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Normal Distribution Calculator

Calculate probabilities and percentiles for the normal (Gaussian) distribution.

Common normal distribution coverages:

Sigma RangeCoveragez
1σ68.27%±1.000
2σ95.45%±2.000
3σ99.73%±3.000
1.96σ95.00%±1.960
2.576σ99.00%±2.576

About this tool

The normal distribution, also known as the Gaussian or bell curve, is one of the most important probability distributions in statistics. It describes how data tends to cluster around a central mean value, with frequencies tapering symmetrically on both sides. This calculator helps you compute probabilities and percentiles for any normal distribution by specifying the mean and standard deviation, making it invaluable for students, researchers, and professionals working with statistical analysis.

To use this tool, enter your desired mean (average) and standard deviation (spread), then either calculate the probability of a value falling within a range, or find the value corresponding to a specific percentile. The calculator instantly provides cumulative probabilities, z-scores, and percentile ranks. Common applications include quality control in manufacturing, standardized testing score interpretation, risk assessment in finance, and hypothesis testing in research.

Frequently Asked Questions

Code Implementation

import math

def norm_cdf(x):
    """Standard normal CDF using math.erfc"""
    return 0.5 * math.erfc(-x / math.sqrt(2))

def normal_cdf(x, mu=0, sigma=1):
    """Normal CDF with given mean and std dev"""
    return norm_cdf((x - mu) / sigma)

def normal_pdf(x, mu=0, sigma=1):
    """Normal probability density function"""
    return (1 / (sigma * math.sqrt(2 * math.pi))) * math.exp(-0.5 * ((x - mu) / sigma) ** 2)

# P(X < 1) for standard normal
print(f"P(X < 1) = {norm_cdf(1):.4%}")       # 84.1345%

# P(0 < X < 1) for N(0,1)
print(f"P(0<X<1) = {norm_cdf(1) - norm_cdf(0):.4%}")  # 34.1345%

# Using scipy for more features
from scipy import stats
mu, sigma = 100, 15   # IQ scores
print(f"P(IQ < 130) = {stats.norm.cdf(130, mu, sigma):.4%}")
print(f"90th percentile IQ = {stats.norm.ppf(0.90, mu, sigma):.1f}")

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