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🔔 Normal Distribution Calculator

Calculate probabilities and z-scores for a normal distribution. Find P(X < x), P(X > x), and P(x₁ < X < x₂) with a visual bell curve.

Normal Distribution Formulas

Probability Density Function (PDF)
f(x) = (1/σ√(2π)) × e^(−(x−μ)²/2σ²)
Z-Score
z = (x − μ) / σ
CDF (using error function)
P(X < x) = ½ × [1 + erf((x−μ) / (σ√2))]

How to Use This Calculator

  1. 1
    Set Distribution Parameters
    Enter the mean (μ) and standard deviation (σ) of your normal distribution.
  2. 2
    Enter x Values
    Enter x for single-tail probabilities, or both x₁ and x₂ for a range probability.
  3. 3
    Read Probabilities
    The calculator shows P(X < x), P(X > x), range probability, and the z-score.
  4. 4
    View the Bell Curve
    The SVG diagram highlights the shaded area corresponding to your probability.

Frequently Asked Questions

A normal distribution is a symmetric, bell-shaped probability distribution defined by its mean (μ) and standard deviation (σ). Many natural phenomena follow a normal distribution, such as heights, test scores, and measurement errors.

A z-score measures how many standard deviations a value is from the mean. z = (x − μ) / σ. A z-score of 1.96 corresponds to the 97.5th percentile of a standard normal distribution.

For any normal distribution: 68% of data falls within ±1σ, 95% within ±2σ, and 99.7% within ±3σ of the mean. This is also called the empirical rule.

The Cumulative Distribution Function (CDF) gives the probability that a random variable X takes a value less than or equal to x. It is computed using the error function (erf) approximation in this calculator.

The standard normal distribution is a special case with μ = 0 and σ = 1. Any normal distribution can be converted to standard normal using the z-score transformation, making z-tables universally applicable.

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