Normal Distribution Calculator

Calculate probabilities, z-scores, and percentiles for the normal distribution. Enter a z-score to find the cumulative probability (area under the curve), or enter a probability to find the corresponding z-score (percentile). Visualize the standard normal distribution with interactive charts.

What Is the Normal Distribution?

The normal distribution (also called the Gaussian distribution) is a continuous probability distribution that is symmetric about the mean. It is one of the most important probability distributions in statistics and is widely used in natural and social sciences.

Standard Normal Distribution: Z ~ N(0, 1)

Mean (μ) = 0, Standard Deviation (σ) = 1

φ(z) = (1/√(2π)) · e^−z²/2

The 68-95-99.7 rule (empirical rule) states that:

Approximately 68% of data falls within 1 standard deviation of the mean.
Approximately 95% falls within 2 standard deviations.
Approximately 99.7% falls within 3 standard deviations.

Key Concepts

Z-Score: The number of standard deviations a data point is from the mean.
Cumulative Probability (CDF): The probability that a random variable Z is less than or equal to a given z-score: P(Z ≤ z).
Right Tail: The probability that Z is greater than z: P(Z ≥ z) = 1 − P(Z ≤ z).
Two-Tailed: The probability that |Z| is greater than |z|: 2 × P(Z ≥ |z|).
Percentile: The z-score corresponding to a given cumulative probability.

Common Z-Score Values

z = 1.96: P(Z ≤ 1.96) = 0.9750 (95% confidence interval)
z = 2.576: P(Z ≤ 2.576) = 0.9950 (99% confidence interval)
z = 1.645: P(Z ≤ 1.645) = 0.9500 (90% confidence interval)
z = 0: P(Z ≤ 0) = 0.5000 (median)

Why Use This Normal Distribution Calculator?

Two Calculation Modes: Convert z-scores to probabilities or probabilities to z-scores.
Between Two Z-Scores: Calculate the probability that Z falls between two values.
Visual Chart: See the shaded area under the normal curve.
Tail Options: Choose left, right, or two-tailed probabilities.
Free & Private: No registration, no data storage.

❓ Normal Distribution Calculator FAQ

What is the standard normal distribution?

The standard normal distribution is a normal distribution with mean μ = 0 and standard deviation σ = 1. It is denoted as Z ~ N(0, 1).

What is a z-score?

A z-score represents the number of standard deviations a data point is from the mean. It is calculated as z = (x − μ) / σ.

What is the 68-95-99.7 rule?

The empirical rule states that approximately 68% of data falls within 1 standard deviation, 95% within 2 standard deviations, and 99.7% within 3 standard deviations of the mean.

How do I find the probability for a given z-score?

Use the "Z-Score → Probability" mode. Enter your z-score and the calculator will return P(Z ≤ z), the cumulative probability.

How do I find the z-score for a given percentile?

Use the "Probability → Z-Score" mode. Enter a probability (between 0 and 1) and the calculator will return the corresponding z-score (percentile).

What is the difference between left tail and right tail?

Left tail gives P(Z ≤ z). Right tail gives P(Z ≥ z) = 1 − P(Z ≤ z). Two-tailed gives P(|Z| ≥ |z|) = 2 × P(Z ≥ |z|).

What is the z-score for a 95% confidence interval?

The z-score for a 95% confidence interval is 1.96. This means P(−1.96 ≤ Z ≤ 1.96) = 0.95.

What is the z-score for a 99% confidence interval?

The z-score for a 99% confidence interval is 2.576. This means P(−2.576 ≤ Z ≤ 2.576) = 0.99.

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.