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Normal Distribution Calculator

Calculate density, cumulative probability and percentiles of the normal distribution with custom mean and standard deviation.

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Calculate PDF/PMF, CDF and percentiles instantly.

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Explanation

The normal distribution models continuous measurements that cluster around a mean value \( \mu \) and become less frequent farther from the center. Its bell-shaped curve is symmetric: values equally far above and below the mean are equally likely. The standard deviation \( \sigma \) controls the spread; the empirical rule states that approximately 68% of values fall within \( \mu \pm \sigma \), 95% within \( \mu \pm 2\sigma \) and 99.7% within \( \mu \pm 3\sigma \).

The Central Limit Theorem guarantees that the mean of a large sample tends to follow a normal distribution regardless of the original distribution of the data, which explains its prominence in inferential statistics. Use it when your variable is continuous, arises as the sum or average of many small independent factors, and doesn't have a hard lower bound limiting its domain. Heights, exam scores, measurement errors and many manufacturing variables are common examples.

Formula

\( f(x)=\frac{1}{\sigma\sqrt{2\pi}}e^{-\frac{(x-\mu)^2}{2\sigma^2}} \)

Parameters

  • μ (mu): mean or center of the distribution.
  • σ (sigma): standard deviation, which controls the spread.

Worked example

Situation: The height of adults in a population follows a normal distribution with mean \(\mu = 170\) cm and standard deviation \(\sigma = 8\) cm. A person is selected at random.

Question 1: What is the probability that they are 178 cm or shorter, that is, \(P(X \leq 178)\)?

Solution: We standardize by subtracting the mean and dividing by the standard deviation: \[ Z = \frac{178 - 170}{8} = \frac{8}{8} = 1.00 \] Looking up the standard normal table (or the CDF): \[ P(X \leq 178) = P(Z \leq 1.00) \approx 0.8413 \] That is, approximately 84.13% of people are 178 cm or shorter.

Question 2: What is the 95th percentile? That is, what height \(x\) satisfies \(P(X \leq x) = 0.95\)?

Solution: The 95th quantile of the standard normal is \(z_{0.95} \approx 1.6449\). Undoing the standardization: \[ x = \mu + z_{0.95} \cdot \sigma = 170 + 1.6449 \times 8 \approx 183.16 \text{ cm} \] Only 5% of the population exceeds 183.16 cm.

Interpretation: The mean of 170 cm is the point of maximum density; the empirical rule indicates that 68% of people are between 162 and 178 cm (\(\mu \pm \sigma\)). The 95th percentile is useful, for example, for sizing doors or ergonomic equipment that should accommodate 95% of the population.

Model assumptions

  • The variable is continuous and can, in principle, take any real value, even if in practice it is bounded by physical limits.
  • There is a structural reason to expect symmetry around a central value: the phenomenon arises as the sum or average of many small, independent factors of comparable magnitude (the Central Limit Theorem then supports the normal approximation even if the original data are not normal), or an equivalent physical/biological argument exists. It is not enough for a histogram to merely "look" bell-shaped.
  • There is no natural bound close to the observed data (for example, a minimum at 0) that truncates the distribution and produces systematic skewness.
  • The spread of the data does not depend strongly on their magnitude: variability is similar at the low and the high end of the observed range (no marked heteroscedasticity).
  • If these conditions do not hold — for example, the variable cannot be negative or the data are strongly skewed — consider other distributions such as the log-normal, the gamma or the Weibull.

Common uses

  • Modeling physical measurements: heights, weights, reaction times or measurement-instrument errors.
  • Statistical quality control and manufacturing processes, defining specification limits and control charts.
  • Approximating the binomial or Poisson distribution when the number of trials or the mean is large.
  • The foundation of confidence intervals and hypothesis tests for sample means, thanks to the Central Limit Theorem.
  • Financial risk models and asset returns, though with caveats: real market tails are usually heavier than those of a theoretical normal.

How to interpret the result

The calculator offers three types of result. The value of the PDF density function, \( f(x) \), is not a direct probability but a density: it indicates how concentrated the distribution is around a specific point. Higher PDF values mean that area is more likely than regions with lower PDF, but you need to integrate the curve over an interval to get an actual probability.

The value of the CDF, \( P(X \leq x) \), is indeed a probability: it represents the fraction of times the variable takes a value less than or equal to \( x \) under the normal model with the chosen parameters. For example, \( P(X \leq 1.96) \approx 0.975 \) in a standard normal means that 97.5% of observations fall below 1.96. The right tail \( P(X > x) = 1 - P(X \leq x) \) measures how extreme the value is in that direction. In the chart, the shaded green area represents exactly that cumulative probability for the range you selected; the horizontal axis shows the domain of the distribution centered on the mean.

A percentile or quantile result reverses the question: given a probability level \( p \), the calculator returns the value \( x \) such that \( P(X \leq x) = p \). The 90th percentile (p = 0.90) separates the lower 90% from the upper 10%. This is useful for setting control thresholds, critical values in hypothesis tests, or specification limits in industrial processes.

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References and further reading

  • Wikipedia: Normal distribution — definition, properties and proofs.
  • Wikipedia: Central limit theorem — why the normal appears so recurrently.
  • Wikipedia: 68–95–99.7 rule — the empirical rule and its usual applications.
  • Casella, G. and Berger, R. L. (2002). Statistical Inference (2nd ed.). Duxbury. — formal treatment of the normal distribution and its properties.

Frequently asked questions

  • How do I convert a value x into a z-score? Subtract the mean and divide by the standard deviation: z = (x − μ)/σ. The z-score tells you how many standard deviations x lies above or below the mean, which lets you work with the standard normal N(0, 1).
  • What is the 68-95-99.7 rule? In any normal distribution, about 68% of values fall within ±1σ of the mean, 95% within ±2σ, and 99.7% within ±3σ. It is a quick reference for judging whether a value is typical or extreme.
  • Why does the normal distribution appear in so many problems? Because of the central limit theorem: the sum or mean of many independent observations is approximately normal regardless of the original distribution. That is also why it approximates the binomial or Poisson when their parameters are large.

Reference: Normal distribution — Wikipedia