distribution, providing results accurate for most statistical applications.