Gaussian Distribution Calculator & Formula

This calculator computes probabilities and percentiles for a normal (Gaussian) distribution. You enter the mean (μ), standard deviation (σ), and a value (x), and it returns the probability density, cumulative probability, and the z-score. It also works in reverse: enter a cumulative probability to find the corresponding x-value (inverse normal).

Formula

The normal distribution is defined by two parameters: the mean (μ) and the standard deviation (σ). The probability density function (PDF) is:

f(x) = (1 / (σ √(2π))) · e^{−((x − μ)² / (2σ²))}

The cumulative distribution function (CDF) gives the probability that a random variable X is less than or equal to a value x:

P(X ≤ x) = 0.5 · [1 + erf((x − μ) / (σ √2))]

where erf is the error function. The z-score standardizes the value:

z = (x − μ) / σ

All variables are unitless; μ and σ must have the same units as x (e.g., kg, cm, dollars). The calculator uses the standard mathematical formulas and does not rely on approximation tables.

Worked Example

Using the default values: μ = 0, σ = 1, x = 1.96.

1. Compute the z-score: z = (1.96 − 0) / 1 = 1.96.
2. Compute the PDF: f(1.96) = (1 / (1·√(2π))) · e^{−(1.96² / 2)} ≈ 0.0584.
3. Compute the CDF: P(X ≤ 1.96) = 0.5 · [1 + erf(1.96 / √2)] ≈ 0.9750.

The calculator shows P(X ≤ 1.96) ≈ 0.9750, which matches the well-known 97.5th percentile of the standard normal distribution.

When to Use the Gaussian Distribution

The normal distribution is the most frequently used probability model in statistics. It applies when data cluster symmetrically around a central mean, with most observations near the center and fewer in the tails. Common uses include:

• Standardizing test scores (IQ, SAT) to compare across different scales.
• Quality control: setting tolerance limits on manufactured parts.
• Finance: modeling asset returns (though real returns often have heavier tails).
• Natural phenomena: heights, blood pressure, measurement errors often approximate normality.

However, not all data are normal. If the data are skewed (e.g., income, waiting times) or have outliers, the normal model may give misleading probabilities.

How to Read the Results

The cumulative probability P(X ≤ x) is the area under the curve to the left of x. For example, if μ = 100 and σ = 15 (IQ scores), then x = 115 gives P ≈ 0.8413, meaning about 84% of the population scores 115 or below. The z-score tells you how many standard deviations x is from the mean; a z of 2 means x is 2σ above the mean. The probability density f(x) is not a probability itself—it is a height on the curve, useful for comparing relative likelihoods at different points.

For the inverse normal (given a probability, find x), you can use the quantile function. For a standard normal, the 97.5th percentile is 1.96, which is why 1.96 is used for 95% confidence intervals.

Common Mistakes and Pitfalls

• Using σ vs. σ²: The standard deviation is the square root of the variance. If you have the variance, take the square root first.
• One-tailed vs. two-tailed: P(X ≤ x) gives the left tail. For the right tail, compute 1 − P. For a two-sided interval, subtract the left tail from the right tail.
• Rounding too early: Round only the final result, not intermediate z-scores, to avoid errors.
• Assuming normality: Always check with a histogram or normality test if you are using real data.

Limitations and Alternatives

The normal distribution is symmetric and unbounded, so it cannot model strictly positive data with heavy tails (e.g., incomes, reaction times). For those, consider the log-normal, exponential, or gamma distributions. Also, the central limit theorem justifies using the normal for sample means, but not always for raw data. For small samples from non-normal populations, use t-distributions or nonparametric methods.

Frequently Asked Questions

What is the difference between a normal distribution and a standard normal distribution?
A normal distribution has any mean and standard deviation; a standard normal has μ = 0 and σ = 1. Any normal variable can be converted to a standard normal via the z-score.

How do I find the probability that X is between two values?
Compute P(X ≤ b) − P(X ≤ a). For example, with μ = 0, σ = 1, P(−1.96 ≤ X ≤ 1.96) = 0.9750 − 0.0250 = 0.9500.

What does the 68-95-99.7 rule say?
For a normal distribution, about 68% of data lies within 1σ of the mean, 95% within 2σ, and 99.7% within 3σ. This is a quick mental check.

Can I use this calculator for sample data?
Yes, but you should use the sample mean and sample standard deviation (with n−1 denominator) as estimates of the population parameters. The calculator assumes you already have those values.

Why is the probability density sometimes greater than 1?
The PDF is a density, not a probability. It can exceed 1 when σ is small, but the area under the curve always equals 1. Probabilities are always between 0 and 1.

What is the inverse normal calculation?
Given a cumulative probability p, you find the x such that P(X ≤ x) = p. For example, the 95th percentile of a standard normal is about 1.645. This calculator does not directly do the inverse, but you can use a search or a z-table.

Related

• Correlation Coefficient Formula & Calculator
• 95% Confidence Interval Calculator: How to Use
• Confidence Interval Calculator: Formula & How-To

in m | Mean (μ) | - | 0 | 0
in s | Standard deviation (σ) | - | 1 | 0
in x | Value (x) | - | 1.96 | 0
out p | Cumulative probability P(X ≤ x) | - | 0.5 * (1 + erf((x - m) / (s * sqrt(2)))) | 4
out z | Z-score | - | (x - m) / s | 2
out pdf | Probability density f(x) | - | (1 / (s * sqrt(2 * pi))) * exp(-0.5 * ((x - m) / s)^2) | 4