Web
Pages that Perform
Statistical Calculations!
( StatPages.org )
(Updated 03/13/2015  Added an entry for
MyCurveFit.com  an easytouse nonlinear curvefitting
page.)
Precision Consulting  Offers
dissertation help, editing, tutoring,
and coaching services on a variety of statistical methods including
ANOVA, Multiple Linear Regression, Structural Equation Modeling,
Confirmatory Factor Analysis, and Hierarchical Linear Modeling. If
you're stuck on your proposal, methodology, or statistical phase of
your dissertation, you might want to contact them. 
The web pages listed below comprise a
powerful,
convenientlyaccessible,
multiplatform 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
email expressing your appreciation for their hard work and generosity
in
making this software freely accessible to the world.
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,
singlepopulation
tests
 Sample comparisons:
ttests, ANOVAs,
nonparametric comparisons
 Contingency tables,
crosstabs, ChiSquare
tests
 Regression,
correlation, least squares
curvefitting, nonparametric correlation
 Analysis of survival data
 Bayesian Methods
 Other statistical tests and
analyses
 Specialized and
disciplinespecific
tests and analyses
 Power, sample size and
experimental design
Other Statistical Resources...
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
"standalone" 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
Version 2.2 
OpenEpi is a free, webbased, open source, operatingsystemindependent
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 popup 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
computergenerated analogs of popular games and processes), Analyses
(collection of common webaccessible tools for statistical data
analysis), Games (interfaces and simulations to reallife processes),
Modeler (tools for distribution, polynomial and spectral modelfitting
and simulation), Graphs, Plots and Charts (comprehensive webbased
tools for exploratory data analysis), Additional Tools (other
statistical tools and resources), SOCR Wiki (collaborative Wiki
resource), Educational Materials and Handson 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
wellorganized output of results, with interpretations and graphical
displays.
 VSUstat
 a free online statistical analysis tool for basic experimental
design. It can do ANOVA and posthoc analysis using Tukey's HSD and
Fisher's LSD for completely randomized designs (CRD), randomized
complete block designs (RCBD), Latin square designs, and twofactor
factorial design for CRD and RCBD. It can also calculate descriptive
statistics (sum, mean, range, min, max, standard deviation, variance,
standard error mean, skewness, standard error skewness, standard error
kurtosis) and analyze Student's ttest (onesample ttest,
independentsamples ttest, pairedsamples ttest). VSUstat can import,
export, and save datasets, access public datasets, and produce output
in PDF, HTML, XLS, and RTF.
 ProtoGenie
 a free
extensible webbased 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.
 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, onesample analysis,
timeseries analysis, twosample 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 singleuser copy or a corporate deployment license can be
purchased.
 The
Calcugator
 a calculator, plotting engine, and programming environment. Also
available as a free standalone 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, printpreviewed 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 userdefined 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 stepbystep
explanation of how it arrived at the answer.
 Expression Evaluators
 type in any
numeric expression; the computer will evaluate it and display the
results...
 Calculators
 pages that look and act
like a pocket calculator...
 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...
 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 webbased interface to the "R" statistical programming
language (similar to S or Splus)
 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 builtin 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,multifunction pages first...
 These pages contain calculations for a very
wide
assortment of probability distribution functions, including
Normal,
Bivariate Normal, Student t, ChiSquare, Fisher F, Bivariate Normal,
Noncentral Student t, Noncentral ChiSquare, Noncentral Fisher F,
Poisson, Lognormal, 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,
chisquare, 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, ChiSquare, 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, ChiSquare, Binomial, and
Poisson distributions
 Statistical
probability
distribution functions: Normal, Student t, ChiSquare, Fisher F
 Pvalues
for the Popular Distributions  Binomial , Chisquare,
Exponential
, Fisher's F, KS: Two Samples , Poisson, Normal , Student's t, and
Uniform distributions.
 Calculate
pvalue from z, t, F, r, or Chi Square; or do the reverse.
 Reverse
computations: enter pvalue (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
 Another
very good Normal Distribution calculator, with nice graphics
 A
very
attractive page for Normal distribution (and inverse), with detailed
explanations
 Normal
area (1tailed)
 Cumulative
area
under the normal curve (integral from minus infinity to z)
 ChiSquare
probabilities, and reverse, with a detailed explanation
 ChiSquare
Distribution
 ChiSquare
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 pvalue calculator.
 Critical
Fisher
F value, given the alpha level, the numerator and denominator
d.f.
 Noncentral
F
value (by Laubscher's square root approximation),
given the
Fvalue, 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, ChiSquare,
Binomial
Test, Wilcoxon
RankedSums Test, Wilcoxon
Signed Ranks Test, and Correlation
Coefficient
 Converting
r to Z
 Statistical Power of: Z
Test, tTest
for One Sample or Two Related Samples, tTest
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, nQueens
 and other puzzles and Miscellanea
 Statiscope
 a beautifullyimplemented 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, AndersonDarling 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 DurbinWatson 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 3dimensional (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 changepoint, that is: estimate when, in a long
sequence of
occurrences, the occurrence rate underwent a sudden change
 Confidence Intervals...
 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
1sided
and 2sided 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
.
 SinglePopulation 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 OneSample Student t Test page  enter or paste
raw
data, or enter mean, SD or SEM, and N
 Onesample
Student t test for Mean vs. a Specified Value  for up to
80
observations, and a postulated population mean.
 Another
Student ttest 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
nonparametric 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 pvalue of the frequentist test.
 Test
observed vs. expected rates of occurrence of events, based on
Poisson distribution; also includes confidence intervals and analysis
of rateratios (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 rateratio, 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 MultiCounts  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 JarqueBera statistic
 Test
for Homogeneity of a Population  enter form 25 to 84
values; page
provides information to test whether histogram is unimodal.
 ShapiroWilk 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 KolmogorovSmirnov 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.
 ChiSquare "Goodness of
Fit" test for
observed vs expected counts (NOT from Contingency
Tables)...
 Measurement Errors and
Error Propagation...
 Student ttest
(for comparing two
samples)...
 a
very
general Student ttest web page  paired or unpaired, equal
or
unequalvariance, from individual observations (which can be
keyentered 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 twogroup 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.
 ttest,
paired
or unpaired
 ttest, paired
or unpaired
 ttest, paired
or unpaired
 ttest,
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
ttest from summary data (N, mean, SD)
 Very
general ttest 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 ...
 OneWay and Factorial
ANOVA for
uncorrelated samples (extension of unpaired Student
ttest to
more than 2 groups)...
 Oneway
ANOVA, with graphical output
 Oneway
ANOVA for 3 Independent Samples
 ANOVA:
Testing the Means  Oneway ANOVA for three groups, each
containing up to 40 subjects.
 Oneway
ANOVA for 4 Independent Samples
 Oneway
ANOVA from summary data
(N, mean, and SD or SEM)  Now also does Tukey HSD posthoc
test!
 Another
1way ANOVA from summary data
 Oneway ANOVA
 Also produces a posthoc analysis (which groups are different from
which others), and a scatterplot of all groups. For a description of
the ANOVA, click
here.
 Twoway
factorial ANOVA for 2 rows by 2 columns
 Twoway
factorial ANOVA for 2 rows by 3 columns.
 TwoWay
ANOVA Test  for blocked designs of up to 4 groups by 6
treatments.
 TwoWay
ANOVA with Replications  for blocked designs of up to 4
groups by
6 treatments, with up to 4 replications.
 Twoway ANOVA  enter data into
the web page, or read it from a text file. For an explanation, click here.
 Twoway
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 oneway ANOVA.
 Very
general nway factorial ANOVA, with interactions, means
table,
interaction plots, Bonferroni posthoc multiple comparisons, and
confidence intervals. (When you get to the Rweb
page, scroll
down to the Analysis Menu and select ANOVA.)
 RepeatedMeasures
ANOVA for
correlated samples (extension of paired Student
ttest to more
than 2 matched measurements)...
 Bartlett's
Test for Equality of Multivariances  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.
 Posthoc
Tests  After doing a twoway (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 posthoc 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
 Nonparametric tests
(use these when
the data is not normally distributed)...
 Sign
test for matched pairs
 Median
test for unmatched pairs
 Wilcoxon
SignedRanks test for matched pairs  a
nonparametric 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 SignedRanks 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 MannWhitney U test, with up
to 80
observations per sample.
 MannWhitney U test
 a nonparametric substitute for the unpaired Student t test when the
data is not normally distributed. This page also produces a dotplot
and a histogram of the data for each group. For a detailed description
of the test, with formulas and examples, click here.
 KS
Test for Equality of Two Populations  Given two sets of
frequencies (using the same grouping intervals), this page calculates
the KolmogorovSmirnov test.
 Twosample KolmogorovSmirnov Test
 Enter numbers into the web page, or read them in from text files.
Also graphs the cumulative distribution of the two samples.
 Wilcoxon
SumofRanks (MannWhitney) test for comparing two unmatched samples
 KruskalWallis
test (nonparametric ANOVA) for 2 or more groups of unpaired data
 This page requires that you first crosstabulate 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.
 KruskalWallis test  This page
also produces a scatterplot of ranks for all groups.
 Least
Significant Difference between mean ranks (posthoc test
after a
significant KruskalWallis test)
 Friedman
test for comparing rankings (nonparametric)
 Twogroup
ordinal comparisons to assess how probable it is that the two groups
come from a single ordering, using WaldWolfowitz, Randomness
Test,
MannWhitney, and KolmogorovSmirnov
 Twogroup
paired comparisons, using Ttest, 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.
 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.
 WebStat
(an integrated (Java) applet)
can perform
Ztests and Ttests (one and twosample) for population means, and
Chisquare and FisherF tests for population variances
 ChiSquare tests...
 2by2
table
analysis
(Chisquare, 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
2by2 table, by Fisher Exact, and by Chi Square (with and
without
Yates' correction), with a good explanation
 for
2by2 table
 2by2
table analysis (Chi Square, Fisher Exact, difference in
proportions, risk ratio, odds ratio, theta, logodds ratio, Poisson
test)
 Diagnostic Test Evaluation
 from a 2x2 crosstab 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
2byN 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.
 Chisquare
Test for Relationship  for up to a 6by6 crosstab.
 for
anysize table
 another
for anysize table
 another
for anysize table (When you get to the Rweb
page, scroll
down to the Analysis Menu and select Two
Way.)
 Exhaustive
analysis of 2by2 tables, with Pearson Chisquare,
Likelyhood
Ratio ChiSquare, Yates Chisquare, Mantel Haenszel Chisquare, Odds
Ratio, Log Odds Ratio, YulesQ, YulesY, Phisquare, 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 nonzero difference.)
 Also see the EvidenceBasedMedicine (EBM) calculator
in the
"Biostatistical Calculators" section of the "Other
Statistical Tests and Analyses" section of this page.
 Threedimensional Tables
(2x2x2)...
 Fisher Exact tests
for contingency
tables...
 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, 5by5 table or less)
 Contingency
table for sequenced categories, 5by2 table, with exact
probability calculations
 Spearman's
correlation from crosstabbed data with sequenced row and column
categories
 McNemar's
test to analyze a matched casecontrol 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 observedvsexpected, and observedvsuniform
situations.
 Comparison of ratings or
rankings by
different raters...
 ChiSquare
test for equality of distributions
 ChiSquare "Goodness of
Fit" test for
observed vs expected counts (NOT from Contingency
Tables)...
 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 2parameter 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 (nonparametric)...
 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 nonparametric 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 ttests, 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 NbyN 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 interrelationships between three partial correlation
coefficients.
 Beyond Simple 2parameter
Curvefitting...
 Very
general
nonlinear leastsquares
curve fitter  almost any function you can write up to 8
nonlinear parameters, up to 10 independent variables.
 MyCurveFit.com
 an easytouse curvefitting page. Offers 13 predefined
functions (no initial guesses required), along with the ability to fit
a general nonlinear function you provide (along with initial guesses).
Displays the results graphically, along with the formula of the fitted
curve. Several types of unequal datapoint 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 nonlinear
leastsquares curvefitter  with an enormous list of
predefined
2D and 3D functions, and extensive graphical and statistical output.
 Another
nonlinear leastsquares curve fitter  with graphical
output!
Choose one of 15 predefined nonlinear functions of one variable and up
to three parameters.
 3D 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 3D 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
commonlyoccurring nonlinear functions (gaussians, sigmoidals,
rationals, sinusoidals, etc.), and then ranks them according to
goodnessoffit.
 Compare
the fit of two models to your data. Which model fits better?
Enter
goodnessoffit (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 curvefitter 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 pointandclick 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 2way and 3way 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 nonoccurrences 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 DurbinWatson 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 onelagapart &
twolagapart
autocorrelations. A related page: Time
Series' Statistics calculates these statistics, and also the
overall mean & variance, and the first & second partial
autocorrelations.
 KaplanMeier
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 pe
 KaplanMeier Survival Plot
 for one or more groups. Draws KM curves with optional confidence
bands (ordinary, log, or loglog 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).
 Two HTML5 web pages by Robert Mening:
 Beginner's guide to HTML_{5}
 Includes a handy "HTML_{5} Cheat Sheet".
2) HTML_{5} Periodic Table
 a clever interactive reference to the components of HTML_{5}.
 Two HTML5 web pages by Robert Mening:
 Beginner's guide to HTML_{5}
 Includes a handy "HTML_{5} Cheat Sheet".
2) HTML_{5} Periodic Table
 a clever interactive reference to the components of HTML_{5}.
 Two HTML5 web pages by Robert Mening:
 Beginner's guide to HTML_{5}
 Includes a handy "HTML_{5} Cheat Sheet".
2) HTML_{5} Periodic Table
 a clever interactive reference to the components of HTML_{5}.
 Two HTML5 web pages by Robert Mening:
 Beginner's guide to HTML_{5}
 Includes a handy "HTML_{5} Cheat Sheet".
2) HTML_{5} Periodic Table
 a clever interactive reference to the components of HTML_{5}.#
Two HTML5 web pages by Robert Mening:
* Beginner's guide to HTML5  Includes a
handy "HTML5 Cheat Sheet".
2) HTML5 Periodic
Table  a clever interactive reference to the components of HTML5.
 Two HTML5 web pages by Robert Mening:
 Beginner's
guide to HTML_{5}  Includes a handy "HTML_{5}
Cheat Sheet".
2) HTML_{5} Periodic Table
 a clever interactive reference to the components of HTML_{5}.Two
HTML5 web pages by Robert Mening:Two HTML5 web pages by
Robert Mening:Beginner's
guide to HTML_{5}  Includes a handy "HTML_{5}
Cheat Sheet".Beginner's guide to HTML_{5}
 Includes a handy "HTML_{5} Cheat Sheet".Beginner's
guide to HTML_{5}  Includes a handy "HTML_{5}
Cheat Sheet".
 Mometrix Academy
 a set of free videos to help you prepare for various kinds
of
standardized tests (ACT, GED, GRE, SAT, LSAT, MCAT, and others), and to
improve your general knowledge in various areas (business &
finance, English, health & fitness, humanities, mathematics,
medical, science, and teaching). These videos are presented as a public
service of Mometrix, The World’s Most Comprehensive
Test Preparation Company™.
The main purpose of this site is to offer free, practical testtaking
advice. If you like the content of these videos and you're taking a
test in the near future, you may want to look at their more
comprehensive study resources at the Mometrix
Home Page.
 David
Kremelberg's statistical resources page  contains resources
for statistics students, including information and free downloadable
Excel templates for Pearson's r, ChiSquare, ttests, and ANOVA. These
templates calculate these statistical tests stepbystep, allowing
students who need to calculate these tests by hand to check their work.
 Peter Rosenmai's EurekaStatistics
web site (a blog about statistics and R).
 C
Programming Tutorial
 A set of short tutorial taking you through the process of creating C
programs, from the simplest "Hello World" application, to more complex
programs using variables and expressions, input and output,
if/then/else and while constructs, and arrays.
 Two HTML5 web pages by Robert Mening:
 SticiGui
(pronounced "stickygooey")  an excellent interactive online
textbook/tutorial that was designed for a "first and last" Statistics
class for Business, Social Science, or liberal arts. In other words,
this site contains the knowledge you'd want to get if you were
going to take only one stats course in your whole life. So it's not
geared toward theory, numerical analysis, sophisticated formulas, or
big collections of specialized tests or probability distributions.
Instead, it helps you think logically about quantitative evidence and
to translate realworld situations into mathematical questions; it
shows you a few important statistical and probabilistic concepts and
some of the difficulties, subjective decisions, and pitfalls, in
analyzing data and making inferences from numbers. The text develops
probability, estimation, and inference using counting arguments: there
is no calculus involved. The web site's creator (a Statistics professor
at U.C. Berkely) hopes that people who study these materials will:
 Read the newspaper with new eyes: become
skilled,
circumspect consumers of qualitative and quantitative information.
 Know that probability in particular, and
numbers in
general, can be used to model some features of the physical world and
human behavior.
 Improve their skills in critical thinking
and logical
reasoning.
 Appreciate the role Statistics plays in many
fields,
from business to economics, law, politics, science and medicine.
 Know that data can be manipulated to tell
many
inconsistent stories, that data analysis is not clear cut, and that
many subjective judgments are involved in analyzing real data.
 Know important questions to ask when faced
with a
quantitative argument—be able to analyze arguments and find their
strengths and weaknesses.
 Understand that untutored intuition tends to
produce
faulty probability judgments and know how to reason about probability.
 Appreciate some of the philosophical
difficulties in
ascribing meaning to probability and in inferring causal relationships
from data.
 Be prepared for more advanced courses in
Statistics—even though they might not take any.
 Mometrix Academy
 a set of free videos to help you prepare for various kinds
of
standardized tests (ACT, GED, GRE, SAT, LSAT, MCAT, and others), and to
improve your general knowledge in various areas (business &
finance, English, health & fitness, humanities, mathematics,
medical, science, and teaching). These videos are presented as a public
service of Mometrix, The World’s Most Comprehensive
Test Preparation Company™.
The main purpose of this site is to offer free, practical testtaking
advice. If you like the content of these videos and you're taking a
test in the near future, you may want to look at their more
comprehensive study resources at the Mometrix
Home Page.
 David
Kremelberg's statistical resources page  contains resources
for statistics students, including information and free downloadable
Excel templates for Pearson's r, ChiSquare, ttests, and ANOVA. These
templates calculate these statistical tests stepbystep, allowing
students who need to calculate these tests by hand to check their work.
 Peter Rosenmai's EurekaStatistics
web site (a blog about statistics and R).
 C
Programming Tutorial
 A set of short tutorial taking you through the process of creating C
programs, from the simplest "Hello World" application, to more complex
programs using variables and expressions, input and output,
if/then/else and while constructs, and arrays.
 Two HTML5 web pages by Robert Mening:
 SticiGui
(pronounced "stickygooey")  an excellent interactive online
textbook/tutorial that was designed for a "first and last" Statistics
class for Business, Social Science, or liberal arts. In other words,
this site contains the knowledge you'd want to get if you were
going to take only one stats course in your whole life. So it's not
geared toward theory, numerical analysis, sophisticated formulas, or
big collections of specialized tests or probability distributions.
Instead, it helps you think logically about quantitative evidence and
to translate realworld situations into mathematical questions; it
shows you a few important statistical and probabilistic concepts and
some of the difficulties, subjective decisions, and pitfalls, in
analyzing data and making inferences from numbers. The text develops
probability, estimation, and inference using counting arguments: there
is no calculus involved. The web site's creator (a Statistics professor
at U.C. Berkely) hopes that people who study these materials will:
 Read the newspaper with new eyes: become
skilled,
circumspect consumers of qualitative and quantitative information.
 Know that probability in particular, and
numbers in
general, can be used to model some features of the physical world and
human behavior.
 Improve their skills in critical thinking
and logical
reasoning.
 Appreciate the role Statistics plays in many
fields,
from business to economics, law, politics, science and medicine.
 Know that data can be manipulated to tell
many
inconsistent stories, that data analysis is not clear cut, and that
many subjective judgments are involved in analyzing real data.
 Know important questions to ask when faced
with a
quantitative argument—be able to analyze arguments and find their
strengths and weaknesses.
 Understand that untutored intuition tends to
produce
faulty probability judgments and know how to reason about probability.
 Appreciate some of the philosophical
difficulties in
ascribing meaning to probability and in inferring causal relationships
from data.
 Be prepared for more advanced courses in
Statistics—even though they might not take any.
 Mometrix Academy
 a set of free videos to help you prepare for various kinds
of
standardized tests (ACT, GED, GRE, SAT, LSAT, MCAT, and others), and to
improve your general knowledge in various areas (business &
finance, English, health & fitness, humanities, mathematics,
medical, science, and teaching). These videos are presented as a public
service of Mometrix, The World’s Most Comprehensive
Test Preparation Company™.
The main purpose of this site is to offer free, practical testtaking
advice. If you like the content of these videos and you're taking a
test in the near future, you may want to look at their more
comprehensive study resources at the Mometrix
Home Page.
 David
Kremelberg's statistical resources page  contains resources
for statistics students, including information and free downloadable
Excel templates for Pearson's r, ChiSquare, ttests, and ANOVA. These
templates calculate these statistical tests stepbystep, allowing
students who need to calculate these tests by hand to check their work.
 Peter Rosenmai's EurekaStatistics
web site (a blog about statistics and R).
 C
Programming Tutorial
 A set of short tutorial taking you through the process of creating C
programs, from the simplest "Hello World" application, to more complex
programs using variables and expressions, input and output,
if/then/else and while constructs, and arrays.
 Two HTML5 web pages by Robert Mening:
 SticiGui
(pronounced "stickygooey")  an excellent interactive online
textbook/tutorial that was designed for a "first and last" Statistics
class for Business, Social Science, or liberal arts. In other words,
this site contains the knowledge you'd want to get if you were
going to take only one stats course in your whole life. So it's not
geared toward theory, numerical analysis, sophisticated formulas, or
big collections of specialized tests or probability distributions.
Instead, it helps you think logically about quantitative evidence and
to translate realworld situations into mathematical questions; it
shows you a few important statistical and probabilistic concepts and
some of the difficulties, subjective decisions, and pitfalls, in
analyzing data and making inferences from numbers. The text develops
probability, estimation, and inference using counting arguments: there
is no calculus involved. The web site's creator (a Statistics professor
at U.C. Berkely) hopes that people who study these materials will:
 Read the newspaper with new eyes: become
skilled,
circumspect consumers of qualitative and quantitative information.
 Know that probability in particular, and
numbers in
general, can be used to model some features of the physical world and
human behavior.
 Improve their skills in critical thinking
and logical
reasoning.
 Appreciate the role Statistics plays in many
fields,
from business to economics, law, politics, science and medicine.
 Know that data can be manipulated to tell
many
inconsistent stories, that data analysis is not clear cut, and that
many subjective judgments are involved in analyzing real data.
 Know important questions to ask when faced
with a
quantitative argument—be able to analyze arguments and find their
strengths and weaknesses.
 Understand that untutored intuition tends to
produce
faulty probability judgments and know how to reason about probability.
 Appreciate some of the philosophical
difficulties in
ascribing meaning to probability and in inferring causal relationships
from data.
 Be prepared for more advanced courses in
Statistics—even though they might not take any.
 Mometrix Academy
 a set of free videos to help you prepare for various kinds
of
standardized tests (ACT, GED, GRE, SAT, LSAT, MCAT, and others), and to
improve your general knowledge in various areas (business &
finance, English, health & fitness, humanities, mathematics,
medical, science, and teaching). These videos are presented as a public
service of Mometrix, The World’s Most Comprehensive
Test Preparation Company™.
The main purpose of this site is to offer free, practical testtaking
advice. If you like the content of these videos and you're taking a
test in the near future, you may want to look at their more
comprehensive study resources at the Mometrix
Home Page.
 David
Kremelberg's statistical resources page  contains resources
for statistics students, including information and free downloadable
Excel templates for Pearson's r, ChiSquare, ttests, and ANOVA. These
templates calculate these statistical tests stepbystep, allowing
students who need to calculate these tests by hand to check their work.
 Peter Rosenmai's EurekaStatistics
web site (a blog about statistics and R).
 C
Programming Tutorial
 A set of short tutorial taking you through the process of creating C
programs, from the simplest "Hello World" application, to more complex
programs using variables and expressions, input and output,
if/then/else and while constructs, and arrays.
 Two HTML5 web pages by Robert Mening:
 SticiGui
(pronounced "stickygooey")  an excellent interactive online
textbook/tutorial that was designed for a "first and last" Statistics
class for Business, Social Science, or liberal arts. In other words,
this site contains the knowledge you'd want to get if you were
going to take only one stats course in your whole life. So it's not
geared toward theory, numerical analysis, sophisticated formulas, or
big collections of specialized tests or probability distributions.
Instead, it helps you think logically about quantitative evidence and
to translate realworld situations into mathematical questions; it
shows you a few important statistical and probabilistic concepts and
some of the difficulties, subjective decisions, and pitfalls, in
analyzing data and making inferences from numbers. The text develops
probability, estimation, and inference using counting arguments: there
is no calculus involved. The web site's creator (a Statistics professor
at U.C. Berkely) hopes that people who study these materials will:
 Read the newspaper with new eyes: become
skilled,
circumspect consumers of qualitative and quantitative information.
 Know that probability in particular, and
numbers in
general, can be used to model some features of the physical world and
human behavior.
 Improve their skills in critical thinking
and logical
reasoning.
 Appreciate the role Statistics plays in many
fields,
from business to economics, law, politics, science and medicine.
 Know that data can be manipulated to tell
many
inconsistent stories, that data analysis is not clear cut, and that
many subjective judgments are involved in analyzing real data.
 Know important questions to ask when faced
with a
quantitative argument—be able to analyze arguments and find their
strengths and weaknesses.
 Understand that untutored intuition tends to
produce
faulty probability judgments and know how to reason about probability.
 Appreciate some of the philosophical
difficulties in
ascribing meaning to probability and in inferring causal relationships
from data.
 Be prepared for more advanced courses in
Statistics—even though they might not take any.

 rform
stratified logrank test.
 KaplanMeier 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
(KaplanMeier)  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 proportionalhazards regression analysis and return
the regression coefficients, their standard errors, hazard (risk)
ratio, and their confidence intervals, and the baseline survivor curve,
along with goodnessoffit 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 pvalue.
 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
exponentialshaped 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 newlydeveloped
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 observedvsexpected, and observedvsuniform
situations.
 Analysis
of
"1degree 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
 2by2
table analysis
(ChiSquare,
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. Payfor plans also provide an
autooptimizer, optimization comparison, manual optimizer, and control
of survey analysis.
 Queueing
Theory Calculator  Performs classic calculations for
singleserver or multiserver 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
CrossValidation  Performs the "leaveoneout"
crossvalidation
inference for: central tendency, leastsquares lines, onedimensional
multinomial tables, twodimensional contingency tables with structural
zeroes, ksample problems, and blockandtreatment designs. The web
page is welldocumented, 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 pvalues when performing multiple comparisons
(has an excellent discussion of this topic)
 Multiple
comparisons correction (Bonferroni adjustment)
 Number
Needed
to Treat, based on a 2by2 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 Casecontrol Studies
 NetMul:
a browser interface to a program that performs:
 Principal Coordinate Analysis (PCO)
 coinertia analysis
 discriminant analysis and within or betweenclass
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 postoptimality analysis of smallsize
linear programming problems (constrained optimization).
 Martindale's
Reference Desk  Calculators OnLine  Statistics
(the
granddaddy of all compendia of calculating web pages)
 Biostatistical Calculators:
 EvidenceBased
Medicine (EBM) calculator  From Warren Goff's interestinglynamed
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, PreOdds, PostProb), and Harm (RR, OR NNH).
Can also compare two different tables.
 Diagnostic Test Evaluation
 from a 2x2 crosstab 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 "Hilltype"
doseresponse relationship). Very useful in doseresponse studies.
 Thorough
analysis of 2by2 table relevant to Predictions and Diagnostic Tests
 sensitivity, specificity, prevalence, diagnostic accuracy, PPV,
posttest 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, 244872
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,
SNAPPE,
SNAP
II / SNAPPE II
 Pediatric, Trauma Scores: Pediatric
Trauma Score
 Calculators
for Clinical Formulas  Aa
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: MS105E / ANSI/ASQC Z1.4,
ISO
2859 (sampling plans for attribute data), MS414 / ANSI/ASQC Z1.9
(sampling plans for measurement data, and MS1235C (sampling
inspection plans for continuous production, Procedure CSP1).
 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 MS105E, Continuous
Sampling CSP1, Western Electric Rules, and System Reliability for
consecutivetype systems)
 Queuing
Theory
Calculator  a remarkably powerful web calculator that can
solve a
wide variety of queueing problems: singleserver, multipleserver,
infiniteserver, 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 705 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 singlelevel buildings, and Wind
Loading Analysis for lowrise buildings.
 SingleCase
analysis tools  an online calculator that can do a number
of
tests and analyses that are especially useful in "singlecase" or
"singlesystem" research: Time Series (handles AB and
multiplebaseline designs, and calculates correlations and the
Cstatistic, with pvalue), Autocorrelation, Chi Square, Testing
Significance of Difference: ttest and MannWhitney U, Binomial
Expansion, and Bayesian Analysis. Also contains a good overview
of singlecase 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.
 Singleperiod
Inventory Analysis  computes the optimal inventory level
over a
single cycle, from upto28 pairs of (number of possible item to sell,
and their associated nonzero 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
realtime
data retrieval system. Includes economic regression analysis.
 BlackScholes
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 upto6x6 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, MiddleoftheRoad, 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 upto6x6 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
evidencebased 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 upto10 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 1sample tests (normal and binomialbased). This site also provides
calculators for noninferiority 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/samplesize
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 chisquare and Poisson tests, and an amazing variety of ANOVAs

1, 2, and 3way; randomized completeblock; Latin and GrecoLatin
squares;
1stage, 2stage, and factorial nested designs; crossover; splitplot;
stripplot; 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 standalone application.
Here's a collection of online power calculator web pages for specific
kinds
of tests:
 For onegroup tests
(comparing the sample to
a specified value) or for paired twogroup tests...
 For designing surveys
(sample size and
confidence intervals for proportions, based on sample size, with or
without corrections for finite populations:
 For twogroup tests...
 Comparing means for two independent samples...
 Difference between two proportions (as, for example,
by a
Chi Square test on a 2by2 crosstab)...
 For ANOVAs and other
multigroup
comparisons...
 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 interrelationships between three partial correlation
coefficients
 Samplesize
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/Samplesize
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.
 Posthoc
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 Determination  For several situations: ANOVA and
2population 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, timetoevent, or a
numerical quantity).
 Survival
Analysis  computes power, sample size, or detectableeffect
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:
 Links
to printable copies (in Adobe Acrobat PDF format) of many
power
tables including: Z
Test, tTest
for One Sample or Two Related Samples, tTest
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 Aoptimal
designs over continuous spaces. Factors may be continuous, fixedlevel,
or qualitative. The site can handle inequality and equality (e.g.,
mixture) constraints; provides color plots; performs oneclick,
runorderrandomization; allows design import/export interoperable with
most 3rdparty analysis software; provides OLS and BLUP fits; and
includes an extensive Classical Design Library(tm), including
factorial, fractionalfactorial, BoxBehnken, centralcomposite,
PlackettBurman, orthogonal array, and uniform designs. All designs may
be evaluated under the I, D, and A, and Soptimality criteria, as
well as for the maximum distance between nearestneighbor pairs of
design points (maximin criterion). The My WebDOE(tm) feature allows
users to store their designs, evaluations, and fits online. There is
no need for userprovided 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, Splitplot, 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 firstorder
carryover (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.
 Samplesize calculations for parallelgroup
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, 10:110,1989]
 compute
drift given power and bound, and
compute probabilities,
all based upon the LanDeMets
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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