Visualizing Statistical Concepts
Dr. Peter Miksza, Indiana University Jacobs School of Music
A Collection of Tools for Emerging Music Education Researchers
The web apps that are available from this page are designed to
demonstrate and familiarize students with statistical concepts that are
commonly encountered in music education research methods courses.
Mastering statistical concepts can often be challenging for graduate
music education students who have most recently been out in the field
working with children in classrooms. Grappling with statistics can often
be overwhelming for those whose last formal mathematics education
experiences may have been many years in the past. In my courses, I have
found that while clear explanations and practice examples are good -
live manipulable demonstrations can be extremely
powerful ways for developing insight in regard to statistics.
Web Apps
Generating Fake
Data for Practice Analyses (link)
- This app allows you to create and download a data set consisting of
two nominal variables, two ordinal variables, a continuous variable of
integer values, and a continuous variable of decimal values. You can set
the proportions for the frequencies of the categorical variables, the
range of the oridinal variables, and the mean and standard deviation for
each of the continuous variables. Once you’ve made your selections, you
can download the resulting data set as a .csv file that can be imported
into any statistical package.
Playing
with Probability (link)
- This app allows for visualizing basic discrete and continuous
probability distributions. You can manipulate a coin-flipping
simulation, a dice-rolling simulation, and a normal distribution to
examine the probabilities of outcomes of your choosing. All simulations
allow you to explore the notion of …in the long run… when thinking about
probability by letting you choose the number of indpendent samples
drawn. It’s also possible to “weight” the coin to be biased in the
flipping simulation.
Correlation
(link)
- This application allows you to visualize correlations of different
strengths for scalar (i.e., interval/ratio), ordinal, and binary nominal
variables. It’s also possible to adjust the means and standard
deviations for the distributions of the scalar variables as well as the
ranges for the ordinal variables. Scatterplots of the scalar and ordinal
variables and a mosaic plot of the nominal variables are produced along
with the resulting coefficient and respective p value.
Standard
Error of the Mean (link)
- This app simulates repeated independent sampling of means from a
population with a mean and standard deviation of your choosing. It
provides a visualization of how the standard error of the mean can vary
as a function given the input parameters for the population and sample
size.
Chi Square Test
of Independence (link)
- This app allows you to specify the proportions for each cell of a
cross-tabulation of two binary nominal variables. The app generates a
summary of a Pearson chi square test of the data, a mosaic plot to
visualize the proportion of cases in each cell, and a table of the
resulting standardized residuals. A plot of a chi square distribution
with the respective degrees of freedom is also produced. The critical
value of chi square and the observed chi square value are annotated upon
the plot.
One-Sample
t-Test (link)
- This tool provides the opportunity to explore many aspects of the
one-sample t-test and inferential statistical tests in general. First,
you are able to generate a random sample of data by specifying the
sample size, mean, and standard distribution. The mean from the random
sample is then compared to a population (i.e., null) mean value of your
choice using a one-sample t-test. The t-test results, a plot of the
distribution of sample data with respect to the null mean, and a plot of
the t distribution for the appropriate degrees of freedom with the
critical values of t and the observed t statistic annotated upon it is
produced as well. You can then see how specifying different properties
of the sample data impacts the confidence interval of the mean, the t
distribution, and the t statistic generated.
Independent
Samples t-Test (link)
- This app allows you to specify the sample size, mean, and standard
deviation for two independent samples and then compares the means via an
independent-samples t-test. The t-test results, a plot of the
distributions of sample data and each mean, as well as a plot of the t
distribution for the appropriate degrees of freedom with the critical
values of t and the observed t statistic annotated upon it is produced
as well.
One-Way ANOVA
(link)
- This app is designed to illustrate the ways variation is partitioned
in the one-way ANOVA procedure. It is based on a generic experimental
design through which a researcher is interested in comparing means of a
control group and two treatment groups. You can specify a sample size
for the groups and the mean and standard deviation of each group. The
app then outputs a stripchart depicting within-group, between-group, and
total variation and a density plot that depicts the overlap among the
distributions of each group. Values for a traditional ANOVA table are
updated with each change in specifications for the groups as well so
that a connection can be made between the visualizations of the
variation and the statistical output. Last, a plot of the F distribution
for the appropriate degrees of freedom with the critical values of F and
the observed F statistic annotated upon it is produced in the second
panel.
OLS
Regression (link)
- This app illustrates some of the basic concepts involved in ordinary
least squares linear regression. Data for generic X and Y variables are
generated according to your selections for sample size as well as
settings for simple linear regression parameters (i.e.,intercept, slope,
and standard deviation of residuals). The first tab in the output panel
displays a scatterplot of the data overlaid with the line of best fit
for the data generated. Underneath the scatterplot is the basic
information for the model that is traditionally reported in
publications. The second tab of the main panel includes plots depicting
the total, residual, and regression variation with a standard ANOVA
table beneath them. The values in the ANOVA table are updated with each
change in specifications for the model parameters so that a connection
can be made between the visualizations of the variation and the
statistical output.