Updated Probability & Statistics Tool

Normal Distribution Calculator

Compute PDF, CDF, Z-scores, tail probabilities and inverse normal values for any normal distribution. Enter a mean μ and standard deviation σ, then evaluate probabilities, convert between X and Z and find critical values.

PDF & CDF Z ↔ X Conversion Tail Probabilities Inverse Normal

Work with Any Normal Distribution N(μ, σ²)

Use this Normal Distribution Calculator to evaluate the probability density function (PDF), cumulative distribution function (CDF), tail probabilities and Z-score conversions for a normal random variable X with mean μ and standard deviation σ. You can also find inverse normal values (quantiles) for common confidence levels and probabilities.

By default, the calculator uses the standard normal distribution N(0, 1). To work with a general normal distribution, change the mean μ and standard deviation σ in any tab. All results use high-quality numerical approximations for the normal CDF and its inverse.

This tab evaluates the PDF f(x) and CDF F(x) for X ∼ N(μ, σ²). It also reports the corresponding Z-score z = (x − μ) / σ.

Enter either x or z and click convert. If you fill both, the calculator prioritizes converting from x to z.

For left or right tail probabilities, only a is used. For between and outside probabilities, both a and b are used and the calculator automatically enforces a ≤ b when interpreting the interval.

Use this tab to find critical values. For example, with tail type "left" and p = 0.975 you obtain the 97.5th percentile; with tail type "central" and p = 0.95 you obtain symmetric bounds around μ that capture 95 percent of the distribution.

This numeric table summarizes standard normal CDF values Φ(z) = P(Z ≤ z) for Z ∼ N(0, 1) at common z-values. You can use it to quickly approximate probabilities or verify results from the other tabs.

What Normal Distribution Calculator Calculates

Use Normal Distribution Calculator to compute PDF, CDF, Z-scores, tail probabilities and inverse normal values for any normal distribution.

Math Calculators groups other calculations in the same subject.

Normal Distribution Calculator Inputs

Normal Distribution Calculator uses Mean μ, Standard Deviation σ, Value x, Value x (optional), Z-Score z (optional).

Normal Distribution Calculator Formula or Method

  • Z-score: z = (x − μ) / σ.
  • Z = \(\frac{X - \mu}{\sigma}\).
  • f(x) = \(\frac{1}{\sigma \sqrt{2\pi}}\) exp\(\,-\frac{(x - \mu)^2}{2\sigma^2}\).

Reading Normal Distribution Calculator Results

The results include Z-Score, PDF f(x), CDF F(x) = P(X ≤ x), Right Tail P(X > x), Value X, CDF P(X ≤ x).

Alongside Normal Distribution Calculator, use Binomial Distribution Calculator to compute binomial probabilities, cumulative probabilities, tail probabilities and summary statistics.

For a different result from Normal Distribution Calculator, Z-Score Calculator can calculate Z-scores, percentiles, raw scores, and standard normal distribution values.

Calculators Related to Normal Distribution Calculator

After checking Normal Distribution Calculator, use Poisson Distribution Calculator to compute Poisson probabilities for event counts in a fixed interval. When you need to calculate mean, median, mode, variance, standard deviation, probability, Z-scores, normal distribution, combinations, permutations, correlation, use Statistics Calculator. Geometric Distribution Calculator covers a separate task: compute probabilities for the geometric distribution using both definitions: failures before the first success and trial number of the first success.

After checking Normal Distribution Calculator, use Sampling Distribution Calculator to explore sampling distributions for sample means, proportions and differences between means. When you need to compute exact binomial probabilities for a given number of trials n, success probability p and number of successes k, use Binomial Probability Calculator. Probability Calculator covers a separate task: work with single events, combined events and conditional probability.

After checking Normal Distribution Calculator, use Standard Deviation Calculator to calculate mean, variance, population and sample standard deviation, weighted standard deviation, grouped data standard deviation.

Frequently Asked Questions About Normal Distribution Calculator

It is used to compute PDF, CDF, Z-scores, tail probabilities and inverse normal values for any normal distribution.

It uses Mean μ, Standard Deviation σ, Value x, Value x (optional), Z-Score z (optional).

Normal Distribution Calculator uses Z-score: z = (x − μ) / σ.

It returns Z-Score, PDF f(x), CDF F(x) = P(X ≤ x), Right Tail P(X > x), Value X, CDF P(X ≤ x).

Check Mean μ, Standard Deviation σ, Value x first. Differences can also come from rounding, unit choices, or a different formula or rule set.