Normal Distribution Calculator

The Normal Distribution Calculator computes probabilities for the normal (Gaussian) distribution — the most important continuous probability distribution in statistics. Enter a mean (μ) and standard deviation (σ) to find the probability that a random variable falls below a value, above a value, between two values, or to find a specific quantile. Results include an interactive bell curve visualization with the shaded probability area, z-score conversion, and a step-by-step calculation breakdown.

What Is the Normal Distribution?

The normal distribution, also called the Gaussian distribution or bell curve, is a symmetric continuous probability distribution centered around its mean (μ). It is completely described by two parameters:

• Mean (μ) — the center of the distribution, where the peak of the bell curve occurs.
• Standard deviation (σ) — controls the spread; a larger σ produces a wider, flatter curve.

Many natural phenomena — heights, test scores, measurement errors, IQ scores — approximately follow a normal distribution. The Central Limit Theorem guarantees that the mean of a sufficiently large sample from any distribution converges to a normal distribution, making it foundational to inferential statistics.

The Normal Distribution Formula

The Probability Density Function (PDF) of a normal distribution is:

The Cumulative Distribution Function (CDF) gives the probability that X is less than or equal to x:

The z-score converts any normal distribution value to the standard normal (mean = 0, std dev = 1):

How to Use This Calculator

1. Select your calculation mode: Choose Left Tail P(X ≤ x), Right Tail P(X ≥ x), Between P(a ≤ X ≤ b), or Inverse (find x from probability).
2. Enter distribution parameters: Input the mean (μ) and standard deviation (σ). For the standard normal distribution, use μ = 0 and σ = 1.
3. Enter your specific values: Depending on the mode, enter the x value, lower/upper bounds, or target probability.
4. Review results: Click Calculate to see the probability, z-score, interactive bell curve with shaded area, and step-by-step breakdown.

Understanding PDF, CDF, and Inverse CDF

• PDF (Probability Density Function): Gives the relative likelihood of a specific value. It represents the height of the bell curve at a given point. For continuous distributions, the PDF itself is not a probability — probabilities come from integrating the PDF over an interval.
• CDF (Cumulative Distribution Function): Gives P(X ≤ x), the probability that the variable is at or below a given value. Graphically, it is the area under the curve to the left of x. The CDF ranges from 0 to 1.
• Inverse CDF (Quantile Function): The reverse of the CDF — given a probability p, it finds the x value such that P(X ≤ x) = p. For example, the inverse CDF at p = 0.975 for the standard normal gives x ≈ 1.96.

The 68-95-99.7 Rule

The empirical rule (also called the three-sigma rule) provides quick probability estimates for any normal distribution:

This means approximately 68% of values fall within one standard deviation of the mean, 95% within two, and nearly all (99.7%) within three. Values beyond 3σ are extremely rare in a normal distribution.

Common Z-Score Reference Table

Common Applications of Normal Distribution

• Quality Control: Monitoring manufacturing processes using control charts and specification limits based on μ ± nσ.
• Hypothesis Testing: Determining p-values and critical values for z-tests and confidence intervals.
• Standardized Testing: SAT, GRE, and IQ scores are designed to follow a normal distribution, allowing percentile comparisons.
• Natural Sciences: Measurement errors, biological traits (height, weight), and many physical quantities are normally distributed.
• Finance: The Black-Scholes model and Value at Risk (VaR) assume normally distributed returns for option pricing and risk assessment.

Frequently Asked Questions

What is a normal distribution?

A normal distribution (also called Gaussian distribution or bell curve) is a symmetric, continuous probability distribution defined by its mean and standard deviation. It is the most important distribution in statistics because many natural phenomena approximately follow it, and the Central Limit Theorem guarantees that sample means converge to it regardless of the underlying distribution.

What is a z-score and how is it used?

A z-score measures how many standard deviations a value is from the mean. It is calculated as z = (x − μ) / σ. Z-scores allow you to compare values from different normal distributions by converting them to the standard normal distribution (mean = 0, standard deviation = 1). A z-score of 1.96 corresponds to the 97.5th percentile.

What is the difference between PDF and CDF?

The PDF (Probability Density Function) gives the relative likelihood of a specific value, representing the height of the bell curve at that point. The CDF (Cumulative Distribution Function) gives the probability that a random variable is less than or equal to a specific value, representing the area under the curve to the left of that point. The CDF always ranges from 0 to 1.

What is the 68-95-99.7 rule?

The 68-95-99.7 rule (also called the empirical rule or three-sigma rule) states that for a normal distribution, approximately 68.27% of values fall within one standard deviation of the mean, 95.45% within two standard deviations, and 99.73% within three standard deviations. This rule helps quickly estimate probabilities without detailed calculations.

How do I find the probability between two values?

To find the probability between two values a and b in a normal distribution, calculate P(a ≤ X ≤ b) = CDF(b) − CDF(a). First convert both values to z-scores using z = (x − mean) / standard deviation, then look up or compute the CDF for each z-score and subtract. This calculator automates this process in the Between mode.