DIGMATH: Dynamic Investigatory Graphical Displays of Math: Graphical Explorations for Statistics and Probability in Excel
Sheldon P. Gordon
Most of the following graphical explorations require the use of macros to operate. In order to use these spreadsheets, Excel must be set to accept macros. To change the security setting on macros:
With Excel 2007 and later:
 When you open the spreadsheet, a new bar appears near the top of the window that says "Security Warning: Active content has been disabled".
 Click on Options.
 Click on "Enable this Content" and then click OK.
The following are the DIGMath explorations that are currently (July, 2017) completed and ready for use. (Many others are under development.) Please feel free to download and use any or all of these files; if you want all of them, you can click on the link: statistics.7z or you can send an email to gordonsp@farmingdale.edu and we will send try sending you a zip file with all of the 68 files currently ready. If you have any problems downloading or running any of these files, please contact us for assistance. If you have any suggestions for improvements or for new topics, please pass them on also.

Fair Coin This DIGMath spreadsheet is a coin flipping simulation in which you have the choice of the number n of repetitions of flipping a fair coin. It shows both graphically and numerically the results of each fun of the simulation to let you visualize the shape of the distribution of outcomes and the associated likelihood (the probability) of each.
 Dice Role A dice rolling simulation in which the user can choose the number of roles of a pair of fair dice.
 Coin Flips A coin flipping simulation in which the user has the choice of the number of fair coins being flipped and the number of repetitions.
 Binomial Distribution This DIGMath module lets you investigate the binomial distribution based on n trials with a probability p of success. You can select values of n and p using sliders and the program draws the histogram for the corresponding binomial distribution. It also shows the mean and standard deviation of the distribution.
 Binomial Simulation This DIGMath spreadsheet is a simulation of binomial probabilities. You have the choice of the number of coins, the probability of success, and the number of repetitions.
 Binomial Probabilities This DIGMath module lets you investigate the probabilities associated with a binomial distribution. The spreadsheet covers six different cases, covering virtually al the standard kinds of problems with binomial probabilities. It will calculate and display the binomial probabilities of getting (1) exactly x successes in a binomial process with n trials with a probability p of success; (2) between two given numbers of successes; (3) at least a given number of successes; (4) more than a given number of successes; (5) at most a given number of successes; and (6) less than a given number of successes. In each case, you select values of n, p, and x using sliders and the program draws the histogram to show the corresponding binomial probability.
 Law of Large Numbers This DIGMath module lets you investigate the Law of Large Numbers  what happens in the long run. You have the choice of the desired probability of success p and the number of repetitions n in a binomial experiment to see the pattern of successes over the long run.
 Chaos of Small Numbers The user has the choice of the desired probability of success and the number of repetitions (up to 25) to see the actual (simulated) outcomes and the cumulative frequency of success to demonstrate the unpredictable nature of the outcomes in the short run.
 The Effects of an Extra Point on the Mean and Standard Deviation This DIGMath module lets you investigate the effect that changing an additional point has on the values for the mean and the standard deviation (both graphically and numerically). The spreadsheet starts with a set of either 5, 10, or 20 points and you can change the value for an additional point to see how it affects the calculations and the extent of the changes depending on the number of points.
 Effect of an Extra Point on Statistics This DIGMath module lets you investigate the effects of an extra point on the mean, the median, the standard deviation, and the InterQuartile Range from two perspectives: depending on how close to or far from the center an extra point lies and depending on the size of the dataset.
 The Normal Distribution This DIGMath module lets you investigate the normal distribution with mean μ and standard deviation σ. You enter both parameters using sliders and the program draws the corresponding normal distribution curve. You can watch the effects of changing the parameter values on the resulting curve.
 Simulating the Normal Distribution This DIGMath spreadsheet lets you investigate the normal distribution in terms of a random simulation. You can enter, via sliders, values for the mean μ and standard deviation σ and then an interval of xvalues from xMin to xMax. You can also select the number of random points you want in this normal distribution and the program will generate those points, plot them along with the graph of the normal distribution, and display graphically, with different colors, those that fall under the designated portion of the normal distribution curve and those that do not. The results are also shown numerically and compared to the theoretical values for the area under the normal curve.
 Normal Approximation to the Binomial Distribution This DIGMath program lets you investigate how well a normal distribution approximates the binomial distribution based on the parameters n and p. You enter the values for n and p via sliders and the program draws the histogram for the binomial distribution and the corresponding normal distribution curve using μ = np and σ = √np(1p) to compare the two distributions.
 The Poisson Distribution This DIGMath spreadsheet lets you investigate two different aspects of the Poisson distribution that expresses the probability of a number of events occurring in a fixed period of time if these events occur with a known average rate and independently of the time since the last event. (1) The first looks at the shape of the Poisson distribution depending on its two parameters, the expected number of outcomes of an event in a given time period and the number of occurrences. You use a slider to vary the first parameters and see the effects on the shape of the distribution. (2) The second aspect is based on the idea that the Poisson distribution can be used to approximate the binomial distribution with probability of success p and number of trials n. Using sliders to change n and p, you can observe which combinations make for a good approximation and which do not.
 Central Limit Theorem Simulation The user can choose any of four underlying populations (normal, uniformly distributed, skewed, and bimodal), the sample size, and the number of random samples. The simulation randomly generates the samples and plots the means of each sample. From the graphical display and the associated numerical displays, it becomes apparent that (1) the distribution of sample means is centered very close to the mean of the underlying population, that (2) the spread in the sample means is a fraction of the standard deviation of the underlying population (about onehalf as large when n = 4, about onethird as large when n = 9, about onequarter as large when n = 16, etc.), so that students quickly conjecture that the formula for the standard deviation of the distribution of sample means is s/√n, and that (3) as the sample size increases, the sampling distribution looks more and more like a normal distribution.
 tDistributions This DIGMath module lets you investigate the properties of the tdistribution based on various numbers of degrees of freedom from 2 up to 31. You enter the desired number of degrees of freedom and the program draws the corresponding tdistribution curve as well as the curves for d.f. = 1, d.f. = 11, and d.f. = 21 and the limiting normal distribution curve when d.f. = 31.
 Distribution of Sample Proportions The user chooses the probability of success π, the sample size n, and the number of random samples. The simulation randomly generates the samples, displays the corresponding proportion of successes, and displays the summary statistics. From these displays, the students quickly conjecture that (1) the mean of the distribution of sample proportions is equal to the proportion of successes in the underlying population, that (2) the simulated results with different values of the sample size n agree with the formula for the standard deviation of this sampling distribution, and that (3) the sampling distribution becomes more and more normal in appearance as the sample size increases.
 Sample Medians This DIGMath simulation is similar to the Central Limit Theorem Simulation, but with sample medians instead of sample means.
 Sample Midranges This DIGMath simulation is similar to the Central Limit Theorem Simulation, but with sample midranges instead of sample means.
 Sample Modes This DIGMath simulation is similar to the Central Limit Theorem Simulation, but with sample modes instead of sample means.
 Distribution of Sample Variances This DIGMath simulation is similar to the Central Limit Theorem Simulation, but instead of simulating sample means from a population, the program now generates and displays the sample variances.
 Distribution of Sample Standard Deviations This DIGMath simulation is similar to the Central Limit Theorem Simulation, but instead of simulating sample means from a population, the program now generates and displays the sample standard deviations.
 Sample IQR's This DIGMath simulation is similar to the Central Limit Theorem Simulation, but instead of simulating sample means from a population, the program now generates and displays the sample InterQuartile Ranges. The IQR is the difference between the first and third quartiles of a set of data and so represents a measure of the spread in the data.
 Distribution of Sample Skewnesses This is similar to the Central Limit Theorem Simulation, but instead of simulating sample means from a population, the program now generates and displays the sample skewnesses  a measure of how far a set of values deviates from a symmetric distribution.
 Simulating Confidence Intervals The user has the choice of the same four underlying populations as in the Central Limit Theorem simulation (to see that the population does not affect the results) and the confidence level (90%, 95%, 98%, 99%), as well as the sample size n (from 10 to 50). The simulation generates random samples from the selected population, calculates and plots the corresponding confidence interval, and summarizes the number and percentage of confidence intervals that actually contain the mean of the underlying population. Students see that the actual (simulated) percentage is typically close to the selected value for the confidence level. They also see that typically the higher the confidence level, the longer the lines are that represent the actual confidence interval. They also see that typically those confidence intervals that do not contain the population mean are nearmisses.
 Constructing Confidence Intervals for Means This DIGMath spreadsheet assists you in constructing a confidence interval for the mean of a population. You enter the sample data  the sample size n, the sample mean, and the sample standard deviation, and select the level of confidence (90%, 95%, 98%, or 99%) you want. The spreadsheet constructs the corresponding confidence interval and displays it, as well as compares it in size, to the confidence intervals with other levels of confidence.
 Simulating Confidence Intervals for Proportions This DIGMath module simulates the notion of confidence intervals for the proportion of successes in a population. The user controls the choice of the population proportion for the underlying population, the confidence level (90%, 95%, 98%, 99%), and the sample size n. The simulation generates a fixed number of random samples from that population, calculates and plots the corresponding confidence interval, and summarizes the number and percentage of confidence intervals that actually contain the proportion of the underlying population. Students see that the actual (simulated) percentage is typically close to the selected value for the confidence level. They also see that typically the higher the confidence level, the longer the lines are that represent the actual confidence interval. They also see that, as the sample size increases, the lengths of the sample confidence intervals decrease and as the sample size decreases, the lengths of the confidence intervals increase. They also see that typically those confidence intervals that do not contain the population proportion are nearmisses.
 Constructing Confidence Intervals for Proportions This DIGMath spreadsheet assists you in constructing a confidence interval for the proportion of a population. You enter the sample data  the sample size n and the number of "successes" in that sample  and select the level of confidence (90%, 95%, 98%, or 99%) you want. The spreadsheet constructs the corresponding confidence interval and displays it, as well as compares it in size, to the confidence intervals with other levels of confidence.
 Visualizing Confidence Intervals for the Mean This DIGMath spreadsheet helps you visualize what confidence intervals are for the mean of a population. You enter the sample data  the sample size n, the sample mean, and the sample standard deviation, and select the level of confidence (90%, 95%, 98%, or 99%) you want. The spreadsheet constructs all four corresponding confidence interval and displays each of them so that you can compare their sizes, their centers, and their lengths to one another as you change the parameters using sliders.
 Visualizing Confidence Intervals for the Proportion This DIGMath spreadsheet helps you visualize what confidence intervals are for the proportion of a population. You enter the sample data  the sample size n and the proportion p of successes, and select the level of confidence (90%, 95%, 98%, or 99%) you want. The spreadsheet constructs all four corresponding confidence interval and displays each of them so that you can compare their sizes, their centers, and their lengths to one another as you change the parameters using sliders.
 Confidence Intervals for the Difference of Means This DIGMath module helps in constructing a confidence interval for the difference of mean based on summary sample data from two samples: the size of the samples, the sample means, and the sample standard deviations. The user can choose the confidence level desired  90%, 95%, 98%, or 99% and the resulting intervals are shown graphically and numerically.
 Confidence Intervals for Difference of Proportions This DIGMath module helps in constructing a confidence interval for the difference in population proportion based on summary sample data from two samples: the size of the samples and the number of "successes" in each sample. The user can choose the confidence level desired  90%, 95%, 98%, or 99% and the resulting intervals are shown graphically and numerically.
 Simulating Hypothesis Testing This DIGMath module provides a visual simulation of the process of testing a hypothesis regarding the population mean π. You have the choice of the same four underlying populations (again, to see that the population does not affect the results) and the level of significance (10%, 5%, 2%, 1%). The simulation generates a fixed number of samples from the selected population, plots the mean of each sample with a vertical line at the appropriate location, and summarizes the number and percentage of sample means that fall into this region. The height of each line is equal to the standard deviation of that sample. Students see that the simulated percentage of sample means that fall in the rejection region is typically close to the selected level of significance. They also see that most of the sample means that fall into the rejection region tend to be quite close to the critical values. They also see that the lines representing the samples whose means are close to the population mean tend to be very tightly clustered compared to those that are near the extreme ends, which are sparsely distributed.
 Visualizing Hypothesis Testing for the Mean This DIGMath spreadsheet helps you visualize the fundamental ideas related to testing a hypothesis for the mean of a population. You enter the sample data  the sample size n, the sample mean, and the sample standard deviation, the choice of a twotailed test or a onetailed test with the tail on the right or the left, and select the significance level α you want highlighted. The spreadsheet displays all four corresponding critical values for a onetailed test or all eight critical values for a twotailed test and shows the position of the sample mean for the data. It also displays the associated z or tvalue and the conclusion of whether you can reject or fail to reject the null hypothesis as the selected significance level.
 Hypothesis Tests for the Mean This DIGMath spreadsheet assists you in testing a hypothesis for the mean of a population. You enter the null hypothesis for the supposed value of m and select the test you want  either two tailed or one tail with either tail. You then enter the sample data  the sample size n, the sample mean m and the sample standard deviation s  and the level of significance α. The spreadsheet displays the corresponding normal or tdistribution, the location of the critical value(s), and the location of the sample mean. It also shows the associated z or tvalue, as well as the corresponding Pvalue, and the conclusion as to whether you Reject or Fail to Reject the null hypothesis.
 Hypothesis Tests for the Proportion This DIGMath spreadsheet assists you in testing a hypothesis for the proportion of a population. You enter the null hypothesis for the supposed value of p and select the test you want  either two tailed or one tail with either tail. You then enter the sample data  the sample size n and the number of successes x in that sample  and the level of significance α. The spreadsheet displays the corresponding normal distribution (when appropriate), the location of the critical value(s), and the location of the sample proportion p. It also shows the associated zvalue, as well as the corresponding Pvalue, and the conclusion as to whether you Reject or Fail to Reject the null hypothesis.
 Hypothesis Test on the Difference of Means This DIGMath spreadsheet assists you in testing a hypothesis for the difference in means of two populations. The null hypothesis is that the two means are equal, and you have to select the alternate hypothesis test you want  either two tailed or one tail with either tail. You then enter the sample data for the two samples  the sample size n, the sample mean m and the sample standard deviation s  and the level of significance α. The spreadsheet displays the corresponding normal or tdistribution for the distribution of differences of sample means, the location of the critical value(s), and the location of the difference in the two sample means. It also shows the associated z or tvalue, as well as the corresponding Pvalue, and the conclusion as to whether you Reject or Fail to Reject the null hypothesis.
 Hypothesis Test on the Difference of Proportions This DIGMath spreadsheet assists you in testing a hypothesis for the difference in proportions of two populations. The null hypothesis is that the two proportions are equal, and you have to select the alternate hypothesis test you want  either two tailed or one tail with either tail. You then enter the sample data for the two samples  the sample size n and the number of successes x in each sample  and the level of significance α. The spreadsheet displays the corresponding normal distribution (if appropriate) for the distribution of differences of sample proportions, the location of the critical value(s), and the location of the difference in the two sample proportions. It also shows the associated zvalue, as well as the corresponding Pvalue, and the conclusion as to whether you Reject or Fail to Reject the null hypothesis.
 The Distribution of the Difference of Means This DIGMath module lets you investigate the distribution of the difference of means based on summary sample data from two samples drawn from the choice of four underlying populations (to see that the population does not affect the results). The user can choose the sample size (from n = 2 to n = 50) from each sample and the number (from 50 to 250) of samples. The simulation generates that number of samples from the selected populations, plots the difference in the sample means of each sample, and displays the mean and standard deviation of the differences in the sample means compared to the theoretical predictions based on the population of differences of means of all possible samples.
 The Distribution of Sample Proportions This DIGMath module lets you investigate the distribution of the difference of proportions based on summary sample data from two samples drawn from two binomial populations. For each population, you can choose probability p of success and the sample size (from n = 2 to n = 100) from each sample, as well as the number of random samples (between 50 and 300). The simulation generates that number of samples from the two populations, plots the difference in the sample proportions of each set of samples, and displays the mean and standard deviation of the differences in the sample proportions compared to the theoretical predictions based on the population of differences of proportions of all possible samples.
 Linear Regression: Fitting a Line to Data This DIGMath module performs a linear regression analysis on any set of up to 50 (x, y) data points. It shows graphically the points and the associated regression line and also displays the equation of the regression line, the value for the correlation coefficient r, and the value for the Sum of the Squares that measures how close the line comes to all the data points.
 Regression Simulation This DIGMath spreadsheet performs a graphical simulation on the regression lines based on a variety of samples drawn from an underlying population. You have the choice of the sample size (n > 2) and the number of samples. The simulation generates the random samples, calculates the equation of and plots the corresponding sample regression line, and also draws the population regression line. It also displays the mean of the slopes and of the vertical intercepts, so that users can compare these values to the slope and intercept of the underlying population. The students quickly see that, with small sample sizes, the likelihood of the sample regression line being close to the population regression line may be very small with widely varying slopes for many of the sample lines. As the sample size increases, the sample regression lines become ever more closely matched to the population line.
 Simulating the Correlation Coefficient This DIGMath spreadsheet lets you investigate the sample distribution for the correlation coefficient r based on repeated random samples drawn from a bivariate population. You can choose between n = 3 and n = 50 random points for each sample and between 50 and 250 such samples from the underlying population. For each sample, it then calculates the correlation coefficient and displays a histogram showing the values of r from the samples. It also calculates and displays the mean of the sample correlation coefficients and compares it to the correlation coefficient for the underlying bivariate population.
 Simulating the Regression Coefficients This DIGMath module lets you investigate the sample distributions for the two regression coefficient a and b in the regression equation y = ax + b based on repeated random samples drawn from a bivariate population. You can choose between n = 3 and n = 40 random points for each sample and between 50 and 250 such samples from the underlying population. For each sample, the program calculates the regression equation and displays the various regression lines along with the regression line for the underlying bivariate population. It then draws two histograms  one showing the distribution of the values of the slope a from the random samples and the other showing the distribution of the values of the vertical intercepts b from those samples. It also calculates and displays the mean of each of the sample regression coefficients and compares it to the regression coefficients for the underlying bivariate population.
 The Effects of an Extra Point on the Regression Line and the Correlation Coefficient This DIGMath module lets you investigate the effect that changing an additional point has on the regression line (both graphically and numerically) and on the correlation coefficient. You have the choice of 5, 10, or 20 fixed points and can move an additional point using sliders to see how it affects the calculations and the extent of the changes depending on the number of points.
 Fitting a MedianMedian Line to Data This DIGMath module fits a medianmedian line to any set of up to 50 (x, y) data points. It shows graphically the points and the associated medianmedian line and also displays the equation of the medianmedian line and the value for the Sum of the Squares that measures how close the line comes to all the data points.
 Simulating the QuartileQuartile Line This DIGMath module lets you investigate the quartilequartile line that fits a set of data via a simulation. The quartilequartile line is based on finding the 1st and 3rd quartiles for both the x and the y values in a set of data and then creating the line that passes through those two points. As such, it is a viable alternative to the usual leastsquares regression line that is conceptually and computationally simpler. You have a choice of the sample size and the number of samples that will be drawn from an underlying population. The spreadsheet generates the random samples and draws all the corresponding quartilequartile lines to help you see the effect of sample size on the consistency of the lines.
 Comparing Lines that Fit Data This DIGMath program lets you compare how well the leastsquares line, the medianmedian line (that is built into many calculators), and the quartilequartile line fit sets of data. You can choose the number of random data points from an underlying population and the spreadsheet generates a random sample and displays the three lines, along with the data points, so that you can compare how well the three lines fit the data and how they compare to one another, particularly as the sample size increases.
 DataFit: Fitting Linear, Exponential, and Power Functions to Data This DIGMath spreadsheet is provided as a visual and computational tool for investigating
the issue of fitting linear, exponential, and power functions to data and the underlying
transformations used to create the nonlinear functions. You can enter a set of data
and the spreadsheet displays six graphs:
(1) For a linear fit: the regression line superimposed over the original (x, y) data;
(2) For an exponential fit: the regression line superimposed over the transformed (x, log y) data values;
(3) The exponential function superimposed over the original (x, y) data;
(4) For a power fit: the regression line superimposed over the transformed (log x, log y) data values;
(5) The power function superimposed over the original (x, y) data;
(6) All three functions superimposed over the original (x, y) data.
The spreadsheet also shows the values for the correlation coefficients associated with all three linear fits and the values for the sums of the squares associated with each of the three fits to the original data. On a separate page, the spreadsheet also shows the residual plots associated with each of the three function fits.  Multivariate Linear Regression This DIGMath spreadsheet lets you perform multivariate linear regression when the dependent variable Y is a function of two independent variables X1 and X2 or a function of three independent variables X1, X2, and X3. You enter the number of data points (up to a maximum of 50) and then the values for the dependent and independent variables in the appropriate columns. The spreadsheet responds with the equation of the associated linear regression equation, the value for the sum of the squares, and the value for the coefficient of determination, R^{2}; note that this value tells you the percentage of the variation that is explained by the linear function.
 The Birthday Problem This DIGMath module lets you investigate the Birthday Problem, which asks for the probability that two people in a group of n people will have the same birthday. The program lets you decide on the number of people in a group (between 1 and 100) and displays the graph of the probability of a match versus the number of people in the group.
 Number of Boys vs. Girls Born in a Family This DIGMath spreadsheet lets you investigate the number of Boys and Girls born into a family based on the fact that 51.2% of all live births are Boys. You can choose the number of children (110) in a family and the number of such families. The program simulates this and displays the outcomes graphically in a histogram and numerically with a table of outcomes and the mean and standard deviation of the results.
 Simulating the Birthday Problem This DIGMath module lets you investigate the Birthday Problem from the point of view of a random simulation. You have the choice of the number of people in a group (from 2 to 50). The program then generates a random sample of birthdates for each of the people and displays the list, including highlighting those that match. It presents the results, including the theoretical probability of a match and the number of matches.
 The Drunkard's (or Random) Walk Simulation This DIGMath module lets you investigate the notion of a random walk in the plane. You have the choice of the number of random steps (between 1 and 1000) and the length of each step. The program then generates a random collection of steps and displays the results graphically, as well as some numerical analysis on the actual distance covered from the starting point compared to the theoretical predictions.
 Buffon Needle Problem This DIGMath module lets you experiment with a graphical simulation of Buffon's Needle Problem  the probability that a needle of length L lands on the seam between parallel strips of flooring of width W when it falls to the floor. You can select the number of random "needles" that fall, the width of the strip of flooring, and the length of the needles and see the results graphically and numerically.
 Buffon's Disk Problem This DIGMath spreadsheet lets you experiment with a graphical simulation of a variation of Buffon's Needle Problem  the probability that a circular disk of radius r lands on the seam between parallel strips of flooring of width W when it falls to the floor. You can select the number of random "disks" that fall, the width of the strip of flooring, and the length of the needles and see the results graphically and numerically.
 Buffon's Needle Problem in Concentric Circles This DIGMath spreadsheet lets you experiment with a graphical simulation of a variation of Buffon's Needle Problem  the probability that a circular disk of radius r lands on the seam between a group of concentric circles when it falls to the floor. You can select the number of random "disks" that fall, the fixed difference in the radii of the concentric circles on the floor, and the length of the needles and see the results graphically and numerically.
 Simulation of Dart Throwing This DIGMath module lets you investigate the process of throwing random darts at a dartboard. You can select between 100 and 1000 random darts and the spreadsheet shows the position that each dart lands and displays the breakdown of how many, and what percentage, of the darts all into each of the rings in the dartboard.
 Product of the Faces of Two Dice This DIGMath simulation lets you investigate the product of the faces on a pair of dice. You can choose up to 720 rolls of the two dice and the spreadsheet shows the distribution of the outcomes and a list of the number and percentage of each possible outcome compared to the theoretical predictions.
 Simulation of Rolling 4Sided Dice This DIGMath spreadsheet lets you investigate the probability experiment of rolling a pair of fair 4sided dice, instead of the usual 6sided dice, so that the possible sums are now 2, 3, ..., 8. The simulation shows the results of repeated trials both graphically in a histogram and numerically in terms of the number of times each of the possible outcomes arises.
 Simulation of Rolling 8Sided Dice This DIGMath spreadsheet lets you investigate the probability experiment of rolling a pair of fair 8sided dice, instead of the usual 6sided dice, so that the possible sums are now 2, 3, ..., 16. The simulation shows the results of repeated trials both graphically in a histogram and numerically in terms of the number of times each of the possible outcomes arises.
 Yahtzee: Rolling Five Dice Simulation The game of Yahtzee^{TM} involves rolling a set of five fair dice. This DIGMath module lets you investigate this experiment by simulating repeated random rolls (up to 720 times) of five dice. It displays the results in a histogram as well as a table showing the simulated outcomes and the expected theoretical outcomes.
 Waiting Time Simulation This DIGMath module lets you investigate the length of time a car will wait at a red light. You can select the total length of the cycle and the length of time that the light is red. The results  the number of times that the wait is 0, 1, 2, ..., seconds  are shown graphically in a histogram and in a table listing the outcomes. The average wait over all repetitions is also shown.
 ChiSquare Analysis This DIGMath spreadsheet is designed to perform a complete chisquare analysis on many different sized contingency tables, including 2 by 2, 2 by 3, 2 by 4, 3 by 2, 3 by 3, and 3 by 4. On the SetUp screen, the user first enters the number of rows and the number of columns of the desired contingency table and is then instructed to click on an appropriate tab to go to the corresponding input screen. On that screen,you then enters the observed values into the various positions in the contingency table. The spreadsheet displays the resulting table of expected frequencies, the number of degrees of freedom, and the value of the chisquare statistic based on the values in the table. It also draws the graph of the corresponding chisquare distribution and indicates the location of the critical value, based on the desired level of significance, separating the rejection region from the region where one cannot reject the null hypothesis. In addition, the program indicates the location of the chisquare statistic corresponding to the data in the contingency table. Finally, the program indicates whether or not one can reject the null hypothesis at that significance level.
 The ChiSquare Distributions This DIGMath module lets you explore the behavior of various chisquare distributions, which depend on the number n of degrees of freedom. You can enter any desired number of degrees of freedom from 2 to 31 and the program draws the graphs of that chisquare distribution as well as those with 3, 7, 11, 15, ..., 27 degrees of freedom. Because the chisquare distributions become more normal in shape as n increases, the program also draws the standard normal distribution with mean μ = 0 and standard deviation σ = 1 for comparison.
 ChiSquare Simulation This DIGMath spreadsheet lets you investigate the variation in the values that can arise for the chisquare statistic via a random simulation based on a two by three contingency table. You define the table by entering the column and row totals and the number of random samples drawn from that population and the spreadsheet generates and graphs the corresponding values of the chisquare statistic.
 OneWay Analysis of Variance (ANOVA) This DIGMath spreadsheet is designed to perform a complete oneway analysis of variance (ANOVA) to test whether the means of two or more (up to 5 sample means) may come from populations with the same mean (the null hypothesis) or from populations with different means (the alternate hypothesis). Each sample can contain up to 10 entries. The spreadsheet displays the resulting ANOVA table, including the value of the Fstatistic. It also draws the graph of the corresponding Fdistribution and indicates the location of the critical value, based on the 5% level of significance, separating the rejection region from the region where one cannot reject the null hypothesis. In addition, the program indicates the location of the Fstatistic corresponding to the data in the table. Finally, the program indicates whether or not one can reject the null hypothesis at the 5% significance level.
 Simulating the Runs Test This DIGMath spreadsheet lets you investigate the Runs Test both graphically and numerically. When there is a collection of outcomes consisting of A's and B's, the object is to see the number of runs that occur. The module lets you select the total number of A's and B's, the number of A's, and the number of random samples. The distribution of the number of runs is drawn as well as numerical measures for the mean and standard deviation.