Statistics

Find the z value from a probability p and tail type (lower, upper, or two sided). This standard normal quantile tool computes critical values such as 1.96 with high precision.

For a normal distribution N(μ, σ²) with a given mean and standard deviation, compute the probability density (PDF), cumulative probability (CDF), upper-tail probability and standardized z value at any x.

Compute the probability P(a≤X≤b) that a normal random variable falls in an interval, given its mean and standard deviation. Also shows lower and upper tail probabilities and z scores, with the distribution curve and the interval shaded on a graph.

Find the percent point x and z-score of a normal distribution N(mu, sigma squared) for a probability p, using lower, upper, or two-sided tails.

Enter the lognormal parameters mu and sigma and a value x to compute the probability density, cumulative probability, upper tail probability, mean, median, mode, and variance, with curve charts.

Find the percent point x=exp(mu+sigma*z) of a lognormal distribution from probability p, a tail method, mu and sigma. Shows the matching z, cumulative probability, mean, median and variance, plus PDF and CDF graphs.

For a chi-square distribution with k degrees of freedom, compute the probability density, cumulative distribution (lower tail), upper-tail probability (p-value), mean, and variance at a given x, with PDF and CDF graphs.

Find the chi-square percent point (the χ² value) from a probability and degrees of freedom. Supports lower and upper tail probabilities for test critical values.

Compute the Student's t-distribution probability density (PDF), lower cumulative probability (CDF), upper tail, and two-sided probability (p-value) from a t value and degrees of freedom, with PDF and CDF graphs.

Find the percent point (critical value t) of the Student t-distribution from a probability p and degrees of freedom. Supports lower, upper, and two-sided tails for building t-test and confidence interval tables.

Compute the probability density, cumulative probability, upper tail p value, and mean of the F distribution with degrees of freedom d1 and d2.

Find the F distribution percent point (critical F value) from a probability and degrees of freedom d1, d2. Supports lower and upper tail quantiles for ANOVA and F tests.

Compute the PDF, CDF, upper tail probability, mean, and variance of the noncentral chi-square distribution for degrees of freedom k and noncentrality lambda at a value x.

Find the percent point (quantile) x of the noncentral chi-square distribution from a probability p, degrees of freedom k, and noncentrality lambda. Supports lower and upper tail modes, with density, mean, variance, and PDF/CDF charts.

Solve for the noncentrality parameter λ of the noncentral chi-square distribution from x, lower probability p, and degrees of freedom k. Useful for power analysis.

Compute the PDF, lower-tail CDF, upper-tail probability, mean, and variance of the noncentral t-distribution for a given degrees of freedom and noncentrality, with charts.

Enter a probability p, tail mode, degrees of freedom nu and noncentrality delta to find the noncentral t quantile t by bisection, plus CDF, upper tail, density, mean and variance.

Back out the noncentrality delta of a noncentral t distribution from a t value, lower probability and degrees of freedom for power analysis.

Compute the probability density, lower cumulative probability, and upper-tail probability of the noncentral F distribution for given degrees of freedom d1, d2 and noncentrality λ, with graphs.

Find the percent point (quantile) F value of the noncentral F distribution from probability p, degrees of freedom d1 and d2, and noncentrality lambda. Supports lower and upper methods with PDF and CDF charts.

Back-solve the noncentrality parameter λ of the noncentral F distribution from a critical value x, lower probability p, and degrees of freedom d1, d2.

Compute the probability density, cumulative probability, upper tail, mean, and variance of the exponential distribution from a rate or mean, with PDF and CDF charts.

Find the percent point (quantile) x of an exponential distribution from a probability p and rate lambda using the closed form. Supports lower and upper tail probabilities and shows mean, variance, and PDF and CDF charts.

Compute the PDF, CDF (lower probability), upper probability, mean, and variance of a continuous uniform distribution U(a,b) and visualize them on a chart.

Compute the joint probability density f(x,y) of a bivariate normal distribution from x, y, means, standard deviations, and correlation.

Compute Owen T function T(h,a) by numerical integration, with bivariate normal probability and standard normal CDF and density.

Compute the PDF, CDF (lower probability), upper probability, mean, and variance of a two-component mixed lognormal distribution, with charts.

Find the percent point (quantile) x for a probability p in a two-component mixed lognormal distribution. Set the weight, each component mu and sigma, and the mixing mode.

Compute the PDF, CDF (lower probability), reliability (survival function), mean, and variance of a Weibull distribution with shape k and scale lambda. Built for reliability engineering and life-data analysis.

Find the Weibull percent point x=λ(−ln(1−p))^(1/k) from probability p, shape k and scale λ, with B10/B50 life and reliability.

For a gamma distribution with shape k and scale theta, compute the probability density, cumulative lower-tail probability, upper-tail probability, mean, and variance at a given x, with charts.

Find the percent point (quantile) x of the gamma distribution from probability p, tail, shape k, and scale theta using the inverse regularized incomplete gamma. Lower and upper tails supported.

Compute the beta distribution PDF, CDF (lower probability), upper probability, mean, and variance from shape parameters α, β and x, with charts.

Find the percent point (quantile) x of a beta distribution from a probability p and shape parameters alpha and beta, via the inverse regularized incomplete beta function. Lower and upper modes.

Compute the probability density, cumulative probability, upper tail, mean, and variance of the Laplace (double exponential) distribution from location and scale.

Find the Laplace distribution percent point (quantile) x from a probability p and the location mu and scale b using the closed-form inverse.

Enter a location x0 and scale gamma to get the Cauchy distribution density, cumulative probability (lower tail), and upper tail probability, with charts.

Find the percent point (quantile) of a Cauchy distribution in closed form from probability p, location x0, and scale gamma. Choose lower or upper tail and view the density and CDF curves.

Compute the probability density, cumulative distribution, upper tail probability, mean, and variance of a logistic distribution with location mu and scale s, with charts.

Find the percent point (quantile) of the logistic distribution from a probability p with location μ and scale s using the logit transform.

Compute the probability density, cumulative probability, and upper-tail probability of the circular von Mises distribution from a mean direction and concentration.

Find the angle percent point (quantile) of the von Mises distribution from probability p, mean direction μ, and concentration κ.

Compute the Pareto distribution PDF, CDF, upper tail probability, mean and variance from scale xm and shape alpha. Visualize the distribution behind the 80-20 rule with charts.

Find the Pareto distribution percent point x = xm / (1 − p)^(1/α) in closed form from probability p, scale xm and shape α, with PDF, CDF, mean and variance plotted.

Enter location mu, scale sigma, and shape xi to compute the probability density, cumulative distribution (lower probability), upper tail probability, mean, and variance of the generalized Pareto distribution, with PDF and CDF charts. Useful for extreme value theory.

Find the percent point (quantile) of the generalized Pareto distribution for a probability p from the location μ, scale σ, and shape ξ using a closed form. Supports lower and upper tail input.

Compute the probability density, cumulative probability (lower tail), upper tail, and median of the Levy distribution with location mu and scale c, with charts.

Find the percent point (quantile) of the Levy distribution from probability p, location and scale using a closed form. Supports lower and upper tail.

From the mean lambda and a count k, compute the Poisson probability mass P(X=k), cumulative P(X less than or equal to k), and upper tail P(X at least k), with a bar chart of the distribution.

Enter a probability p and a mean lambda to find the smallest integer k whose cumulative probability is at least p, with the actual CDF, probability mass, mean and variance.

Back-solve the Poisson mean lambda from an observed count and a cumulative probability. Supports lower and upper confidence limits and a general inverse, with PMF and cumulative probabilities.

For a binomial distribution with n trials and success probability p, compute the probability mass P(X=k), cumulative probability, upper tail, mean, and variance, with a bar chart of the distribution.

Find the smallest integer k whose cumulative probability reaches q, from probability q, trials n, and success probability p.

Enter the number of successes r and success probability p to compute the negative binomial probability mass P(X=k), cumulative probability, upper tail, mean, and variance, visualized as a bar chart.

From probability q, number of successes r, and success probability p, find the smallest integer k where the negative binomial cumulative probability reaches q. Shows CDF, probability mass, upper tail, mean, variance and a PMF bar chart.

From a success probability p and a number of failures k, compute the geometric distribution probability mass P(X=k), cumulative probability, upper tail, mean, and variance, with a bar chart.

Find the smallest failure count k where the cumulative probability P(X less or equal k) reaches q for a geometric distribution. Shows PMF, CDF, upper tail, mean, variance and bar charts.

From population N, successes K, and sample size n, compute the probability of drawing exactly k successes without replacement (PMF, cumulative, upper) plus mean and variance, with a bar chart.

Compute the lower and upper tail probabilities of the studentized range Q for a given number of groups k and degrees of freedom nu. Useful for Tukey HSD critical values.

Find the critical value q of the studentized range distribution (the Tukey HSD percent point) from probability p, number of groups k, and degrees of freedom, using numerical integration and bisection.

Fit a least-squares regression line y = a + bx to your (x, y) data points and get the slope, intercept, R squared, and correlation, with a scatter plot and fitted line.

Fit a logarithmic model y = a + b ln(x) to your (x, y) data using least squares. See the coefficients a and b, the coefficient of determination R², a scatter plot with the fitted curve, and a table of predicted values.

Fit y = a e^(b x) to your (x, y) data points using least squares. Shows the coefficients a and b, R-squared, the correlation coefficient, a scatter plot with the fitted curve, and predicted values.

Fit y=ab^x to your (x,y) data by least squares. Get the coefficient a, base b, and R squared with a scatter plot and regression curve.

Enter your (x, y) data points and fit a power model y = a x^b by least squares. Shows the coefficient a, exponent b, R squared and correlation, a scatter plot with the fitted curve, and a table of predictions and residuals.

Fit the reciprocal model y=a+b/x to your (x, y) data points by least squares. Get coefficients a and b, the coefficient of determination R-squared, a scatter plot with the fitted curve, and a residual table.

Fit a quadratic regression y=a+bx+cx² to your (x, y) data points by least squares. Get the coefficients, R², and the parabola vertex, with a scatter plot and fitted curve.

Fit six models (linear, logarithmic, exponential, power, reciprocal, quadratic) to your data and compare them by R squared to find the best regression equation, with a scatter plot and fitted curve.

Fit a regression model to your data points and predict y from a given x, or x from y. Supports linear, quadratic, logarithmic, exponential and power models with R squared and a scatter plot.

Fit the line y = a + b x by weighted least squares, giving each data point a frequency (weight). Shows slope, intercept, R squared, and a scatter plot.

Fit a logarithmic model y = a + b ln(x) to frequency-weighted data points using weighted least squares. Get the coefficients a and b, the coefficient of determination R squared, the correlation, and the total weight, plus a scatter plot with the fitted curve.

Fit the exponential model y = a e to the bx power to frequency weighted (x, y) data using weighted least squares. Get coefficients a and b, the coefficient of determination, the correlation, total weight, and a scatter plot with the fitted curve.

Fit the exponential model y = a times b to the power x to frequency weighted data points. Weighted least squares on log transformed values gives the coefficient a, base b and coefficient of determination, with a scatter plot sized by frequency.

Fit a power model y = a x^b to frequency-weighted data points using least squares, showing the coefficient, exponent, and R-squared with a scatter plot.

Fit the reciprocal model y = a + b / x to frequency-weighted (x, y) data using weighted least squares. Returns the coefficients a and b, the coefficient of determination R², and the total weight, with a scatter chart whose point size reflects each frequency.

Fit a quadratic curve y = a + b x + c x² to weighted data points using weighted least squares. Shows the coefficients a, b, c, the coefficient of determination R², the vertex, and the total weight, plus a scatter plot whose point sizes scale with the weight and the fitted parabola.

Fit six models (linear, logarithmic, exponential, power, reciprocal, quadratic) to frequency-weighted data, compare them by weighted R squared, and show the best model and its equation.

Fit weighted (frequency) data with a chosen model and predict y at a given x. Supports linear, quadratic, power, exponential and logarithmic models, with weighted R squared and a scatter plot.