(Updated 12/20/2015 -- Updated link to Soren Merser's Survival Analysis page, and added a link to his new Cox Regression page,)

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The web pages listed below comprise a powerful, conveniently-accessible, multi-platform statistical software package. There are also links to online statistics books, tutorials, downloadable software, and related resources.

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.

- 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

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:

- 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.
*Choosing a Statistical Test*, Chapter 37 of Dr. Harvey Motulsky's book*Intuitive Biostatistics*.- "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.

As you can see from looking at the StatPages.org 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:

- Statigraphics Stratus-- a browser-based version of Stratigraphics statistical software. Provides plotting, probability distributions, summary statistics, one-, two-, and multiple-sample analysis, time-series analysis, regression analysis, curve-fitting, attribute estimates, ANOVAs, Statistical Process Control, smoothing, power/sample-size calculations, and other statistical analyses.Also provides access to over 50 applets in which you can enter data, compute statistics, create tables and graphs, and print out the results. The free "Guest" version supports up to 100 rows and 10 columns of data. For larger data sets, a single-user copy or a corporate deployment license can be purchased. To use the Guest version, click here.
- OpenEpi Version 2.2 -- OpenEpi is a free, web-based, open source, operating-system-independent series of programs for use in public health and medicine, providing a number of epidemiologic and statistical tools. Version 2 (4/25/2007) has a new interface that presents results without using pop-up windows, and has better installation methods so that it can be run without an internet connection. Version 2.2 (2007/11/09) lets users run the software in English, French, Spanish, or Italian.
- SOCR -- Statistics Online Computational Resource. A very comprehensive collection of online calculators and other interactive resources, including: Distributions (interactive graphs and calculators), Experiments (virtual computer-generated analogs of popular games and processes), Analyses (collection of common web-accessible tools for statistical data analysis), Games (interfaces and simulations to real-life processes), Modeler (tools for distribution, polynomial and spectral model-fitting and simulation), Graphs, Plots and Charts (comprehensive web-based tools for exploratory data analysis), Additional Tools (other statistical tools and resources), SOCR Wiki (collaborative Wiki resource), Educational Materials and Hands-on Activities (varieties of SOCR educational materials), SOCR Statistical Consulting and Statistical Computing Libraries.
- SciStatCalc -- a very good web site (thank you Alijah Ahmed!), with online calculators for many of the common statistical probability functions and significance tests, and pages that explain the concepts and formulas behind the tests. Calculating pages provide flexible input options (type the numbers in, or read them from a text fiile), and well--organized output of results, with interpretations and graphical displays.
- MedCalc -- Two different web sites:
- MedCalc.org -- a collection of seven free calculating web pages: tests for one mean or one proportion, comparisons of means or proportions, relative risk, odds ratio, and diagnostic test evaluation.
- MedCalc.net -- a pay-for browser-based statistical system (5-day free trial available) providing a wide range of statistical analyses (click here for list).

- ProtoGenie -- a free extensible web-based environment for research design and data collection for surveys, experiments, clinical trials, time series, cognitive and vision research, and methods courses. Lets you specify groups and define measurement and treatment events and their sequencing. The goal is to let users move smoothly from research design and data collection to interim and final statistical analysis.
- 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.
- SISA (Simple Interactive Statistical Analysis) -- SISA allows you to do statistical analysis directly on the Internet. Click on one of the procedure names below, fill in the form, click the button, and the analysis will take place on the spot. Study the user friendly guides to statistical procedures to see what procedure is appropriate for your problem.
- 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.
- AppOnFly has a different approach -- the company maintains Windows computers with a wide selection of Windows software, including utilities for accessing PDF, ZIP, graphics and other files, web browsers, various office suites (MS Office, Libre Office, Apache OpenOffice, WordPerfect Office), and statistical packages (SPSS, NCSS, R), to mention a few. They then provide apps for accessing this Windows desktop environment from any device -- computer, smartphone, tablet (Windows, Mac OS, iOS, Android). This is a pay-for service, but they offer one-month free trials.

- 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
- 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 -- Produces a small 3D plot of 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

- Indefinite
Integrals -- using the
- 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)
- 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 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:- Xuru's page to calculate PDFs, CDFs, and their complementary and inverse, along with expected values, mean, variance, skewness & kurtosis, for binomial, geometric, negative binomial, Poisson, hypergeometric, normal, chi-square, Student t and Fisher F distributions.
- Distribution/density calculators, plotters and random number generators
- Very sophisticated interactive page for over 65 continuous and discreet statistical distribution functions -- Select a function from a menu, and it will show you a graph of that function along with its properties. You can click on the graph to select limits, and it will show you the left, middle and right integrals.

**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**.) - 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

- Normal, t, Chi-Square, and Binomial (density and cumulative)
probabilities; (When you get to the
**These pages each compute probabilities and/or inverses for a specific distributions**:- Normal distribution areas, with nice graphical interpretations
- Another very good Normal Distribution calculator, with nice
graphics

- 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 Distribution
- Chi-Square Distribution
- Student t Distribution
- Student t Distribution and its inverse (t value from p value)
- 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

- 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...

- 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
- 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:
- generates M groups of N numbers each by distributing the numbers from 1 to M*N randomly into the M groups
- generates M blocks of N numbers each by randomly shuffling the numbers from 1 to N in each block
- 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

- 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
- Xuru's page for single variable descriptive statistics: mean, median, sd, variance, mean abs deviation, geometric mean & sd, skewness, kurtosis, quartiles, standard errors, Anderson-Darling normality test, and some confidence intervals of the mean and sd. You can copy and paste data directly from a spreadsheet or a tabulated data file, or enter numbers manually.
- 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.
- 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
- Determination and Removal of Outliers -- Given a set of numbers, this page iteratively isolates potential outliers for removal.
- 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

- 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, and as an Excel Add-In.)
- Exact and "modified Wald" C.I.'s for observed proportion or count, with a good explanation
- Bayesian "credible" intervals around an observed proportion. Somewhat comparable to the "classic" confidence intervals, but tend to be slightly narrower.
- 95% or 99% C.I. for proportions for any specified sample size and population size
- 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 .**

- Tolerance Intervals for the Normal
Distribution. (Don't confuse
- Single-Population Tests...
- Sign and Binomial test -- test an observed proportion against a proposed population proportion
- 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.
- Another Student t-test of a single mean (vs specified value) 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.
- Shapiro-Wilk Test for Normality -- enter numbers into page, or read them from a text file. Performs normality test, also shows a histogram of the data. For a description of the test, along with the formulas and programming, click here.
- 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.

- 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.
- a very polished calculator for two-group Student t test, with graphical display of means and confidence intervals, and an interpretation of the results. Can take individual values or summary statistics (N, mean, SD) for each group.
- t-test, paired or unpaired
- t-test, paired or unpaired
- t-test, paired or unpaired
- t-test, paired
- Paired Student t test -- enter data into the page, or read it from a text file. This page also produces histograms of the data (each group, and paired differences). For a detailed description of the test, with formulas and examples, click here.
- 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.
- 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) -- Now also does Tukey HSD post-hoc test!
- Another 1-way ANOVA from summary data
- One-way ANOVA -- Also produces a post-hoc analysis (which groups are different from which others), and a scatterplot of all groups. For a description of the ANOVA, click here.
- 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 ANOVA -- enter data into the web page, or read it from a text file. For an explanation, click here.
- 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].
- Bartlett's test for equality/homogeneity of variances for three or more groups. Also produces a scatter plot of all the groups. For a description of the test, along with the formulas and programming, click here.
- 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

- One-Way and Factorial ANOVA for uncorrelated
samples (extension of
- 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 -- a non-parametric substitute for the paired Student t test when the data is not normally distributed. This page also produces histograms of the data (each group, and paired differences). For a detailed description of the test, with formulas and examples, click here.
- 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.
- Mann-Whitney U test -- a non-parametric substitute for the unpaired Student t test when the data is not normally distributed. This page also produces a dot-plot and a histogram of the data for each group. For a detailed description of the test, with formulas and examples, click here.
- 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.
- Two-sample Kolmogorov-Smirnov Test -- Enter numbers into the web page, or read them in from text files. Also graphs the cumulative distribution of the two samples.
- 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.
- Kruskal-Wallis test -- This page also produces a scatterplot of ranks for all groups.
- 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
- Comparison of two proportions between two groups (each given as # successes / # of trials). Shows confidence intervals, and interprets the results of the comparison.
- Paired Preferences Test -- Enter the sample size, and the two percentages (preferring A and preferring B), and this program will calculate the T score and significance level. This page is based on a normal approximation to the binomial distribution, and should not be used if the sample size is less than 30.

- Comparison of Event Rates between two groups...
- a very polished calculator for comparing two event rates (number of events in a certain
amount of time). Shows confidence intervals around each event rate, and
interprets the significance of the difference between the rates of the two
groups.

- a very polished calculator for comparing two event rates (number of events in a certain
amount of time). Shows confidence intervals around each event rate, and
interprets the significance of the difference between the rates of the two
groups.
- 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

- 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**. Also see Andrew Mackinnon's DAG_Stat -- an Excel spreadsheet that contains even more quantities (with confidence intervals) that can be derived from a 2x2 table). - EpiMax Table Calculator -- similar to the above, but with a clearer screen layout.
- 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)
- Diagnostic Test Evaluation -- from a 2x2 cross-tab of diagnostic test results (positive or negative) vs. true disease state (present or absent), calculates sensitivity, specificity, positive and negative likelihood ratios and predictive values, and disease prevalence, along with their 95% confidence intervals.
- 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.
- Chi-square Test for Relationship -- for up to a 6-by-6 cross-tab.
- 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.

- 2-by-2 table analysis
(Chi-square, Fisher Exact Test, sensitivity, odds ratio, relative risk,
difference in proportions, number needed to treat, etc.)
- 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)
- Fisher Exact (2x2)

- Test differences between two observed proportions, based on the Binomial distribution
- Barnards Test (2x2) -- An exact test for 2x2 tables that is exact (like the Fisher test), but can be more powerful than the Fisher test (more likely to produce significance). For an explanation, click here.
- 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
- McNemar's test for 2x2 paired tables -- For a background explanation, with formulas and examples, click here.
- Cochrane's Q Test -- An extension of the McNemar test to 2xN tables. For an explanation, click here.
- 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.
- Online Kappa Calculator -- calculates free-marginal and fixed-marginal variations of birater and multirater Kappas (chance-adjusted measures of interrater agreement).
- 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, 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)

- Straight Lines and Correlation Coefficients...

- Least squares regression. (nice interface)
- Linear correlation and regression (nicely designed)
- Linear regression to data copy/pasted from a spreadsheet or tabular file.
- Linear Regression -- enter X and Y into the web page, or read them in from a text file. Produces regression coefficients, coefficient of determination, and other quantities, along with a graph of the observed data points and fitted line. For a description of the concepts of linear regression, click here.
- Several variations on 2-parameter linear regression (logarithmic regression, exponential regression, and power regression)
- Simple Linear Regression -- for up to 84 points, with extensive output and residual analysis.
- The Data Applet provides descriptive statistics, histograms, boxplots, and scatterplots
- 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)...
- Correlation test
- Pearson Correlation Coefficient -- also produces a scatterplot of the data. For a description of correlation coefficients, click here.
- Spearman Rank Correlation Coefficient -- a non-parametric substitute for the Pearson correlation coefficient. This page also produces a scatterplot of the data. For a description of correlation coefficients, click here.
- 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 correlation coefficient from summary data (N, mean, SD) of the X and Y variables
- 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.

- 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.
- MyCurveFit.com -- an easy-to-use curve-fitting page. Offers 13 pre-defined functions (no initial guesses required), along with the ability to fit a general non-linear function you provide (along with initial guesses). Displays the results graphically, along with the formula of the fitted curve. Several types of unequal data-point weighting are provided. Lets you generate predicted values (interpolated and extrapolated) from the fitted curve. Lets you save results in Excel and PDF formats.
- ZunZun non-linear least-squares curve-fitter -- with an enormous list of pre-defined 2-D and 3-D functions, and extensive graphical and statistical output.
- 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.
- 3-D Regression and Interactive Graph (by MiaBella LLC) -- a powerful web page that fits a linear function of two predictor variables (Z = a + b*X + c*Y), and displays a very elegant 3-D scatterchart of the {X,Y,Z} points and the fitted plane. You can rotate the graph in three dimensions using the mouse, and you can see the X, Y, and Z values of any point (say, an outlier) by clicking on the point.
- Polynomial Regression -- fit polynomials of degree 2 through 10.
- Multiple Linear Regression -- fit functions of more than one predictor variable.
- Multiple Polynomial Regression -- fit functions of one or more predictors, each expressed as polynomials, up to the order you specify.
- Nonlinear Regression -- Automatically fits over 100 of the most commonly-occurring non-linear functions (gaussians, sigmoidals, rationals, sinusoidals, etc.), and then ranks them according to goodness-of-fit.
- 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.
- 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, 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...
- Fit any of five families of curves (linear, polynomial, exponential, descending exponential, Gaussian) and draw a graph
- Logistic Regression, if the dependent variable is restricted to two values (such as whether an event did or did not occur)
- 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.

- Kaplan-Meier Survival Plot and LogRank Test -- Type or copy/paste data, or read it in from a file. Prepares tables, graphs (with 95% confidence intervals), and statistical comparison output. Can accommodate two or more groups, and can perform stratified log-rank test. Uses the R statistical engine on the host server to provide very high-quality output.
- Kaplan-Meier Survival Plot -- for one or more groups. Draws K-M curves with optional confidence bands (ordinary, log, or log-log type, at the 50, 80, 90, or 95% conf. level). This is part of Peter Rosenmai's EurekaStatistics web site (a blog about statistics and R).
- Kaplan-Meier Survival Plot and LogRank -- calculates survival curves (with confidence bands), and performs a LogRank test test to comparing survival curves between two groups.
- 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.
- Cox Proportional Hazards Regression -- you can copy/paste data from Excel, or upload a CSV file. Produces a regression table report, survival plot, survival table, log-rank test, and a predicted survival plot for specified covariable patterns. Uses the R statistical engine on the host server to provide very high-quality output.
- 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.
- Compare Average Survival Time between two distributions -- Enter the number of events and the average time to event for each of two groups. The calculator will display the confidence interval around each mean time, and will compare the two mean times. (Assumes an exponential-shaped survival curve.)

- 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.
- 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.

- ReliCheck -- an online reliability analysis tool that allows users to check the reliability of the scores on their survey. The free option provides reliability score, statistical strength of survey, general item analysis, and a statistical summary of the survey. Pay-for plans also provide an auto-optimizer, optimization comparison, manual optimizer, and control of survey analysis.
- Queueing Theory Calculator -- Performs classic calculations
for single-server or multi-server queues (queue length, waiting time,
etc.).

- 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).
- Interactive Cross-Validation -- Performs the "leave-one-out" cross-validation inference for: central tendency, least-squares lines, one-dimensional multinomial tables, two-dimensional contingency tables with structural zeroes, k-sample problems, and block-and-treatment designs. The web page is well-documented, with about a dozen examples worked out and explained.
- Fittestmodel --an
online forum, on which statistical evidence can be presented that is always
replicable, testable and extendible at the 'click of a button'. The name
*Fittestmodel*encompasses both the goal and the means of science, namely to find the fittestmodel by fitting, testing and modelling. Users may discuss statistical evidence online or query for results based on search criteria such as dataseries, methods or criteria that measure the 'quality' of results. Publicly available datasets from various sources may be combined into new statistical evidence and statistical techniques will be added on a continuous basis, by user request or otherwise. - 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.

- 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).

- Martindale's Reference Desk - Calculators On-Line - Statistics (the grand-daddy of all compendia of calculating web pages)
- Biostatistical Calculators:
- 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.
- Diagnostic Test Evaluation -- from a 2x2 cross-tab of diagnostic test results (positive or negative) vs. true disease state (present or absent), calculates sensitivity, specificity, positive and negative likelihood ratios and predictive values, and disease prevalence, along with their 95% confidence intervals.
- Number Needed to Treat (NTT) Calculator -- you enter the risk of outcome for the treated and control groups, and this page will calculate the absolute risk reduction (ARR), relative risk (RR), relative risk reduction (RRR), and number needed to treat (NNT). This page also contains a handy table showing NNT for various values of control group risk and relative risk, and a chart illustrating the relationship.
- 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 EC_{50}_{ }(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)

- 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.
- Structural Engineering Calculators, from BuildingsGuide.com -- currently includes ASCE 7-05 Code calculators for Snow Loading Analysis, Ice Loading Analysis for WT, MT & ST shapes, for W, M, S & HP shapes, and for C & MC shapes; Seismic Base Shear for single-level buildings, and Wind Loading Analysis for low-rise buildings.
- 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".
- 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.

Check out the PowerAndSampleSize.com web site, which contains (at last count) 19 interactive calculators for power or required sample size for many different types of statistical tests: testing 1 mean, comparing 2 or more means, testing 1 proportion, comparing 2 or more proportions, testing odds ratios, and two 1-sample tests (normal and binomial-based). This site also provides calculators for non-inferiority and equivalence studies. The web pages display graphs that dynamically show how power varies with various design parameters as you change other parameters.

Also, look at 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...- Required sample size or power for testing one mean for equality, or non-inferiority or superiority, or equivalence.
- Required sample size or power for a one-sample normal-based test of a mean.
- 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
- Required sample size or power for a one-sample binomial test of a proportion.
- Required sample size or power for testing one proportion for equality, or non-inferiority or superiority, or equivalence.
- 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. - 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.
- Find the required sample size or statistical power for comparing an observed proportion with a specific value
- 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
- Sample size or confidence interval of a proportion

- For two-group tests...
- Comparing means for two independent samples...
- Required sample size or power for testing two means for equality, or non-inferiority or superiority, or equivalence.
- Required sample size, or the statistical power when comparing the means of two samples (can have different standard deviations)
- Same-size samples
- Same-size samples; equal or unequal variances
- 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. - A graphical interactive power/sample-size calculator for equal-variance two-group test.
- Sample size for parallel-group equivalence and superiority trials, with continuous outcome variables.

- Difference between two proportions (as, for example, by a Chi Square
test on a 2-by-2 cross-tab)...
- Required sample size or power for testing two proportions for equality, or non-inferiority or superiority, or equivalence.
- Equal sample sizes (easy to use; with a good explanation)
- Equal sample sizes -- easy to use, with graphical display
- 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
- Sample size for parallel-group equivalence and superiority trials, with binary outcome variables.
- Test an odds ratio (from a 2x2 table) for equality, or non-inferiority or superiority, or equivalence.

- Comparing means for two independent samples...
- 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
- Required sample size or power for comparing N groups, two at a time, by a 1-way ANOVA.
- 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. - 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. - Required sample size or power for compairing N proportions, two at a time, by a 1-way ANOVA.

- For regressions and correlation tests...

- 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 f
^{2}), and the desired statistical power level. If you know the effect size as R^{2}, you can calculate f^{2}from R^{2}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 R
^{2}, 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 R
^{2}, 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) Before doing a retrospective power analyses, check out Richard Stevens' humorous web page to see if you really need to do one.
- Sample size
calculator for pilot or safety studies where the goal is to
**detect the occurrence of one or more events**that might indicate possible problems in study design (ambiguous inclusion criteria, misinterpretation of questionnaire items) or product safety (occurrence of serious adverse events, abnormal lab tests, or other safety markers). Enter the prevalence rate you want to be able to detect (like one in a hundred), and the probabibility you want to have of seeing at least one such event in your sample (like 95% probability), and the page will tell you how many subjects you need in the study. - 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 -- this page contains a table for selecting appropriate calculator, based on the type of study (parallel, crossover, or test of association) and the type of outcome measurement (success/failure, time-to-event, or a numerical quantity).
- 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
- Find sample size, power and minimal detectable difference for a:
- study on the effect of one variable on another (linear regression)
- cross-over study were the outcome is a measurement (paired t-test or non-parametric equivalent)
- parallel trial where the outcome is a measurement (unpaired t-test or non-parametric equivalent)

- 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.
- Sample-size calculations for parallel-group equivalence and superiority trials with continuous or binary outcome variables.
- 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
- Type I & II error criteria. [see Simon, Controlled Clin Trials, 10:1-10,1989]
- 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.

- WebDOE