Z Distribution Calculator
Calculate probabilities, Z-scores, and percentiles for the standard normal distribution. This calculator supports left-tail, right-tail, between, and inverse (percentile to Z-score) calculations with a visual bell curve and step-by-step explanations.
What Is the Z Distribution?
The Z distribution, also known as the standard normal distribution, is a special case of the normal distribution with mean (μ) = 0 and standard deviation (σ) = 1. It is used to standardize any normal distribution by converting values into Z-scores, which represent the number of standard deviations a value is from the mean.
Z = (x − μ) / σ
Where x is the raw score, μ is the population mean, and σ is the population standard deviation.
The total area under the standard normal curve is 1 (100%). The curve is symmetric about zero, with 50% of the area to the left and 50% to the right.
Common Z-Score Values
- z = 0: Exactly at the mean (50th percentile)
- z = ±1: About 68% of data falls within 1 standard deviation
- z = ±1.96: About 95% of data falls within 1.96 standard deviations
- z = ±2.576: About 99% of data falls within 2.576 standard deviations
- z = 1.645: 90th percentile (or 5% in upper tail for one-tailed test)
The 68-95-99.7 Rule (Empirical Rule)
- 68.27% of values fall within ±1 standard deviation of the mean
- 95.45% of values fall within ±2 standard deviations of the mean
- 99.73% of values fall within ±3 standard deviations of the mean
How the Calculator Works
- Select a mode: Left-tail, right-tail, between, or inverse (percentile to Z-score).
- Enter the required value(s): Z-score(s) or percentile.
- Click Calculate: The calculator computes the probability, percentile, and shows the shaded area on the bell curve.
- Review the steps: See the formula and intermediate calculations.
Why Use This Z Distribution Calculator?
- Four Calculation Modes: Left-tail, right-tail, between, and inverse.
- Visual Bell Curve: See the shaded area representing the probability.
- Step-by-Step Explanations: Understand the calculation process.
- Interactive Z-Table: Quickly look up Z-score probabilities.
- Free & Private: No registration, no data storage.
❓ Z Distribution Calculator FAQ
What is a Z-score?
A Z-score measures how many standard deviations a value is from the mean. A Z-score of 0 means the value is exactly at the mean, 1 means one standard deviation above, and -1 means one standard deviation below.
What is the standard normal distribution?
The standard normal distribution is a normal distribution with mean 0 and standard deviation 1. It is used to standardize any normal distribution and is the basis for Z-tables.
What is the difference between left-tail and right-tail probability?
Left-tail probability (P(Z ≤ z)) is the area under the curve to the left of z. Right-tail probability (P(Z ≥ z)) is the area to the right of z. For example, P(Z ≤ 1.96) ≈ 0.975 and P(Z ≥ 1.96) ≈ 0.025.
What is the inverse (percentile to Z) calculation?
The inverse calculation finds the Z-score that corresponds to a given percentile. For example, the 95th percentile corresponds to Z ≈ 1.645 (left-tail).
What is the 68-95-99.7 rule?
The empirical rule states that for a normal distribution: 68% of data falls within ±1 standard deviation, 95% within ±2 standard deviations, and 99.7% within ±3 standard deviations of the mean.
When should I use the Z distribution?
Use the Z distribution when you have a normal distribution and you want to find probabilities, percentiles, or standardize values. It is also used in hypothesis testing and confidence intervals.
Can this calculator handle negative Z-scores?
Yes, the calculator fully supports negative Z-scores. The standard normal distribution is symmetric, so probabilities for negative Z-scores are computed using the CDF.
Is this calculator free?
Yes, this calculator is completely free to use. No registration or personal data storage is required. All calculations are performed in your browser.