Probability
Random
Patterns
Birthday
Problem
Let's
Make a Deal
Univariate
Distributions
Examples
of Histograms
Histogram
Bin Size
Cumulative
Distributions
Boxplots
Mean
and Median
Normal
Distribution Shapes
Normal
Distribution Probabilities
Normal
Approximation to Real Data
Bivariate
Relationships
Guessing
the Size of a Correlation
Correlation
and Regression
Sampling
and Sampling Distributions
Randomization
Sampling
Distribution
Law
of Large Numbers
Central
Limit Theorem
Normal
Approximation to the Binomial
The
Standard Deviation is Biased (but not the Variance)
Statistical
Inference for Means
Difference
Between t and Z
Confidence
Interval for a Mean
Hypothesis
Test for a Mean
P-Value
Calculation for a t-Test
Randomization
Tests
Power
Effect
Size
Relative
Precision of the Mean and Median
Analysis
of Variance
Statistical
Inference for Categorical Data
Binomial
Distribution
Confidence
Interval for a Proportion
Hypothesis
Test for a Proportion
Power
Margin
of Error and Sample Size Calculations for a Finite Population
Unimportance
of Population Size
Chi-Square
Power
and Sample Size Calculations
Random Patterns:
This applet can be used to illustrate the patterns that appear in random processes.http://users.ece.gatech.edu/~gtz/ee3340/java/random/
(Seems to work a little better in Microsoft Internet Explorer. Choose the "uniform" distribution option. Random patterns are nicely produced with about N=300)
Birthday Problem:
These applets use simulations to assess the probability that two (or more) out of N people have the same birthday.http://www-stat.stanford.edu/~susan/surprise/Birthday.html
(You can vary N, and you can run the simulation in either a fast or slow mode. This applet has a very attractive display.)www.math.uah.edu/stat/java/BirthdayExperiment.html
(You can vary N. You can also run the simulation either one trial at a time or many repetitions at a time, and the results accumulate in either case.)
Let's Make a Deal:
These web sites and applets use simulation to assess the probabilities of winning in the "Let's Make a Deal" game, which is also called the "Monty Hall" game.htttp://cartalk.cars.com/About/Monty/
(An entertaining introduction to the puzzle. Lets you play one game at a time and also presents a running total of the results from everyone who has ever played the game at this site. To get to the running total, first click on "go play," and then click on "Statistics.")www.intergalact.com/threedoor/threedoor.html
(Provides an introduction, plays one game at a time, and keeps a running total of your results; all with nice visuals.)www.math.uah.edu/stat/java/MontyGame.html
(Plays one game at a time and keeps a running total of your results.)www.math.uah.edu/stat/java/MontyExperiment.html
(Lets you run many repetitions of the game at a time and lets you vary the probability of switching doors to see which probability maximizes your chances of winning. If you use this applet, you may first want to run the applet immediately above which better explicates how the game works.)www.stat.sc.edu/~west/javahtml/LetsMakeaDeal.html
(Plays one game at a time and keeps a running total of your results.
Examples of Histograms:
These web sites present histograms for a variety of data sets and variables.http://stat-www.berkeley.edu/users/stark/Java/HistHiLite.htmHistogram Bin Size:
(Seems to work best in Microsoft Internet Explorer. You can also change the interval width -- i.e., the bin size.)http://research.ed.asu.edu/siip/data.gal/
(Scroll down and click on any of the options under the heading "Graphics by dataset.")
These applets adjust the shape of the histogram as you change the interval width (i.e., the bin size).www.stat.sc.edu/~west/javahtml/Histogram.html
(A histogram of the times between eruptions of Old Faithful.)www.psychstat.smsu.edu/introbook/exercises/groupedfrequency.htm
(Requires Microsoft Internet Explorer. Displays histograms for a variety of what appear to be "hypothetical" data sets.)
Cumulative Distributions:
These applets illustrate cumulative frequency histograms.www.psychstat.smsu.edu/introbook/exercises/groupedfrequency.htm
(Requires Microsoft Internet Explorer. Displays cumulative distributions for a variety of what appear to be "hypothetical" data sets.)www.stat.ucla.edu/textbook/demos/distributions.phtml
(Requires Xlisp-Stat which can be downloaded for free via anonymous ftp [instructions] [file=densdemo]. This applet displays the density curves and cumulative density curves for a variety of distributions including the normal. Very Useful.)
Boxplots:
This applet presents a variety of histograms and boxplots side by side.http://research.ed.asu.edu/cgi-bin/sidebyside.pl
Mean and Median:
These applets allow you to manipulate the scores in a histogram and view the resulting effects on the mean and median.http://surfstat.newcastle.edu.au/surfstat/main/1-2-2.html
(Scroll down to the box entitled "Applet Centres.")www.ruf.rice.edu/~lane/stat_sim/descriptive/
(Click the "Begin" button.)http://stat-www.berkeley.edu/users/stark/Java/HistHiLite.htm
(Seems to work best in Microsoft Internet Explorer. Let's you present histograms for a variety of data sets and let's you highlight a section of the histogram. By highlighting either the upper or lower half of a histogram you can get a visual image of the median, and compare it to the value of the mean.)www.stat.ucla.edu/textbook/demos/center.phtml
(Requires Xlisp-Stat which can be downloaded for free via anonymous ftp [instructions] [file=meanmed]. In the applet, you can click and drag any of the data points.)
Normal Distribution Shapes:
These applets illustrate the variety of shapes of a normal distribution.http://stat-www.Berkeley.edu/users/stark/Java/StandardNormal.htm
(Seems to work better using the "scroll bars" rather than the "arrow keys.")www.stat.ucla.edu/textbook/demos/distributions.phtml
(Requires Xlisp-Stat which can be downloaded for free via anonymous ftp [instructions] [file=densdemo].)
Normal Distribution Probabilities:
These applets allow you to calculate the areas under various portions of standard and nonstandard normal distributions.http://u2.newcastle.edu.au/surfstat/main/tables.html
(Calculates Z scores given areas or calculates areas given Z scores. You can also choose other distributions besides the normal distribtuion.)http://www-stat.stanford.edu/~naras/jsm/FindProbability.html
(Calculates areas given Z scores and Z scores given percentiles.)http://psych.colorado.edu/~mcclella/java/zcalc.html
(Four options:
1. Calculates areas given Z scores
2. Calculates Z scores given percentiles or percentiles given Z scores
3. Calculates probabilities given a score, mean, and standard deviation
4. Calculates the area between a Z score and the mean in four different ways.)
Normal Approximation to Real Data:
This applet shows how well real data (100 measurements of the acceleration of gravity) approximate a normal distribution.http://stat-www.berkeley.edu/users/stark/Java/NormApprox.htm
(You can highlight different portions of the histogram to assess the accuracy of the approximation.)
Guessing the Size of a Correlation:
These applets present scatterplots and ask you to guess the size of the correlation.www.ruf.rice.edu/~lane/stat_sim/reg_by_eye/ Correlation and Regression:
(You are presented with scatterplots of data at random and asked, in multiple choice format, to guess the size of the correlation. You can learn the correct answer with a click of a button. You can also try to position the regression line by hand given the MSE for each position, and you can do so with or without knowing the minimum MSE. With a click of a button, you can then learn the correct positioning of the regression line. Very Useful.)www.psychstat.smsu.edu/introbook/exercises/scattertest.htm
(Requires Microsoft Internet Explorer. Generates scatterplots at random, gives you multiple choice options for the size of the correlation, and then tells you the correct answer.)
These applets present scatterplots that you can use to illustrate a variety of concepts including: how the appearance of a scatterplot differs for data with different degrees of correlation, the positioning of the regression line within a scatterplot, and how both the regression line and the correlation are altered when data points are added or altered.www.stat.uiuc.edu/~stat100/java/guess/PPApplet.html
(Starts with a blank scatterplot to which you can add, and erase, individual data points by clicking the mouse. Alternatively, you can create a scatterplot for randomly generated sets of data, for any size N and for any (target) correlation value. The applet calculates both the correlation and the coefficients of the regression line, and lets you plot the regression line and residuals. Very useful.)http://stat-www.berkeley.edu/users/stark/Java/ScatterPlot.htm
(Let's you draw scatterplots for a variety of sets of real data -- or import your own data. The applet calculates the correlation and let's you plot the regression line, a graph of averages, and the residuals, among other options. You can also add data points and toggle between the results with and without these additional points. Very useful.)http://stat-www.berkeley.edu/users/stark/Java/Correlation.htm
(You can vary both the correlation and N with scroll bars and a scatterplot is automatically updated -- in this regard, note that the scroll bars seem to work better than the arrow keys. The applet let's you plot the regression line, a graph of averages, and the residuals. You can also add data points and toggle between the results with and without these additional points. Very useful.)www.ruf.rice.edu/~lane/stat_sim/reg_by_eye/
(You are presented with scatterplots of data at random. You can try to position the regression line by hand given the MSE for each position, and you can do so with or without knowing the minimum MSE. With a click of a button, you can then learn the correct positioning of the regression line. Very Useful.)www.stat.ucla.edu/textbook/demos/correlation.phtml
(Requires Xlisp-Stat which can be downloaded for free via anonymous ftp [instructions] [file=showcorr]. Presents scatterplots with regression lines for any degree of correlation and for any size N.)www.math.csusb.edu/faculty/stanton/m262/regress/regress.html
(Starts with a blank scatterplot to which you can add individual data points. The applet calculates the regression coefficients, plots the residuals and gives you the option to plot the regression line with or without the data.)
Sampling and Sampling Distributions
Randomization:
With these applets, you can select random samples from a population or make random assignments to treatment conditions.
www.randomizer.org/
(Simple random sampling)
www.stat.ucla.edu/calculators/perm.phtml
(Random permutations
of N digits.)
www.ruf.rice.edu/~lane/stat_sim/sampling_dist/
(On separate axes,
one over the other, this applet plots [a] the population density (which
can take different forms), [b] a histogram of sample data, and [c] two
sampling distributions for any of a variety of sample sizes and estimators
(including the mean, median, standard deviation, and variance. Very
well crafted and informative.)
www.ruf.rice.edu/~lane/stat_sim/normal_approx/
(Contrasts the binomial
histogram to the normal density and calculates the binomial and normal
probabilities for any range of outcomes. Allows for a variety of
binomial distributions.)
www.ms.uky.edu/~mai/java/stat/GaltonMachine.html
(An attractive, but
time-consuming, quincunx.)
Statistical Inference for Means
Difference Between t and Z:
These applets plot a density curve for the standardized normal distribution overlayed on a density curve for the t distribution with a specified degrees of freedom.http://www-stat.stanford.edu/~naras/jsm/TDensity/TDensity.html
(Seems to work best in Microsoft Internet Explorer. The degrees of freedom for the t distribution can be varied from 1 to 49. The resolution is such that you won't see any difference between the normal and t curves except for the smaller degrees of freedom.)www.stat.ucla.edu/textbook/demos/distributions.phtml
(Requires Xlisp-Stat which can be downloaded for free via anonymous ftp [instructions] [file=densdemo]. Choose the "T" distribution in the first window.)
www.stat.sc.edu/~west/javahtml/ConfidenceInterval.html
(Draws 50 samples
at random, plots the position of the confidence intervals relative to the
population mean, and counts how many fail to cover the population mean.
You can ask for more sets of 50 random samples to be drawn and you can
vary
alpha to show the effect that has on the width of the confidence intervals.)
http://stat-www.berkeley.edu/users/stark/Java/Ci.htm
(Uses a simulation
to calculate the proportion of 68% confidence intervals that contain the
population mean. Let's you draw 1 to 500 samples at a time and plots
the position of each confidence interval relative to the population mean.)
www.stat.ucla.edu/textbook/demos/testing.phtml
(Requires Xlisp-Stat
which can be downloaded for free via anonymous ftp [instructions]
[file=t-test]. This applet draws samples at random, and plots them
on an axis that also shows the sample distribution of the mean. You
can vary the null and alternative hypotheses, the population standard deviation,
and N. The applet reports the sample mean, the critical values, the
p-value, and power. Very Useful.)
Randomization Tests:
This applet illustrates the logic of a randomization test using a variety of data. You can also enter your own data.www.stat.psu.edu/~rho/tools/twogroup.htm
(Requires the Neuron plug-in which can be downloaded for free from www.asymetrix.com/products/toolbook2/neuron/dowload.html.)
www.stat.psu.edu/~rho/tools/power.htm
(This applet plots
the sampling distribution of the mean both assuming the null hypothesis
is true and assuming the alternative hypotheis is true. The applet
calculates power and shades both the area representing the (one-tailed)
rejection region under the null hypothesis distribution and the area representing
power under the alternative hypothesis distribution. You can vary
N, the true mean, and alpha to see how this affects power. Requires
the Neuron plug-in which can be downloaded for free from www.asymetrix.com/products/toolbook2/neuron/dowload.html.
Very
useful.)
www.stat.ucla.edu/textbook/demos/testing.phtml
(Requires Xlisp-Stat
which can be downloaded for free via anonymous ftp [instructions]
[file=t-test]. This applet draws samples at random, and plots them
on an axis that also shows the sample distribution of the mean. You
can vary the null and alternative hypotheses, the population standard deviation,
and N. The applet reports the sample mean, the critical values, the
p-value, and power. Very Useful.)
www.ruf.rice.edu/~lane/stat_sim/repeated_measures/
(Runs simulations
to estimate power for both within and between-subject designs. You
can vary N, the population means, the population standard deviations, and
the correlation between pairs of scores. The results of any single
simulation can be presented in a scatterplot. A creative way to
come to understand power.)
http://acad.cgu.edu/wise/hypothesis/hypoth_applet.html
(On separate axes,
one above the other, this applet plots a density curve for a normal population
and for the sampling distribution of the mean, both assuming the null hypothesis
is true and assuming the alternative hypothesis is true. Clicking
a button plots a histogram for a random sample on top of the true
population density and plots the sample mean on top of the corresponding
sampling distribution. The z-score test statistic and p-value
for the sample are also reported. Repeating the simulation adds further
means to the sampling distribution. Among other things, this applet
can be used to show how power varies with the size of N, the population
means, the population standard deviation, and alpha.)
www.stat.sc.edu/~ogden/javahtml/power/power.html
(Scroll down.
This applet plots two sampling distributions of the mean; one assuming
the null hypothesis is true and other assuming the alternative hypothesis
is true. The applet shades both the area representing the rejection regions
under the null hypothesis distribution and the area representing power
under the alternative hypothesis distribution. You can vary N,
the null hypothesis value, the true population mean, and the standard deviation
of the population. This is a beautifully crafted applet but note
that changes in N and the other values cause a rescaling of the axis, which
may or may be confusing to students.)
www.rug.rice.edu/~lane/stat_sim/group_diff.html
(Calculates the relative
proportion of individuals who are above a specified cut-off point for any
size mean difference between two populations.)
Statistical Inference for Categorical Data
Binomial Distribtuion:
These applets plot a histogram for the binomial distribution for any value of N and any value of p.www.ruf.rice.edu/~lane/stat_sim/binom_demo.html
(This is an older version of the applet listed immediately below, but gives a nicer overall view of the contrast between the binomial histogram and the normal density.)www.ruf.rice.edu/~lane/stat_sim/normal_approx/
(Contrasts the binomial histogram to the normal density and calculates the binomial and normal probabilities for any range of outcomes.)www.stat.ucla.edu/textbook/demos/distributions.phtml
(Requires Xlisp-Stat which can be downloaded for free via anonymous ftp [instructions] [file=densdemo]. Very Useful.)
Confidence Interval for a Proportion:
This applet illustrate the likelihood that a confidence interval for a proportion contains the population proportion.www.ruf.rice.edu/~lane/stat_sim/normal_approx_conf/
(Let's you investigate how well an approximate confidence interval contains the population proportion for any value of N and for any valur of the population proportion.)
www.math.csusb.edu/faculty/stanton/m262/proportions/proportions.html
www.researchinfo.com/calculators/sscalc.htm
(This site contains
the same sample size calculator as in the applet immediately above.)
www.researchinfo.com/calculators/sscalc.htm
(This site contains
the same sample size calculator as in the applet immediately above.)
http://media.tasc.ac.uk/sobol/survey5/
(This web site lets
you (a) choose among pairs of variables from a variety of data sets, (b)
plot the bivariate frequency table for any given pair and calculate a chi-square
test of the relationship, and (c) then see how the results change when
you collapse cells from either or both of the two variables -- which is
especially useful when you collapse cells that have small Ns. Follow
the directions at the web site.)
___________
Revised 10/18/99.
Support from the Center for Teaching and Learning at the University of
Denver is gratefully acknowledged. Please send suggestions for additions
or other comments to Chip Reichardt (creichar@du.edu
).