1. What is Z-Score to Probability Conversion?

The Z-score to probability conversion calculates the cumulative probability (P) from a standard normal distribution Z-score using the formula P = Φ(Z). This represents the probability that a standard normal random variable is less than or equal to the given Z-score.

2. How Does the Calculator Work?

The calculator uses the standard normal cumulative distribution function:

Where:

• \( Z \) — Standard normal Z-score (unitless)
• \( \Phi \) — Cumulative distribution function of standard normal distribution
• \( P \) — Probability value between 0 and 1 (unitless)

Explanation: The calculator uses an approximation of the error function to compute the cumulative probability from the standard normal distribution.

3. Importance of Probability Calculation

Details: Converting Z-scores to probabilities is essential in statistics for hypothesis testing, confidence intervals, and determining statistical significance in various research and data analysis applications.

4. Using the Calculator

Tips: Enter the Z-score value (positive or negative). The calculator will return the corresponding cumulative probability from the standard normal distribution.

5. Frequently Asked Questions (FAQ)

Q1: What is a Z-score?
A: A Z-score measures how many standard deviations a data point is from the mean of a standard normal distribution.

Q2: What does the probability value represent?
A: The probability represents the area under the standard normal curve to the left of the given Z-score.

Q3: How accurate is the approximation?
A: The error function approximation used provides accuracy to about 7 decimal places, which is sufficient for most statistical applications.

Q4: Can I use this for two-tailed probabilities?
A: For two-tailed probabilities, you would need to calculate 2 × min(P, 1-P) where P is the one-tailed probability.

Q5: What are typical Z-score ranges?
A: Z-scores typically range from -3 to +3 in most applications, covering about 99.7% of the standard normal distribution.