1. What is Z-Score Probability Calculation?

The Z-score probability calculation determines the probability that a random variable falls between two values in a normal distribution. It standardizes values to show how many standard deviations they are from the mean.

2. How Does the Calculator Work?

The calculator uses these formulas:

Where:

• \( x_1 \) — Lower bound value
• \( x_2 \) — Upper bound value
• \( \mu \) — Mean of the distribution
• \( \sigma \) — Standard deviation of the distribution
• \( \Phi \) — Standard normal cumulative distribution function

Explanation: The calculator first converts your values to Z-scores, then calculates the area under the normal curve between these Z-scores.

3. Importance of Z-Score Calculation

Details: Z-scores are fundamental in statistics for comparing different normal distributions, hypothesis testing, and determining probabilities of events.

4. Using the Calculator

Tips: Enter your lower and upper bounds, the mean of your distribution, and standard deviation. The calculator will show the corresponding Z-scores and probability between them.

5. Frequently Asked Questions (FAQ)

Q1: What does a negative Z-score mean?
A: A negative Z-score indicates the value is below the mean of the distribution.

Q2: What's the probability if Z1 = -1.96 and Z2 = 1.96?
A: Approximately 95%, as this covers 95% of the area under a normal curve.

Q3: Can I use this for non-normal distributions?
A: The calculation assumes normality. For non-normal distributions, other methods may be needed.

Q4: What's the difference between Z-scores and T-scores?
A: T-scores are similar but used when sample sizes are small and population standard deviation is unknown.

Q5: How accurate is this calculator?
A: It provides good approximations for most practical purposes, using a polynomial approximation of the normal CDF.