Web Pages that Perform Statistical Calculations!
( StatPages.net )

Over 600 Links (including 380 Calculating Pages) -- And Growing!

(Updated 04/08/2006 -- check out What's New, and the Awards and Recognition this site has received.)

The web pages listed here comprise a powerful, conveniently-accessible, multi-platform statistical software package. There are also links to online statistics books, tutorials, downloadable software, and related resources. All of these resources are freely accessible, once you can get onto the Internet. FREE dial-up Internet access is available from NetZero and Juno.

These pages are located on servers all over the world, and are the result of much cleverness and hard work on the part of some very talented individuals. So if you find a page useful it would be nice to send the authors a short e-mail expressing your appreciation for their hard work and generosity in making this software freely accessible to the world. Please let me know of any dead links, computational errors, or other problems you might encounter (e-mail me at johnp71@aol.com).

Table of Contents for this page...

• Selecting the right kind of analysis
• "Online Software" Package websites
• Calculators, plotters, function integrators, and interactive programming environments
• Probability distribution functions: tables, graphs, random number generators
• Descriptive statistics, histograms, charts
• Confidence intervals, single-population tests
• Sample comparisons: t-tests, ANOVAs, non-parametric comparisons
• Contingency tables, cross-tabs, Chi-Square tests
• Regression, correlation, least squares curve-fitting, non-parametric correlation
• Analysis of survival data
• Bayesian Methods
• Other statistical tests and analyses
• Specialized and discipline-specific tests and analyses
• Power, sample size and experimental design

Other Statistical Resources...

• Free statistical software packages

• Online statistics textbooks, reference manuals, and journals
• Interactive statistical demonstrations and tutorials
• Links to other statistics-related pages
• About this Project

First -- Choose the right test! [return to Table of Contents]

There are a bewildering number of statistical analyses out there, and choosing the right one for a particular set of data can be a daunting task. Here are some web pages that can help:

1. Statistical Decision Tree, from the developers of the MicrOsiris package. This is an interactive set of web pages to help you select the right kind of analysis to perform on your data. It asks you a simple series of questions about your data (how many variables, etc.), then makes recommendations about the best test to perform.
2. Choosing a Statistical Test, Chapter 37 of Dr. Harvey Motulsky's book Intuitive Biotatistics.
3. "Selecting Statistics", by Bill Trochim (Cornell). Another  interactive set of web pages to help you select the right kind of analysis to perform on your data.
4. The very extensive test-selection routine used in Dr. Robert Knodt's MODSTAT statistical package.

"Online Software Package" web sites [return to Table of Contents]

As you can see from looking at the StatPages.net web site, there are many "stand-alone" web pages that are each designed to perform only a single test or calculation. In addition, some talented individuals and groups have created coherent website that perform an entire suite of calculations, with a logical organization and consistent user interface. Each of these web sites is really a fairly complete online statistical software package in itself. Here are some of these "comprehensive" statistical analysis web sites:

• OpenEpi -- a project to create open source software for Epidemiologic statistics in JavaScript and HTML. Can be run from a web site or downloaded to a local computer and run without a web connection. Currently includes calculations for: Proportion, Two by Two Table, Paired Case Control, 1 Person Time Rate, 2 Person Time Rates, R by C Table, Diagnostic Test Eval, Dose-Response/Trend, Sample Size, Random numbers.
• Statlets -- an "online statistical computing center" providing access to over 50 applets in which you can enter data, compute statistics, create tables and graphs, and print out the results. Provides basic plotting, probability distributions, summary statistics, one-sample analysis, time-series analysis, two-sample comparisons, regression analysis, attribute estimates, ANOVAs, and Statistical Process Control. This public version supports up to 50 rows and 8 columns of data. For larger data sets, a single-user copy or a corporate deployment license can be purchased.
• The Calcugator -- a calculator, plotting engine, and programming environment. Also available as a free stand-alone downloadable program. Simple to use; rivals programs like MATLAB, with 200 functions/operators to perform real, integer, rational, complex, boolean, statistical, vector, array and matrix computations. Both the input and output of the program are displayed on standard windows which can be further edited, saved, merged, print-previewed and printed. Allows rapid creation of 2D and 3D plots of functions, polar and parametric displays, bar, pie, pareto and xy charts. All plots can be configured using the mouse (zooming, panning, selecting). Titles and labels are supported, and all figures created by the Calcugator can be exported into popular file formats or pasted into an editable window. As a programming environment it has a simple and compact language with identical syntax to Java/C/C++, and allows user-defined functions.
• The WebMath page performs a large number of numeric calculations and symbolic algebraic manipulations of the type that might arise in high school / college algebra and calculus, including some elementary statistical calculations. In doing so, it provides a detailed step-by-step explanation of how it arrived at the answer.

Calculators, Plotters, Function Integrators, and Interactive Programming Environments...[return to Table of Contents]

• Expression Evaluators -- type in any numeric expression; the computer will evaluate it and display the results...
  • Scientific Calculator (numeric expression evaluator)
  • Expression Evaluator, similar to above, but doesn't require Java or JavaScript capability
  • Visible Memory Kalculator -- provides a growing visible memory of all values inputed or computed for use at any time later (just click on it). Can also read text (ascii) files.
  • Evaluates various sums, cross-products, and other "building block" expressions that arise in statistical formulas
  • The Vanderbilt MathServe Calculus Toolkit has separate calculating/graphing pages for: Factoring Polynomials, Partial Fractions, Polynomial Equations, Graphs of Functions, Graphs of Equations, Limits, Derivatives, Antiderivatives (Indefinite Integrals), Definite Integrals, Inverse Functions, Newton's Method, Polynomial Interpolation, Sums, Parametric Equations, and Polar Functions
  • Inverse Symbolic Calculator -- tells you where a number came from. For example, if you type in 1.55838744, this program will tell you that it's really the square root of 17/7.
• Calculators -- pages that look and act like a pocket calculator...
  • A simple 5-function calculator, with memory
  • CoCalc -- an advanced RPN scientific calculator from CoHort Software. Has log, trig, boolean, binary/hex, and basic statistics functions. Implemented as a Java applet, so it can be run from the Web or stored on your computer for "offline" running when not connected to the Web. Also available in a larger-font version.
  • Links to Other Online Calculators, and calculator-related resources, from Essential Links (el.com).
• Plotters -- type in any algebraic function; it displays the graph...
  • Function plotter -- Lets you zoom in and out to view any portion of the graph. (Needs Java.)
  • Function plotter -- Plot almost any function or relation found in high school and undergraduate college mathematics. Plots functions of the form y = f(x), such as y = x2 or y = 3x + 1, or relations of the form f(x,y) = g(x,y), such as x2 + y2 = 4. (No Java needed.)
  • Linear Programming Grapher-- Enter a linear function of two variables to be minimized, and any number of linear inequality expressions, and the page will instantly solve it and display a graph showing the feasible region and the constraints.
  • Simplex Tool -- Similar to the Linear Programming Grapher, but works with functions of more than two variables, and doesn't graph the results.
• Integrators -- type in any function; the computer displays the indefinite integral function (if one exists) and/or the value of the definite integral (area under the curve) between two endpoints...
  • Indefinite Integrals -- using the Mathematica engine
• Interactive Programming Environments -- These pages implement various mathematical programming languages. You can enter commands or entire programs (type or copy/paste) into the web page, and they will be executed immediately.
  • Rweb -- an interactive web-based interface to the "R" statistical programming language (similar to S or S-plus)
  • SHAZAM -- a programming environment for econometricians, statisticians, and others who use statistical techniques. Its primary strength is estimating and testing many types of regression models. Provides a flexible command language and capabilities for programming procedures. Has an interface to the GNUPLOT package for high quality graphics.
  • Mx  -- a matrix algebra interpreter and numerical optimizer for exploration of matrix algebra. Many built-in fit fuctions for structural equation modeling and other statistical modeling. Has fitting fuctions like those in LISREL, LISCOMP, EQS and CALIS, along with facilities for maximum likelihood estimation of parameters from missing data structures, under normal theory. Users can easily specify complex 'nonstandard' models, define their own fit functions, and perform optimization subject to linear and nonlinear equality or boundary constraints.

Probability Distribution Functions: Tables, Graphs, Random Number Generators... [return to Table of Contents]

• Probability Integrals -- these pages take the place of a handbook of statistical functions. They're arranged with the most comprehensive,multi-function pages first...
  • These pages contain calculations for a very wide assortment of probability distribution functions, including Normal, Bivariate Normal, Student t, Chi-Square, Fisher F, Bivariate Normal, Noncentral Student t, Non-central Chi-Square, Non-central Fisher F, Poisson, Log-normal, Exponential, Beta, Gamma, Logistic, Binomial, Negative Binomial, Multinomial, Cauchy, Gumbel, Laplace, Pareto, Weibull, Uniform (continuous and discrete), Triangular, Geometric, and Hypergeometric:
    • Distribution/density calculators, plotters and random number generators
    • Calculate and plot probability distributions, cumulative PDF, survivor, hazard, tail areas
    • Calculate tail areas and left-integrals
    • Tables for 16 continuous and discreet distribution functions
  • These pages each compute probabilities for the four most common probability distributions:
    • Normal, t, Chi-Square, and Binomial (density and cumulative) probabilities; (When you get to the Rweb page, scroll down to the Analysis Menu and select Probability.)
    • Normal, t, F, Chi-Square probabilities and inverses, with nice graphical representation.
    • Normal, t, F, Chi-Square, Binomial, and Uniform probabilities and inverses
    • Normal, Student t, Chi-Square, and Binomial probabilities and inverses. Provides an easy graphical way to specify whether you want upper / lower tail or interior integrals.
    • Central and tail areas for Normal, Student, F, Chi-Square, Binomial, and Poisson distributions
    • Statistical probability distribution functions: Normal, Student t, Chi-Square, Fisher F
    • P-values for the Popular Distributions -- Binomial , Chi-square, Exponential , Fisher's F, K-S: Two Samples , Poisson, Normal , Student's t, and Uniform distributions.
    • Calculate p-value from z, t, F, r, or Chi Square; or do the reverse.
    • Reverse computations: enter p-value (and, if necessary, sample sizes and/or d.f.); program will compute z, t, F, Chi Square, and correlation coefficient
  • These pages each compute probabilities and/or inverses for a specific distributions:
    • Normal distribution areas, with nice graphical interpretations
    • A very attractive page for Normal distribution (and inverse), with detailed explanations
    • Normal area (1-tailed)
    • Cumulative area under the normal curve (integral from minus infinity to z)
    • Chi-Square probabilities, and reverse, with a detailed explanation
    • Chi Square probabilities and reverse
    • Chi-Square Distribution
    • Chi-Square Distribution
    • Student t Distribution
    • Student t Distribution and its inverse (t  value from p value)
    • Studentized Range -- the probability of a studentized range being less than or equal to value x with v d.f. from r sample
    • Probabilities for the Fisher F distribution
    • Another Fisher F distribution p-value calculator.
    • Critical Fisher F value, given the alpha level, the numerator and denominator d.f.
    • Non-central F value (by Laubscher's square root approximation), given the F-value, numerator and denominator d.f., and the noncentrality parameter.
    • Binomial, Poisson and Gaussian distribution probabilities
    • Binomial probability calculator
    • Binomial Approximation of the Normal Distribution
    • Cumulative frequency for the Binomial distribution
    • Probabilities for Gamma, complete Beta, and Incomplete Beta distributions
    • Multinomial Distribution
  • This page contains links to printable copies (in Adobe Acrobat PDF format) of many statistical tables including some for which no "calculating pages" are available
    • Normal Curve
    • Critical Values for: Student t, Fisher F, Studentized Range Statistic and Dunnett's Test, Chi-Square, Binomial Test, Wilcoxon Ranked-Sums Test, Wilcoxon Signed Ranks Test, and Correlation Coefficient
    • Converting r to Z
    • Statistical Power of: Z Test, t-Test for One Sample or Two Related Samples, t-Test for Two Independent Samples, Analysis of Variance, and Correlation Coefficient
    • Required Sample Size for various tests
• Random Number Generators...
  • Distribution/density calculators, plotters and random number generators for a large number of distributions
  • Random Decimal Digits, in batches of 500 at a time.
  • Random integers -- generates any number of random integers, uniformly distributed between any two limits
  • Generate tables of random integers from any specified range, or random values from a Normal distribution with any specified mean and SD.
  • Random fractional numbers -- generates any number of random numbers, each a fraction between 0 and 1 with 8 digits after the decimal point
  • Random Permutations -- generates N sets of random permutations of integers from 1 to M
  • Another random permutation generator
  • Randomly assign subjects to treatment groups -- can randomly choose a group for each subject, or randomly shuffle subjects within groups.
  • Research Randomizer -- generates one or more sets of random numbers from a specified range, with or without repeats, sorted or unsorted.
  • Block Randomizer -- assigns subjects randomly to different groups, with multiple blocking to ensure that imbalances are kept under control if the study is terminated prematurely
  • Random assignment of subjects to one or more groups -- three variations:
    1. generates M groups of N numbers each by distributing the numbers from 1 to M*N randomly into the M groups
    2. generates M blocks of N numbers each by randomly shuffling the numbers from 1 to N in each block
    3. generates a subset of N numbers by random selection from a list of the numbers from 1 to M
• Combinatorial Objects Server -- generates an incredible assortment of...
  • Permutations and their restrictions
  • Subsets or Combinations
  • Permutations or Combinations of a Multiset
  • Set Partitions
  • Numerical Partitions and relatives
  • Binary, rooted, free and other trees
  • Necklaces, Lyndon words, DeBruijn Sequences
  • Irreducible and Primitive Polynomials over GF(2)
  • Ideals or Linear Extensions of a Poset
  • Spanning Trees and other Subgraphs of a Graph
  • Unlabelled Graphs
  • Pentomino Puzzles, Polyominoes, n-Queens
  • and other puzzles and Miscellanea

Descriptive Statistics, Histograms, Charts... [return to Table of Contents]

• Statiscope -- a beautifully-implemented page for calculating and displaying a large number of descriptive statistics from a set of numbers you enter
• WebStat (an integrated applet) can generate summary statistics, as well as histograms, stem and leaf plots, boxplots, dotplots, parallel coordinate plots, means plots, scatterplots, QQ plots, and time series plots
• Descriptive Sampling Statistics -- Enter up to 80 numbers; this page will calculate the mean, variance, SD, CV, skewness and kurtosis.
• Descriptive statistics (mean, SD, SEM, and CI of mean). Can enter or paste raw data, or enter mean, SD or SEM, and N to get CI.
• Descriptive Statistics -- Enter up to 80 values; page calculates: N, mean, variance, SD, CV, skewness, kurtosis, SEM, median, min, max, range, 1^st & 3^rd quartiles, interquartile range, quartile deviation, coeff of quartile var, and absolute deviation.
• Measuring for Accuracy -- Given a set of observed and predicted values, this page calculates the SD of errors, mean absolute & relative error, and Durbin-Watson statistic.
• Arithmetic, Geometric, and Harmonic Means -- of up to 80 values.
• Rweb - extensive tabular and graphical descriptive summarization: mean, quartiles, histograms, scatterplot matrices (with smoothers), QQ plots (normal and pairwise), time series, box plots. (When you get to the Rweb page, scroll down to the Analysis Menu and select Summary.)
• The Data Applet provides descriptive statistics, histograms, boxplots, and scatterplots
• A variety of descriptive statistics and a stem and leaf display
• Detect Outliers -- this calculator performs Grubbs' test, also called the ESD method (extreme studentized deviate), to determine whether one of the values in the list you enter is a signficant outlier from the rest. Also contains an excellent discussion of what to do about outliers.
• Combine Subgroups -- calculate the mean and SD of a combination of groups from the N, mean and SD of each group.
• Computes summary statistics for one variable, draws a crude histogram, and sorts a list of values. Given pairs of values, it computes the least squares regression line and Pearson correlation coefficient.
• Basic descriptive statistics (mean, sum of squares, variance, standard deviation, minimum, 25^th percentile, median, 75^th percentile, and maximum for up to 500 numbers
• Empirical Distribution Function -- from up to 42 sets of [value, frequency].
• Multinomial Distributions -- Enter up to 12 values and their corresponding probabilities, and this page will calculate Expected Value, Variance, Standard Deviation, & Coefficient of Variation
• Paired Data Sets Statistics -- Enter up to 28 sample paired data sets, and this page will calculate means, variances, and covariance 
• Histogram -- Enter up to 80 numbers, and this page will display a histogram. 
• Histogram from a set of numbers, lets you dynamically alter the interval width and see the effect immediately
• Histogram -- type in or upload a data set or give a URL; submit; returns a colored histogram that you can copy from the page; also does polygons and cumulative
• Determination and Removal of Outliers -- Given a set of numbers, this page iteratively isolates potential outliers for removal.
• Point Pattern Analysis -- used to describe and help analyze point patterns. It consists of 14 different analysis routines for a variety of basic descriptive statistics: nearest neighbor analysis, K-function, space-time Knox, Join-Count statistics, Global Moran’s I and Geary’s c, general Getis-Ord’s G, local K-function, and more.
• Draw a scattergram from {x,y} data
• Draw a 3-dimensional scattergram from {x,y,z} data
• Generate a VRML file to view 3-dimensional (x,y,z) data. To view the resulting files requires a VRML viewer.
• Compute and plot a Kernel Density Estimate from a set of points, using Epanechnikov, triangular, biweight or Gaussian kernels
• Compute Poisson change-point, that is: estimate when, in a long sequence of occurrences, the occurrence rate underwent a sudden change
• Boxplot -- type in or upload a data set, or give a URL; submit; returns a colored boxplot that you can copy from the page
• Parallel Boxplot -- type in or upload a bivariate data set with a continuous variable and a group indicator; submit; returns a colored parallel boxplots that you can copy from the page
• Q-Q Plot -- type in or upload a data set, or give a URL; submit; returns a colored q-q plot that you can copy from the page
• Plot up to 10 x,y data points

Confidence Intervals, Single-Population Tests, Measurement Errors... [return to Table of Contents]

• Confidence Intervals...
  • for the difference between two means, given N, mean, SD for each group
  • Exact C.I.'s for Binomial (observed proportion) and Poisson (observed count) (also available as an Excel spreadsheet)
  • Exact and "modified Wald" C.I.'s for observed proportion or count, with a good explanation
  • Confidence interval around a proportion, given the population size, the sample size, the sample percentage and the confidence level. Has interesting animation, plus a good explanation of the concepts. A related page has no animation, but can be printed out.
  • 95% C.I. around an observed proportion
  • Bayesian "credible" intervals around an observed proportion. Somewhat comparable to the "classic" confidence intervals, but tend to be somewhat narrower.
  • 95% or 99% C.I. for proportions for any specified sample size and population size
  • 95% C.I. around an observed sample mean
  • Confidence interval around an observed sample SD, assuming the data are sampled from a Normal distribution
  • Percentage: Estimation & Testing -- calculates exact binomial confidence intervals and tests of hypothesis for population proportion, from infinite or finite populations.
• Tolerance Intervals...
  • Tolerance Intervals for the Normal Distribution. (Don't confuse tolerance intervals with confidence intervals!) A tolerance interval for a measured quantity is the interval in which there is a specified likelihood that a specified fraction of the population's values lie. This page will calculate 1-sided and 2-sided tolerance intervals for any specified population fraction, and for any specified level of confidence, from the mean and standard deviation of a finite sample, under the assumption that the population is normally distributed. These calculations are also available in a downloadable Excel spreadsheet: tolintvl.xls .
• Single-Population Tests...
  • Sign and Binomial test -- test an observed proportion against a proposed population proportion
  • Another Sign  and Binomial test
  • Mean, SD, confidence interval, etc. for a set of values
  • An excellent One-Sample Student t Test page -- enter or paste raw data, or enter mean, SD or SEM, and N
  • One-sample Student t test for Mean vs. a Specified Value -- for up to 80 observations, and a postulated population mean.
  • Student t-test of a single mean (vs specified value) from N, mean, SD
  • Another Student t-test of a single mean (vs specified value) from N, mean, SD
  • Similar test of single mean vs 0 (equivalent to a paired Student t) from N, mean, SD
  • Test for Asymmetry around zero -- Enter a set of numbers (usually a mix of positive and negative numbers), and the program will apply a non-parametric test (originally created by R. A. Fisher) of whether the numbers are consistent with a population frequency distribution that is symmetrical around zero (but does not necessarily have to be normal). It is a frequentist test to work Darwin's experiment with matched pairs, and experiments like it.
  • Test for the mean being greater than some specified value. This unusual test is Bayesian and frequentist at the same time. The null hypothesis asserts some value for the mean of a population of positive numbers; the alternative hypothesis says the mean is higher than that. This test gives a Bayesian likelihood ratio that is also an upper bound on the p-value of the frequentist test.
  • Test observed vs. expected rates of occurrence of events, based on Poisson distribution; also includes confidence intervals and analysis of rate-ratios (such as Standardized Mortality Ratio, Morbidity Ratio, and Comparative Mortality Figure)
  • Similar to above, but used to study the distribution of accidents and events at the individual level
  • Exact confidence intervals around a rate-ratio, using Liddell's method (also contains a number of common approximations, for comparison)
  • Test observed vs expected proportions, based on the Binomial distribution
  • Binomial Test -- whether the number of "successes" differ from what was expected based on the number of trials and the probability of success.
  • Similar to above, but deals with the probability of a particular sample size, given an observed 'x' number positive (or white, or car crashes) vs. an expected 'U' proportion positive
  • Compatibility of Multi-Counts -- tests whether up to 14 observed event counts (each over the same amount of time) are consistent with a single expected event rate.
  • Runs Test for Randomness -- Enter up to 80 numbers, and this page will calculate a runs test to see if the numbers form a random sequence
  • Testing the Variance -- of up to 80 observations against a postulated population variance.
  • Analyze observed proportions in samples from finite populations, based on the Hypergeometric distribution
  • Test for Normality -- Enter up to 80 numbers, and this page will test for normality based on the Jarque-Bera statistic
  • Test for Homogeneity of a Population -- enter form 25 to 84 values; page provides information to test whether histogram is unimodal.
  • Test for Normality -- enter up to 42 sets of [value, frequency]; page will calculate skewness, kurtosis, and Liliefors test for consistency with a normal distribution.
  • Test for Uniform Distribution -- enter up to 42 sets of [value, frequency]; page will calculate the Kolmogorov-Smirnov test for consistency with a uniform distribution.
  • Testing Poisson Process -- enter up to 14 sets of [value, frequency]; page will calculate a Chi square test for consistency with a Poisson distribution.
  • Lilliefors Test for Exponential Distribution -- tests whether a set of observed values are consistent with an exponential distribution.
• Chi-Square "Goodness of Fit" test for observed vs expected counts (NOT from Contingency Tables)...
  • Chi Square test -- takes observed values, and expected values that can be specified as expected occurrences, or percentages or fractions of the total. Data can be typed in or copied and pasted.
  • Chi-Square test
  • Chi-Square test
  • Chi-Square test (for up to 8 categories)
  • Goodness-of-Fit for Discrete Variables -- Chi square test for up to 14 sets of [Observed, Expected] frequencies.
• Measurement Errors and Error Propagation...
  • Calculate how the standard error of one or two variables propagates through any function of those variables
  • Compute confidence intervals of a sum, difference, quotient or product of two means, assuming both groups follow a Gaussian distribution.

Sample Comparisons: t-Tests, ANOVAs, Non-parametric Comparisons... [return to Table of Contents]

• Student t-test (for comparing two samples)...
  • a very general Student t-test web page -- paired or unpaired, equal- or unequal-variance, from individual observations (which can be key-entered or copy/pasted) or summary data (N, Mean, SD or SEM). Includes explanations and advice on carrying out this type of test.
  • t-test, paired or unpaired
  • t-test, paired or unpaired
  • t-test, paired or unpaired
  • t-test, paired
  • Paired Student t Test -- on up to 42 pairs of values, along with a postulated population mean difference.
  • Testing Two Populations -- Unpaired Student t test for up to 80 observations in each sample. Also accepts a postulated difference between the two population means, which can be different from 0.
  • A general 2-sample comparison calculator, for paired, unpaired, equal-variance, obtaining its p-values from table lookup or from resampling
  • Unpaired t-test from summary data (N, mean, SD)
  • Very general t-test program for comparing measured quantities, observed counts, and proportions between two unpaired samples; also produces risk ratio, odds ratio, number needed to treat, and population analysis.
• ANOVA (Analysis of Variance) -- comparison of two or more samples ...
  • One-Way and Factorial ANOVA for uncorrelated samples (extension of unpaired Student t-test to more than 2 groups)...
    • One-way ANOVA, with graphical output
    • One-way ANOVA for 3 Independent Samples
    • ANOVA: Testing the Means -- One-way ANOVA for three groups, each containing up to 40 subjects.
    • One-way ANOVA for 4 Independent Samples
    • One-way ANOVA from summary data (N, mean, and SD or SEM)
    • Another 1-way ANOVA from summary data
    • Two-way factorial ANOVA for 2 rows by 2 columns
    • Two-way factorial ANOVA for 2 rows by 3 columns.
    • Two-Way ANOVA Test -- for blocked designs of up to 4 groups by 6 treatments.
    • Two-Way ANOVA with Replications -- for blocked designs of up to 4 groups by 6 treatments, with up to 4 replications.
    • Two-way factorial ANOVA for 2 rows by 2 columns, from summary data (N, mean, SD)
    • ANOVA for Condensed Data Sets -- Enter up to 10 sets of (N, mean, SD); page calculates a one-way ANOVA.
    • Very general n-way factorial ANOVA, with interactions, means table, interaction plots, Bonferroni post-hoc multiple comparisons, and confidence intervals. (When you get to the Rweb page, scroll down to the Analysis Menu and select ANOVA.)
  • Repeated-Measures ANOVA for correlated samples (extension of paired Student t-test to more than 2 matched measurements)...
    • One-way repeated-measures ANOVA for 3 correlated samples
    • One-way repeated-measures ANOVA for 4 correlated samples
    • ANOVA for repeated-measures or matched measurements -- Enter three sets of matched measurements (up to 40 points each); page calculates a repeated-measures ANOVA.
  • Bartlett's Test for Equality of Multi-variances -- for up to 14 sets of [N, variance].
  • Post-hoc Tests -- After doing a two-way (or other) ANOVA, post -hoc tests (also called post tests) compare individual pairs of groups. This calculator does not perform the ANOVA calculations, but takes the output from an ANOVA (residual means square error, degrees of freedom) performs a post-hoc test between any pairs of cells that you select (using cell means and N's), at whatever alpha you specify.
  • Tukey LSD (Least Significant Difference), using the standard table produced by an ANOVA
  • Scheffe Least Significant Difference, using data from a standard ANOVA table and the N's for the two groups being compared
• Non-parametric tests (use these when the data is not normally distributed)...
  • Sign test for matched pairs
  • Median test for unmatched pairs
  • Wilcoxon Signed-Ranks test for matched pairs -- This page takes case-by-case pairs of matched data
  • Another Wilcoxon Signed-Ranks test for matched pairs -- This page takes summarized, tabulated data: how many cases had differences of +1, +2, +3, etc., and -1, -2, -3, etc.
  • Comparing Two Random Variables -- by the Mann-Whitney U test, with up to 80 observations per sample.
  • K-S Test for Equality of Two Populations -- Given two sets of frequencies (using the same grouping intervals), this page calculates the Kolmogorov-Smirnov test.
  • Wilcoxon Sum-of-Ranks (Mann-Whitney) test for comparing two unmatched samples
  • Kruskal-Wallis test (non-parametric ANOVA) for 2 or more groups of unpaired data -- This page requires that you first cross-tabulate your data into a matrix, with a row for every group and a column for every different numeric value that any subject had; the cell of the matrix tell how many subjects (if any) in that group had exactly that numeric value.
  • Least Significant Difference between mean ranks (post-hoc test after a significant Kruskal-Wallis test)
  • Friedman test for comparing rankings (non-parametric)
  • Two-group ordinal comparisons to assess how probable it is that the two groups come from a single ordering, using Wald-Wolfowitz, Randomness Test, Mann-Whitney, and Kolmogorov-Smirnov
  • Two-group paired comparisons, using T-test, Wilcoxon, Signs test, and McNemar test
  • McNemar's test for the paired comparison of proportions (or for matched pairs of labels)
• Comparison of proportions between two groups...
  • Comparison of Binomial proportions
• Sequential Analysis -- each subject's data (usually paired comparisons) is tested as it becomes available, and a decision is made to accept or to reject the null hypothesis or to keep testing.
  • by Paired Preferences -- Each pair of observations is compared and rated qualitatively as "preferring A" or "preferring B"
  • by Paired Differences -- Each pair of numbers is subtracted to obtain a difference
• WebStat (an integrated (Java) applet) can perform Z-tests and T-tests (one- and two-sample) for population means, and Chi-square and Fisher-F tests for population variances

Contingency Tables, Cross-tabs, Chi-Square Tests... [return to Table of Contents]

• Chi-Square tests...
  • 2-by-2 table analysis (Chi-square, Fisher Exact Test, sensitivity, odds ratio, relative risk, difference in proportions, number needed to treat, etc.) with confidence intervals
  • for 2-by-2 table, by Fisher Exact, and by Chi Square (with and without Yates' correction), with a good explanation
  • for 2-by-2 table
  • 2-by-2 table analysis (Chi Square, Fisher Exact, difference in proportions, risk ratio, odds ratio, theta, log-odds ratio, Poisson test)
  • for 2-by-N table, where the two rows represent dichotomies like lived/died, present/absent, yes/no. This can test for a trend in the probability of an event when you have counts of the two categories over a set of time intervals.
  • for table up to about 30 cells
  • Chi-square Test for Relationship -- for up to a 6-by-6 cross-tab.
  • for up to 10-by-10 tables. This page also has a section for comparing observed with explicitly-specified frequencies.
  • for any-size table
  • another for any-size table
  • another for any-size table (When you get to the Rweb page, scroll down to the Analysis Menu and select Two Way.)
  • Exhaustive analysis of 2-by-2 tables, with Pearson Chi-square, Likelyhood Ratio Chi-Square, Yates Chi-square, Mantel Haenszel Chi-square, Odds Ratio, Log Odds Ratio, Yules-Q, Yules-Y, Phi-square, Pearson correlation, and McNemar Test
  • Paired Proportion Test -- for testing whether the proportion of subjects having some characteristic is the same in two matched groups or in one group before and after some intervention. (Also can test against a null hypothesis specifying some non-zero difference.)
  • Also see the Evidence-Based-Medicine (EBM) calculator in the "Biostatistical Calculators" section of the "Other Statistical Tests and Analyses" section of this page.
• Three-dimensional Tables (2x2x2)...
  • Three-dimensional 2x2x2 table
  • Log-Linear Analysis for a 2x2x2 Table of Cross-Categorized Frequency Data [Calculates the values of G2 for first- and second-order interaction effects for a table of observed frequency data cross-classified according to three categorical variables, A, B, and C, each of which has two levels or subcategories (a1, a2; b1, b2; c1, c2)]
• Fisher Exact tests for contingency tables...
  • Fisher exact (2x2)
  • Fisher exact (2x2)
  • Fisher exact (2x2)
  • Fisher Exact, with good Help discussion
  • Fisher Exact (2x5)
• Exact unconditional homogeneity/independence tests for 2-by-2 tables
  (said to be more powerful than the Fisher exact test!)
• Test differences between two observed proportions, based on the Binomial distribution
• Contingency table for sequenced categories (Ordinal by Ordinal, 5-by-5 table or less)
• Contingency table for sequenced categories, 5-by-2 table, with exact probability calculations
• Spearman's correlation from cross-tabbed data with sequenced row and column categories
• McNemar's test to analyze a matched case-control study, with a good explanation
• McNemar's test for paired contingency tables
• Exact Bayes test for independence in r by c contingency tables -- Can also handle comparison of observed-vs-expected, and observed-vs-uniform situations.
• Comparison of ratings or rankings by different raters...
  • Friedman test for comparing rankings (Ordinal by Nominal)
  • Quantify agreement with kappa -- assesses how well two observers, or two methods, classify subjects into groups. For up to a 12-by-12 table.
  • Cohen's Kappa for comparing the way two raters scored each of a number of items, using case-by-case data entry
  • Another Cohen's Kappa, for case-by-case data
  • Another Cohen's Kappa, using already-tabulated data
  • Kappa for nominal data as concordance between multiple raters -- Each of several raters puts each of several entities into one of several categories
  • Intraclass correlation for concordance between multiple raters, using a data matrix that tells how each rater scored each case
• Chi-Square test for equality of distributions
• Chi-Square "Goodness of Fit" test for observed vs expected counts (NOT from Contingency Tables)...
  • Chi Square test -- takes observed values, and expected values that can be specified as expected occurrences, or percentages or fractions of the total. Data can be typed in or copied and pasted.
  • Chi-Square test
  • Chi-Square test
  • Chi-Square test (for up to 8 categories)
  • Chi-Square test for up to 10 categories. This page also has a section for up to a 10-by-10 contingency table.

Regression, Correlation, Least Squares Curve-fitting, Non-parametric Correlation... [return to Table of Contents]

• Straight Lines and Correlation Coefficients...
  • Least squares regression line and Pearson correlation coefficient.
  • Variations on straight-line fitting, when X and Y have error
  • Least squares regression. (nice interface)
  • Linear correlation and regression (nicely designed)
  • Simple Linear Regression -- for up to 84 points, with extensive output and residual analysis.
  • Correlation and regression calculator -- input two sets of numbers (or upload a file); computes the means, variances, covariance, correlation coefficient and regression coefficients; also gives a scatterplot with the two regression lines
  • The Data Applet provides descriptive statistics, histograms, boxplots, and scatterplots
  • Least squares straight line (also allows some simple transformations), with an interesting tutorial on the topic
  • Least squares straight line, also creates a high-quality Postscript graph of your data and the fitted line
  • Least squares straight line, allows several common types of y-value weighting (constant, proportional, or Poisson errors); also allows you to recall recently-entered data (for a limited time)
  • Scatter Diagram and Test for Outliers -- for up to 84 points.
  • Bivariate Sampling Statistics -- calculates means, variances, and covariance for up to 42 [x,y] measurements.
  • Calculate partial correlation coefficients r_bc.a, r_ac.b, r_ab.c from r_ab, r_ac, r_bc
  • WebStat (an integrated (Java) applet) can perform simple regression analysis
• Correlation Tests...
  • Spearman's rank correlation (non-parametric)...
    • Spearman
    • Spearman
    • Spearman (newer version here)
    • Spearman's correlation from cross-tabbed data with sequenced row and column categories
  • Correlation test
  • Significance level corresponding to a correlation coefficient
  • Testing the Correlation Coefficient -- enter up to 42 r values, along with a postulated population r value.
  • Minimum significant correlation coefficient for a given sample size
  • 95% Confidence Interval around an observed correlation coefficient.
  • Comparison of two correlation coefficients
  • Comparison of two or more correlation coefficients
  • Comparison of two sets of (X,Y) data to see if they are consistent with the same straight line (tests whether the slopes are different, and whether the lines are vertically distinct)
  • Comparing Two Linear Regressions -- Enter two sets of [x,y] values; page calculates two straight lines, then compares slopes and intercepts.
  • Test for Several Correlation Coefficients -- enter up to 14 sets of [N, r]; page will test whether all r's are consistent with a single population r value.
  • Biserial and point-biserial correlation analysis
  • Biserial correlation coefficient from summary data (N, mean, SD) of the X and Y variables
  • Point-biserial correlation analysis
  • Lin's "concordance correlation coefficient"  -- first proposed by Lin (1989) for assessment of concordance in continuous data. A breakthrough in assessing agreement between alternative methods for continuous data. Seems to avoid the shortcomings of correlation coefficient r, paired t-tests, least squares analysis for slope and intercept, coefficient of variation, intraclass correlation coefficient.. It is robust on as few as 10 pairs of data.
  • Manipulation of a correlation matrix -- you enter the N-by-N correlation matrix, the page computes all Partial Correlation Coefficients, all Standardized Partial Regression Coefficients, and the Multiple Correlation Coefficient for each variable.
  • A versatile page for calculating the significance of a correlation (rho<>0), significance of the difference between two correlations, power and sample size requirements for correlations testing, and the inter-relationships between three partial correlation coefficients.
  • Sobel's test to determine the extent to which an intermediate variable ("mediator") carries the influence of an independent variable (predictor) on a dependent variable (outcome).
• Beyond Simple 2-parameter Curve-fitting...
  • Very general nonlinear least-squares curve fitter -- almost any function you can write-- up to 8 nonlinear parameters, up to 10 independent variables.
  • Another non-linear least-squares curve fitter -- with graphical output! Choose one of 15 pre-defined nonlinear functions of one variable and up to three parameters.
  • Compare the fit of two models to your data. Which model fits better? Enter goodness-of-fit (SSQ, or weighted SSQ) and # of data points and # of parameters for each model. The calculator will compare the models using Akaike's method, , then the F test .
  • Linear, parabolic, or cubic fit, with graphics (newer version here)
  • Multivariate linear or univariate polynomial regression, with graphical output. Has a good discussion of the relevant mathematics and computational accuracy.
  • Fit "rational functions" (also called "Pade functions") to {X,Y} data.  A rational function is a fraction whose numerator and denominator are both polynomials in X. They can fit a broader range of functions than polynomials alone can -- they can fit data where the Y value "levels off" to a horizontal line for very large or small X, and can fit functions that have "singularities" (Y shoots to infinity at some value of x). This curve-fitter is part of an extensive set of online calculators to solve problems in structural engineering (bending and buckling of beams and plates, etc.) at the Software for Structures web site.
  • Univariate and multiple regression, with very extensive graphical output (histograms, scatterplots, scatterplot matrices) and residual analysis (QQ, histogram, residuals vs dependent or predictors). Very intuitive point-and-click interface, dynamically customized for your data. (When you get to the Rweb page, scroll down to the Analysis Menu and select Regression.)
  • Automatic Multiple Regression, (New Web Address!) automatically builds a model or regression equation! You merely supply the dependent and independent variables and it does the rest. It will find which variables are important enough to include in the model, determine the proper transformation of each of those variables, then look for 2-way and 3-way interaction terms important enough to include in the model, and transform them appropriately.
  • Multiple Linear Regression -- up to 16 data points and up to 4 independent variables; calculates fitted model, and a large number of residual analysis statistics.
  • Quadratic Regression -- Fits a least squares parabola to up to 84 data points, and provides extensive residual analysis.
  • Multiple regression, if you already have the correlation coefficient matrix between all independent and dependent variables...
    • for 2 independent variables
    • for 3 independent variables
    • for 4 independent variables
  • Fit any of five families of curves (linear, polynomial, exponential, descending exponential, Gaussian) and draw a graph
  • Curve fitting, smoothing, parameter estimating, data correlating and forecasting utility
  • Logistic Regression, if the dependent variable is restricted to two values (such as whether an event did or did not occur)
  • Regression and GLM Calculator -- performs linear, Poisson, binomial and Gamma regression, with canonical, identity, logit, log, probit, inverse, cloglog, and sqrt link functions
  • Cox Proportional Hazards Survival Regression Analysis
  • A faster version of Cox Proportional Hazards Analysis
  • Regression by Prevalence -- when you have data on the number of occurrences and non-occurrences of something over a set of time intervals. Tests whether the probability of the occurrence shows a trend over time.
  • Test Bias Assessment Program, computes statistics to help you decide if test scores predict a criterion differently across subgroups
• Time Series Analysis...
  • Autoregressive Time Series -- tools for the identification, estimation, and forecasting based on autoregressive order obtained from a time series.
  • Detecting Trend & Autocrrelation in Time Series -- Given a set of numbers, this page tests for trend by Sign Test, and for autocorrelation by Durbin-Watson test.
  • Plot of a Time Series -- generates a graph of a time series with up to 144 points.
  • Seasonal Index -- Calculates a set of seasonal index values from a set of values forming a time series. A related page performs a Test for Seasonality on the index values.
  • Forecasting by Smoothing -- Given a set of numbers forming a time series, this page estimates the next number, using Moving Avg & Exponential Smoothing, Weighted Moving Avg, and Double & Triple Exponential Smoothing.
  • Runs Test for Random Fluctuations -- in a time series.
  • Test for Stationary Time Series -- Given a set of numbers forming a time series, this page calculates the mean & variance of the first & second half, and calculates one-lag-apart & two-lag-apart autocorrelations. A related page: Time Series' Statistics calculates these statistics, and also the overall mean & variance, and the first & second partial autocorrelations.

Analysis of Survival Data... [return to Table of Contents]

• Life Table (Kaplan-Meier) -- Enter the number died and censored at each time period, and the page calculates the cumulative survival probability and 95% confidence intervals. Also graphs the survival curve, and exports the data, so you can create a better graph using another program.
• Cox Proportional Hazards Survival Regression Analysis -- specify each subject's observation time and status (last seen alive or dead), and any number of independent variables (predictors, confounders, and other covariates). This web page will perform a proportional-hazards regression analysis and return the regression coefficients, their standard errors, hazard (risk) ratio, and their confidence intervals, and the baseline survivor curve, along with goodness-of-fit information. You can also use a faster version by Ronald Brand (Leiden University), or an enhanced version by Kevin Sullivan (Emory University) that has illustrative examples and explanatory material.
• Comparison of Two Survival Distributions, using data from a data file in your computer (many different file types are supported). A graph is returned to your browser with the two survival curves plotted, along with the estimated relative risk, standard error and p-value.

Bayesian Methods... [return to Table of Contents]

• Bayesian Credibililty Analysis -- allows the credibility of a clinical trial finding to be assessed in the light of current knowledge. This page takes the odds ratio and its confidence interval from a clinical trial, and uses a newly-developed Bayesian method to calculate a quantity called the critical odds ratio (COR). If odds ratios at least as impressive as that indicated by the COR can be justified by existing knowledge, then the results of the clinical trial can be deemed credible.
• Etiologic Predictive Value (EPV) -- a new statistical method developed for determining the probability of symptoms being caused by a bacteriological finding, while taking carriers into consideration. To calculate EPV, one must know the number of positive and negative tests among patients and healthy controls as well as the sensitivity of the test. This enables calculating the positive and negative EPV with a 95% confidence interval.
• Exact Bayes test for independence in r by c contingency tables -- Can also handle comparison of observed-vs-expected, and observed-vs-uniform situations.
• Analysis of "1-degree of freedom" data -- performs interactive frequentist and Bayesian conditional tests for counts data having one degree of freedom. That is, it does hypergeometric, binomial, Poisson, Bessel, and related distributions (for double dichotomies, sign tests, a special kind of structural zero design, etc.).
• Bayes' theorem calculations -- takes prior probabilities and conditional probabilities, and calculates revised probabilities. (great for solving certain kinds of brain teaser puzzles)
• Interpret P values -- Compute post test probability to take into account the context of the experiment, as expressed by the prior probability that your hypothesis is true.
• Bayesian calculations for diagnostic tests -- computes interrelationships among true pos, true neg, false pos, false neg, prevalence, sensitivity, specificity, predictive values, and likelihood ratios.
• Calculate the post-test probability of an outcome (disease) from prior probability (prevalence) of the disease, and from the sensitivity and specificity of the test
• Sequential Experimental Design for testing the probability ratios
• 2-by-2 table analysis (Chi-Square, sensitivity, odds ratio, relative risk, etc. with confidence intervals
• Wald's Sequential Probability Ratio's -- for designing a sequential experiment in which a decision is made after each observation either to accept the null hypothesis, accept the alternate hypothesis, or acquire more observations.

Other Statistical Tests and Analyses... [return to Table of Contents]

• Universal Inventory/Test Scorer will instantly and automatically score ANY objective  test or personality inventory/questionnaire. For any particular questionnaire, you create a text file that describes the scores associated with each possible answer to each question (True/False, A/B/C/D/E, Likert Scale, etc.). It is available as a Java implementation and as JavaScript implementation. These will run online, or can be downloaded to be run locally on your computer (offline from the Internet). 
• Bonferroni adjustment of critical p-values when performing multiple comparisons (has an excellent discussion of this topic)
• Multiple comparisons correction (Bonferroni adjustment)
• Number Needed to Treat, based on a 2-by-2 table
• Detect Outliers -- this calculator performs Grubbs' test, also called the ESD method (extreme studentized deviate), to determine whether one of the values in the list you enter is a signficant outlier from the rest.
• Selection Bias Calculator for Prevalence Estimates
• Calculate and plot an ROC Curve (for grouped predictor data)
• Clustering Calculator generates tree structures of data clustering, and much more
• Misclassification Bias in Prevalence Studies
• Predictive Value from Sensitivity, Specificity and Prevalence, (when analyzing a clinical test), with a nice explanation
• Selection Bias in Case-control Studies
• NetMul: a browser interface to a program that performs:
  • Principal Coordinate Analysis (PCO)
  • co-inertia analysis
  • discriminant analysis and within- or between-class analyses
  • analyses on distance matrices or neighboring graphs.
• A Web-Based SAS Code Developer for Experimental Designs
• Simultaneous Equations and Matrix Inversion -- up to 10 equations (or 10x10 matrix).
• Linear Optimization with Tools for Sensitivity Regions -- This page finds the optimal solution, and does a post-optimality analysis of small-size linear programming problems (constrained optimization).

Specialized and Discipline-Specific Tests and Analyses... [return to Table of Contents]

• Martindale's Reference Desk -  Calculators On-Line - Statistics (the grand-daddy of all compendia of calculating web pages)
• Biostatistical Calculators:
  • All-purpose Four-fold Table Calculator: for Cohort or Case-control studies. Calculates Rx parameters: CER, EER, ARR, RRR, NNT; Dx parameters: Sensitivity, Specificity, LR:, LR-, Prevalence; Hyopthesis-testing parameters: RR, OR, NNH, Chi Square.
  • Evidence-Based Medicine (EBM) calculator -- From Warren Goff's interestingly-named web site. Analyzes one or more fourfold (2x2) tables; calculates Chi Square, CER, EER, and RR, and parameters related to treatment (RRR, ARR, NNT, NNH, with 95% confidence intervals), diagnosis (Sensitivity, Specificity, PPV, NPV, Prevalence, LR+, LR-, OR, Pre-Odds, Post-Prob), and Harm (RR, OR NNH). Can also compare two different tables.
  • Number Needed to Treat, also Normal, Student t, Chi-Square, Binomial, and Random Digits
  • Clinical Significance Calculator -- For two groups (control and treatment), enter the group size and incidence rate; the page will calculate absolute and relative risk reductions, odds ratio, and number needed to treat, along with 95% confidence intervals for each result
  • Compute EC_anything from EC₅₀ (assuming a standard "Hill-type" dose-response relationship). Very useful in dose-response studies.
  • Thorough analysis of 2-by-2 table relevant to Predictions and Diagnostic Tests -- sensitivity, specificity, prevalence, diagnostic accuracy, PPV, post-test probabilities, likelihood ratio tests
  • Calculation of posttest probability from Likelihood Ratio and pretest probability
  • Conversion of Sensitivity and Specificity to Likelihood Ratios
  • Calculator to predict the probability of a successful outcome to lumbar disc surgery (based on a logistic model)
  • LODS - Logistic Organ Dysfunction System calculator
  • Scoring systems for ICU and surgical patients -- Online calculation of scores used in general or specialized Intensive Care or Anesthesia, including:
    • Adult, General scores: SAPS II, APACHE II, SOFA, MODS , ODIN, MPM (on admission , 24 hrs, 48 hrs , MPM Over Time) , MPM II (on admission, 24-48-72 hrs) , LODS, and TRIOS
    • Adult, Specialized and Surgical Intensive Care - Preoperative evaluation: EUROSCORE, ONTARIO, Parsonnet, System 97, QMMI, MPM, POSSUM, and Portsmouth POSSUM
    • Adult, Trauma scores: ISS/RTS/TRISS, and 24 h - ICU Trauma Score
    • Adult, Therapeutic intervention, nursing ICU scores: TISS
    • Pediatric, General scores: PRISM, DORA, PELOD, and PIM
    • Pediatric, Specialized (Neonatal, Surgical): CRIB, SNAP, SNAP-PE, SNAP II / SNAPPE II
    • Pediatric, Trauma Scores: Pediatric Trauma Score
  • Calculators for Clinical Formulas -- A-a Gradient, Anion Gap, Body Surface Area, Body Mass Index, Estimated Creatinine Clearance, Fractional Excretion of Sodium, Heart Disease Risk, Ingested Substance Blood Level, Pregnancy Due Date , Serum Osmolality , and Weights and Measures (converts lbs. to kgs. and F to C)
• Disparate Impact Analysis
• Item Analysis -- for multiple choice questionnaires
• Statistical Quality Control (SQC) Online -- Online calculators and tutorials to perform SQC annd Statistical Process Control (SPC). Contains:
• • Online versions of Military & Civilian Standard Tables: MS-105E / ANSI/ASQC Z1.4, ISO 2859 (sampling plans for attribute data), MS-414 / ANSI/ASQC Z1.9 (sampling plans for measurement data, and MS-1235C (sampling inspection plans for continuous production, Procedure CSP-1).
  • Online Calculators for Process Capability Index (Cp), MTBF Calculator for a system given the part (component) failure rate, and Control Charts and Runs Rules (Switching Rules for MS-105E, Continuous Sampling CSP-1, Western Electric Rules, and System Reliability for consecutive-type systems)
• Queuing Theory Calculator -- a remarkably powerful web calculator that can solve a wide variety of queueing problems: single-server, multiple-server, infinite-server, infinite or finite waiting room, Erlang loss model, and machine interference model (with or without spare machines). Provides detailed output in the form of averages, standard deviations, and frequency distributions in the form of tables and graphs.
• Single-Case analysis tools -- an online calculator that can do a number of tests and analyses that are especially useful in "single-case" or "single-system" research: Time Series (handles A-B and multiple-baseline designs, and calculates correlations and the C-statistic, with p-value), Autocorrelation, Chi Square, Testing Significance of Difference: t-test and Mann-Whitney U, Binomial Expansion, and Bayesian Analysis. Also contains a good overview of single-case methods.
• Decision Making in Economics and Finance:
• • ABC Inventory Classification -- an analysis of a range of items, such as finished products or customers into three "importance" categories: A, B, and C as a basis for a control scheme. This pageconstructs an empirical cumulative distribution function (ECDF) as a measuring tool and decision procedure for the ABC inventory classification.
  • Inventory Control Models -- Given the costs of holding stock, placing an order, and running short of stock, this page optimizes decision parameters (order point, order quantity, etc.) using four models: Classical, Shortages Permitted , Production & Consumption, Production & Consumption with Shortages.
  • Optimal Age for Replacement -- Given yearly figures for resale value and running costs, this page calculates the replacement optimal age and average cost.
  • Single-period Inventory Analysis -- computes the optimal inventory level over a single cycle, from up-to-28 pairs of (number of possible item to sell, and their associated non-zero probabilities), together with the "not sold unit batch cost", and the "net profit of a batch sold".
  • Theoretical Expectancy Calculator -- estimates amount of workforce improvement from implementing a valid selection procedure in an organization. Computes institutional expectancies under three different models.
  • Investment Derivative Calculations -- A very elaborate online calculator and real-time data retrieval system. Includes economic regression analysis.
  • Black-Scholes Calculator -- to place a value on stock options.
  • Bardahl Calculator -- to compute the reasonable working capital needs of a corporation.
• Probabilistic Modeling:
  • Bayes' Revised Probability -- computes the posterior probabilities to "sharpen" your uncertainties by incorporating an expert judgement's reliability matrix with your prior probability vector. Can accommodate up to nine states of nature.
  • Decision Making Under Uncertainty -- Enter up-to-6x6 payoff matrix of decision alternatives (choices) by states of nature, along with a coefficient of optimism; the page will calculate Action & Payoff for Pessimism, Optimism, Middle-of-the-Road, Minimize Regret, and Insufficient Reason.
  • Determination of Utility Function -- Takes two monetary values and their known utility, and calculates the utility of another amount, under two different strategies: certain & uncertain.
  • Making Risky Decisions -- Enter up-to-6x6 payoff matrix of decision alternatives (choices) by states of nature, along with subjective estimates of occurrence probability for each states of nature; the page will calculate action & payoff (expected, and for most likely event), min expected regret , return of perfect information, value of perfect information, and efficiency.
  • Multinomial Distributions -- for up to 36 probabilities and associated outcomes, calculates expected value, variance, SD, and CV.
  • Revising the Mean and the Variance -- to combine subjectivity and evidence-based estimates. Takes up to 14 pairs of means and variances; calculates combined estimates of mean, variance, and CV.
  • Subjective Assessment of Estimates -- (relative precision as a measuring tool for inaccuracy assessment among estimates), tests the claim that at least one estimate is away from the parameter by more than r times (i.e., a relative precision), where r is a subjective positive number less than one. Takes up-to-10 sample estimates, and a subjective relative precision (r<1); the page indicates whether at least one measurement is unacceptable.
  • Subjectivity in Hypothesis Testing -- Takes the profit/loss measure of various correct or incorrect conclusions regarding the hypothesis, along with probabilities of Type I and II errors (alpha & beta), total sampling cost, and subjective estimate of probability that null hypothesis is true; returns the expected net profit.

Power, Sample Size and Experimental Design Calculations... [return to Table of Contents]

Check out the large number of power and sample size calculators at the UCLA Statistics website. Many of them are included below.

Also, check out the very general and elegant power/sample-size calculator by Russel Lenth (U of Iowa). It handles tests of means (one or two samples), tests of proportions (one or two samples), linear regression, generic chi-square and Poisson tests, and an amazing variety of ANOVAs -- 1-, 2-, and 3-way; randomized complete-block; Latin and Greco-Latin squares; 1-stage, 2-stage, and factorial nested designs; crossover; split-plot; strip-plot; and more! This calculator is implemented in Java, and can be run as a web page, or can be downloaded to your computer to run offline as a stand-alone application.

Here's a collection of online power calculator web pages for specific kinds of tests:

• For one-group tests (comparing the sample to a specified value) or for paired two-group tests...
  • Find the required sample size or the statistical power when comparing the mean of a sample to a specific value.
  • Power/Sample-size for One-sample or Paired t test -- select the One-sample t test (or paired t) option, then click the Run Selection button.
  • Confidence Interval around a mean -- select the CI for one mean option, then click the Run Selection button.
  • Comparing a mean to a specified value
  • Power or sample size for one-group comparison, when you can assume that the values come from a normal distribution.
  • Comparing a proportion to a specified value
  • Comparing a proportion to a specified value
  • Power/Sample size to compare a proportion to a specific value -- select the Test of one proportion option, then click the Run Selection button.
  • Sample-size for Conf Interval around a proportion -- select the CI for one proportion option, then click the Run Selection button.
  • Comparing on observed number of occurrences to a specified rate of occurrence
  • Sample size and 95% confidence interval for a variable, knowing the population standard deviation
• For designing surveys (sample size and confidence intervals for proportions, based on sample size, with or without corrections for finite populations:
  • Calculates sample size for given population size, confidence interval (margin of error), confidence level, and population proportion. Also displays margin of error for three other specified sample sizes (your choice), and sample sizes for three other specified confidence levels.
  • Compute the sample size, given the population size, the confidence interval and the confidence level. A related "advanced" page also allows you to specify the postulated proportion (rather than assuming 50%).
  • Find the required sample size or statistical power for comparing an observed proportion with a specific value
  • Confidence Interval Calculator to compute the margin of error and confidence interval given the population size, sample size, sample percentage, and confidence level. An animated version is also available, which displays the calculation of the confidence interval graphically.
  • Find the sampling error in an observed proportion
  • Calculate sample size required for a given confidence interval, or confidence interval for a given sample size. Can handle finite populations. Downloadable program also available.
  • Another sample-size / confidence interval calculator for proportions in finite samples
  • Power vs sample size for survey questionnaire results, with graphical output
  • Sample size or confidence interval of a proportion
• For two-group tests...
  • Comparing means for two independent samples...
    • Find the required sample size, or the statistical power when comparing the means of two samples (can have different standard deviations)
    • Same-size samples (also available at another website)
    • Same-size samples; equal or unequal variances
    • Power or sample size for equal-variance two-group tests, when the values come from a normal distribution
    • Power or sample size for unequal-variance two-group tests, when the values come from a normal distribution
    • Power/Sample-size for equal-variance or unequal-variance t tests -- select the Two-sample t test (pooled or Satterthwaite) option, then click the Run Selection button.
  • Difference between two proportions (as, for example, by a Chi Square test on a 2-by-2 cross-tab)...
    • Equal sample sizes (easy to use; with a good explanation)
    • Equal- or unequal-size samples
    • Find the required sample size or statistical power for comparing two observed proportions
    • Power/Sample-size for comparing two proportions -- select the Test comparing two proportions option, then click the Run Selection button.
    • T independent samples (also available at another website)
  • Difference between two rates of occurrence of events
  • Survival outcome analysis -- computes power or sample size in a two group design with a survival outcome; allows for unequal sample sizes, loss to follow-up rates for each group, cross overs, and specifying the patient entry distribution into the study
• For ANOVAs and other multi-group comparisons...
  • Sample size needed for comparison of 2 or more groups, knowing the SD's with a group, and the expected difference between groups
  • Sample size computation for multiple comparisons -- said to be more "realistic" than ordinary ANOVA sample size calculations (explained on the web page)
  • Very general power calculator -- select the Balanced ANOVA (any model) option, then click the Run Selection button.
  • Two-way ANOVA power calculation
  • Simplified power analysis for multi-way ANOVA designs
  • Power/Sample-size for Chi-square tests of tables larger than 2x2 -- select the Generic chi-square test option, then click the Run Selection button.
• For regressions and correlation tests...
  • Power or sample size for comparing an observed correlation coefficient with a specified value
  • A versatile page for calculating the significance of a correlation (rho<>0), significance of the difference between two correlations, power and sample size requirements for correlations testing, and the inter-relationships between three partial correlation coefficients
  • Sample-size for multiple regression -- will tell you the minimum required sample size for your study, given the alpha level, the number of predictors, the anticipated effect size (as f²), and the desired statistical power level. If you know the effect size as R², you can calculate f² from R² with this calculator.
  • Power/Sample-size for simple or multiple linear regression -- select the Linear regression option, then click the Run Selection button.
  • Beta level  for multiple regression (i.e., the Type II error rate, which is 1 minusPower), given the observed alpha level, the number of predictors, the observed R², and the sample size.
  • Post-hoc power for multiple regression -- calculates the observed power for your study, given the observed alpha level, the number of predictors, the observed R², and the sample size.
  • Power calculations for logistic regression with a continuous exposure variable and an additional continuous covariate or confounding variable. Also accommodates measurement error in the exposure variable. Has graphical output.
• Other power calculations...
  • Retrospective power analysis (after doing the test)
  • Sample Size Determination -- For several situations: ANOVA and 2-population economic sampling, correlation with acceptable absolute precision, estimating the mean or proportion with acceptable absolute or relative Precision, estimating the mean or proportion from finite populations, and testing the mean or proportion based on the Null and an Alternative.
  • Power calculations for clinical trials and scientific experiments
  • Survival Analysis -- computes power, sample size, or detectable-effect size in a two group design with a survival outcome.
  • Generic Poisson Test -- select the Generic Poisson test option, then click the Run Selection button.
  • Exact power for the Fisher exact test
  • A large, well-organized collection of of power and sample size calculators, containing many of the above links
  • Find sample size, power and minimal detectable difference for a:
    • study on the effect of one variable on another
    • cross-over study were the outcome is a measurement
    • parallel trial where the outcome is a measurement
• Links to printable copies (in Adobe Acrobat PDF format) of many power tables including: Z Test, t-Test for One Sample or Two Related Samples, t-Test for Two Independent Samples, Analysis of Variance, Correlation Coefficient, and Required Sample Size for various tests
• Wald's Sequential Probability Ratio's -- for designing a sequential experiment in which a decision is made after each observation either to accept the null hypothesis, accept the alternate hypothesis, or acquire more observations.
• Experimental Design...
  • WebDOE^(tm) -- for "design of experiments". Searches for I-, D- and A-optimal designs over continuous spaces. Factors may be continuous, fixed-level, or qualitative. The site can handle inequality and equality (e.g., mixture) constraints; provides color plots; performs one-click, run-order-randomization; allows design import/export interoperable with most 3rd-party analysis software; provides OLS and BLUP fits; and includes an extensive Classical Design Library(tm), including factorial, fractional-factorial, Box-Behnken, central-composite, Plackett-Burman, orthogonal array, and uniform designs. All designs may be evaluated under the I-, D-, and A-, and S-optimality criteria, as well as for the maximum distance between nearest-neighbor pairs of design points (maximin criterion). The My WebDOE(tm) feature allows users to store their designs, evaluations, and fits on-line. There is no need for user-provided candidatepoints.
  • Design and Analysis of Comparative Experiments website by Horticulture Research International -- provides facilities for the design and analysis of of comparative experiments for biological and agicultural research based on a range of experimental block and treatment structures. Constructs simple experimental designs interactively and also constructs appropriate statistical software for the analysis of the designs. Handles Randomised block, Split-plot, Latin and incomplete Latin square, Trojan and incomplete Trojan square designs.
  • Tables of Latin Squares for constructing "Williams design" experiments, in which every subject receives every treatment. These designs are balanced for first-order carry-over (residual effects). Tables are provided for experiments ranging from 2 to 26 treatments. Tables can also be downloaded as a text file and as an Excel spreadsheet.
  • EDGAR -- generates experimental designs and randomizes the position of experimental treatments in the design, so that the subsequent analysis of the data is comparatively straightforward
  • Gehan/Simon Two-Stage Designs approximating the power and significance level specified in the input.
  • Find Optimal/MiniMax Phase-II 2-stage designs, where H0: p=p0 and H1: p=p1>p0, subject to a fixed maximum sample size, N. Finds all the designs that satisfy Type I & II error criteria. [see Simon, Controlled Clin Trials, 10:1-10,1989]
  • Compute boundaries for a specified alpha spending function,
    compute drift given power and bound, and
    compute probabilities,
    all based upon the Lan-DeMets method. Allows computation of boundaries at any time during the monitoring of a study. It is valid for any normal test statistic with independent increments. The information time is the ratio of accrued sample size to the total sample size for normal data.

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