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.

*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.

The web apps accessable from this page were built with the free, open-source statistical computing and graphics language **R**. R is an amazingly flexible platform for organizing, visualizing, and analyzing data. Setting out to work in R can involve a bit of a learning curve since itrequires acquiring some fluency with basic programming language. However, there is an astounding amount of free learning resources available on the web in the form of websites, books, blogs, tutorials, etc. There are also great web-based courses that often have free trials.

Many recommend also using **R Studio**, a platform to help organize workflow when using R. The free version of R-Studio is what I use, and I’m fairly sure it has all that most of the folks in music education would likely every need.

More to the point… …R Studio has an integrated web app builder called **shiny**. Shiny makes building data visualization tools and web-based dashboards for exploring data fairly straight-forward. The apps that are produced with shiny are built in roughly the same type of code that is used to run analyses and make plots in R in general.