About

Statistical analysis relies heavily on the Normal Distribution, also known as the Gaussian distribution. This tool computes the area under the bell curve, representing the probability that a random variable falls within a specific range. It is essential for determining statistical significance, quality control limits in manufacturing, and grading on a curve in educational settings.

The calculator solves for the Cumulative Distribution Function (CDF) and the Probability Density Function (PDF) using high-precision error function approximations. It handles both standard problems (finding probability from a score) and inverse problems (finding a raw score or Z-score from a desired probability percentile). Correctly interpreting these values is critical in fields ranging from finance (Value at Risk models) to psychometrics.

Formulas

The Z-score standardizes any normal distribution by subtracting the mean and dividing by the standard deviation:

Z = x − μσ

The Probability Density Function (PDF) describing the shape of the curve is:

f(x) = 1σ√2π e^−0.5⋅Z²

The Cumulative Distribution Function (CDF), which calculates the probability P(X ≤ x), requires integration and is approximated using the Error Function (erf):

CDF(x) = 0.5 ⋅ (1 + erf(Z√2))

Reference Data

Confidence Level	Z-Score (2-sided)	P(Value inside)	P(Value outside)
50%	0.6745	0.5000	0.5000
80%	1.2816	0.8000	0.2000
90%	1.6449	0.9000	0.1000
95%	1.9600	0.9500	0.0500
98%	2.3263	0.9800	0.0200
99%	2.5758	0.9900	0.0100
99.5%	2.8070	0.9950	0.0050
99.9%	3.2905	0.9990	0.0010
Six Sigma (6σ)	6.0000	0.999999998	2e-9

Frequently Asked Questions

A one-tailed test looks for statistical significance in a single direction (e.g., is the value greater than X?). A two-tailed test looks for extreme values in both directions (e.g., is the value significantly different from the mean, either higher or lower?).

The normal distribution is asymptotic. Although the probability becomes infinitesimally small as you move away from the mean, it never mathematically reaches zero. This implies there is always a theoretical non-zero chance, however slight, of extreme outliers.

Use the inverse mode when you have a target percentile (e.g., 'Top 10% of students') and need to find the cutoff score required to qualify for that bracket.

No. This calculator assumes a perfect Gaussian bell curve. If your data is heavily skewed (e.g., income distribution) or has fat tails, using Z-scores based on normal approximation will yield incorrect probabilities.