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Normal Distribution Calculator

Calculate probabilities, Z-scores, and critical values for the normal distribution, also known as the Gaussian or bell curve distribution.

Distribution Parameters

Find Probability

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What is the Normal Distribution?

The normal distribution, also known as the Gaussian distribution, is a continuous probability distribution that is symmetric about the mean. It is characterized by its bell-shaped curve and is completely determined by two parameters: the mean (μ) and the standard deviation (σ).

Key Properties

  • The distribution is symmetric around the mean.
  • The mean, median, and mode are all equal.
  • Approximately 68% of the data falls within one standard deviation of the mean.
  • Approximately 95% of the data falls within two standard deviations of the mean.
  • Approximately 99.7% of the data falls within three standard deviations of the mean (the 68-95-99.7 rule).

Z-scores and Standardization

Z-scores (or standard scores) represent the number of standard deviations a value is from the mean. Converting to Z-scores standardizes the normal distribution to a standard normal distribution with mean 0 and standard deviation 1, making it easier to calculate probabilities.

The formula for calculating a Z-score is:

Z = (X - μ) / σ

Applications

The normal distribution is widely used in:

  • Statistical inference and hypothesis testing
  • Quality control and process monitoring
  • Risk assessment in finance and insurance
  • Natural and social sciences to model random variables
  • Measurement errors in experimental sciences

Using the Calculator

This calculator offers three main functionalities:

  1. Probability: Calculate the probability that a random variable follows a specified condition (between two values, less than a value, or greater than a value).
  2. Z-score: Convert a value to its corresponding Z-score and provide associated probabilities.
  3. Inverse: Find a value (critical value) given a probability, which is useful for constructing confidence intervals and conducting hypothesis tests.

Formula Sheet

Normal PDF

f(x) = (1/(σ√(2π))) * e^(-(x-μ)²/(2σ²))

Z-score

Z = (X - μ) / σ

Standard Normal CDF

Φ(z) = P(Z ≤ z)

Probability Calculations

P(a < X < b) = Φ((b-μ)/σ) - Φ((a-μ)/σ)

P(X < a) = Φ((a-μ)/σ)

P(X > a) = 1 - Φ((a-μ)/σ)

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