A z-score is the number of standard deviations a value lies above or below the mean. It is calculated as (x minus the mean) divided by the standard deviation.

What is the difference between probability and density?

A cumulative probability is an area under the normal curve. Probability density is the height of the curve at a point and is not the probability of that exact point.

What is the 68-95-99.7 rule?

For a normal distribution, approximately 68.27 percent of observations lie within one standard deviation of the mean, 95.45 percent within two, and 99.73 percent within three.

Normal Distribution Calculator

Calculate probabilities, z-scores, probability density, and percentiles for a normal distribution. Enter the mean and standard deviation, then choose the calculation you need. Results are calculated on the server and displayed on this page.

Normal Distribution Calculator

Calculation

Mean (μ)

The center of the distribution.

Standard deviation (σ)

Must be greater than zero.

Value (x)

Enter the value to evaluate.

Upper bound (b)

Used for probability between two values.

Cumulative probability (p)

For example, 0.95 means the 95th percentile.

Choose a calculation mode and enter the required parameters.

Reset

Normal Distribution Formulas

A normal random variable is written as X ~ N(μ, σ²), where μ is the mean and σ is the positive standard deviation. The distribution is symmetric about the mean and has a bell-shaped probability curve.

1. Z-score

z = (x − μ) / σ

Standardization converts a value from the original distribution into standard deviation units.

2. Probability density function

f(x) = [1 / (σ√(2π))] × exp(−(x − μ)² / (2σ²))

Density measures the height of the curve at x. For a continuous distribution, the probability of one exact value is zero; probabilities are represented by areas under the curve.

3. Left-tail probability

P(X ≤ x) = Φ((x − μ) / σ)

This is the area under the normal curve to the left of x.

4. Right-tail probability

P(X > x) = 1 − Φ((x − μ) / σ)

This is the area to the right of x.

5. Probability between two values

P(a ≤ X ≤ b) = Φ((b − μ) / σ) − Φ((a − μ) / σ)

6. Percentile or inverse normal calculation

x = μ + σΦ⁻¹(p)

This finds the value x with cumulative probability p to its left. For example, p = 0.95 gives the 95th percentile.

Example: Standard Normal Distribution

For the standard normal distribution, the mean is 0 and the standard deviation is 1. A value x = 1 has z-score 1, and its left-tail probability is approximately 84.13%.

Interval	Approximate probability
Within μ ± 1σ	68.27%
Within μ ± 2σ	95.45%
Within μ ± 3σ	99.73%

These are commonly known as the empirical rule or the 68–95–99.7 rule. They are useful approximations for normally distributed data.

Frequently Asked Questions

What is a normal distribution?

A normal distribution is a continuous probability distribution that is symmetric around its mean. Its shape is determined by its mean and standard deviation.

How do I calculate a z-score?

Subtract the mean from your value and divide by the standard deviation. A positive z-score is above the mean, and a negative z-score is below it.

What is the difference between density and probability?

Density is the height of the probability curve at a point. Probability is the area under the curve over an interval. The density at a point is not the probability of that exact value.

What does the 95th percentile mean?

The 95th percentile is the value at or below which approximately 95% of the distribution lies. The remaining 5% lies above it.

Can the mean or standard deviation be negative?

The mean can be any finite real number. The standard deviation must be strictly positive because a normal distribution with zero standard deviation is degenerate and is not handled by this calculator.

Are the results exact?

The calculator uses numerical approximations for the cumulative normal distribution and inverse CDF. Results are displayed with limited decimal precision, so they should be treated as numerical estimates rather than symbolic exact values.

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