Ivo D. Dinov, Nicolas Christou, and Juana Sanchez
University of California, Los Angeles
Journal of Statistics Education Volume 16, Number 2 (2008), www.amstat.org/publications/jse/v16n2/dinov.html
Copyright © 2008 by Ivo D. Dinov, Nicolas Christou, and Juana Sanchez all rights reserved. This text may be freely shared among individuals, but it may not be republished in any medium without express written consent from the authors and advance notification of the editor.
Key Words: Statistics education; Technology-based blended instruction; Applets; Central Limit Theorem; SOCR.
Modern approaches for information technology based blended education utilize a variety of novel instructional, computational and network resources. Such attempts employ technology to deliver integrated, dynamically linked, interactive content and multi-faceted learning environments, which may facilitate student comprehension and information retention. In this manuscript, we describe one such innovative effort of using technological tools for improving student motivation and learning of the theory, practice and usability of the Central Limit Theorem (CLT) in probability and statistics courses. Our approach is based on harnessing the computational libraries developed by the Statistics Online Computational Resource (SOCR) to design a new interactive Java applet and a corresponding demonstration activity that illustrate the meaning and the power of the CLT. The CLT applet and activity have clear common goals; to provide graphical representation of the CLT, to improve student intuition, and to empirically validate and establish the limits of the CLT. The SOCR CLT activity consists of four experiments that demonstrate the assumptions, meaning and implications of the CLT and ties these to specific hands-on simulations. We include a number of examples illustrating the theory and applications of the CLT. Both the SOCR CLT applet and activity are freely available online to the community to test, validate and extend (Applet: http://www.socr.ucla.edu/htmls/SOCR_Experiments.html and Activity: http://wiki.stat.ucla.edu/socr/index.php/SOCR_EduMaterials_Activities_GeneralCentralLimitTheorem).
Contemporary Information Technology (IT) based educational tools are much more than simple collections of static lecture notes, homework assignments, and web-based applets posted on course-specific Internet sites. Over the past five years, a number of technologies have emerged that provide dynamic, linked, and interactive learning content with heterogeneous points-of-access to educational materials (Dinov, 2006c). Examples of such new IT resources include common web-places for course materials (BlackBoard, 2006; MOODLE, 2006), complete online courses(UCLAX, 2006), Wikis (SOCRWiki, 2006), interactive video streams (ClickTV, 2006; IVTWeb, 2006; LetsTalk, 2006), audio-visual classrooms, real-time educational blogs (Brescia & Miller, 2006; PBSBlog, 2006), web-based resources for blended instruction (WikiBooks, 2006), virtual office hours with instructors (UCLAVOH, 2006), collaborative learning environments (SAKAI), test-banks and exam-building tools (TCEXAM) and resources for monitoring and assessment (ARTIST, 2006; WebWork).
This explosion of tools and means of integrating science, education and technology has fueled an unprecedented variety of novel methods for learning and communication. Many recent attempts (Blasi & Alfonso, 2006; Dinov, 2006c; Mishra & Koehler, 2006) have demonstrated the power of this new paradigm of IT-based blended instruction. For instance, in statistics education there are a number of excellent examples where fusing new pedagogical approaches with technological infrastructure allowed instructors and students to improve motivation and enhance the learning process (Forster, 2006; Lunsford, Holmes-Rowell, & Goodson-Espy, 2006; Symanzik & Vukasinovic, 2006). In this manuscript, we build on these and other similar efforts and introduce a general, functional and dynamic central limit theorem (CLT) applet along with a corresponding hands-on activity.
There are various statements of the central limit theorem, but all of them represent weak-convergence results regarding (mostly) the sums of independent identically-distributed (random) variables (Davidson, 1994; Lyapunov, 1906). There are also versions of the CLT for dependent variables (Merlev`ede, Peligrad, & Utev, 2006). Perhaps, the most frequently used and practically important statement of the central limit theorem is in terms of (arithmetic) averages of random variables sampled from a process with well-defined and finite first two moments. In this case, CLT implies that the average will follow approximately a normal distribution, as the sample-size increases (Aberson, Berger, Healy, Kyle, & Romero, 2000). Various generalized versions of the CLT may be phrased in terms of products of random variables (DeGroot, 1970), convolution of densities (Chung et al., 2001), or as the product of characteristic functions. (Formanov, 2002) The CLT is the statistical analogue of the Grand Unified Theory in physics (Hall & Nomura, 2002). As such, the CLT is one of the most celebrated results in the field. The history of the CLT is equally as impressive as its deep scientific value (Tijms, 2004).
Many undergraduate and graduate classes use the following statement of the central limit theorem:
CLT: Let X1,…,Xn, … be a random sample (independent and identically distributed) from a (native) distribution with well-defined mean μX and variance σX2( |mX| < ∞ and sX2 < ∞). Then as n increases, the sampling distributions of the sample average
and the total sum
approach Normal distributions with corresponding
In essence, the CLT implies that the Normal distribution is the center of the universe of all nice distributions. This is the reason why we encounter so frequently estimates involving arithmetic-averaging. In a way, the pathway from a nice distribution to Normal distribution is paved by sample averages. In other words, the CLT provides a unifying framework for all (nice) distributions by facilitating explicit roadmaps from them to Normal distribution. A useful analogue to CLT is the physics Grand Unifying Theory, which similarly attempts to unite the theory behind the fundamental forces in nature.
The ramifications of the CLT go beyond the scope of this interpretation. For example, one may wonder if there are other types of population-parameters or sample-statistics that have similar limiting properties. How large does the sample size have to be to ensure normality of the sample average or total sum? Does the convergence depend on the characteristics of the native distribution (e.g., shape, center, dispersion)? How about weighted averages, non-linear combinations, or more general functions of the random sample? People exposed to the CLT frequently ask many such interesting questions. Some may have known theoretical answers (exact or approximate); other questions may be better addressed empirically by simulations and experiments (Garfield, 2002).
Several other groups have developed hands-on activities paired with interactive applets that demonstrate the foundation of the CLT. Among these efforts are the Cal Poly ISCAT collection (Lunsford et al., 2006; Rossman, Chance, & Ballman, 1999), Cal State CLT Applet (Stanton, 2005); Vienna University Learning by Simulation (Lohninger, 2008); Northwestern NetLogo modeling-and-simulation environment (Abrahamson & Wilensky, 2004); Star Library CLT Activity (Andrews, 2005); Rand CLT applet (RANDCLT, 2007), WebStat (West & Ogden, 1998) and many others. A "CLT" keyword search at www.causeweb.org yields a number of instructional plans, activities, aids and resources for technology-enhanced materials for teaching the central limit theorem.
Each of these applets, activities and resources has unique features that make it useful in its specific context. Some affirm ones intuitive beliefs, while others reinforce practice learning or address specific CLT-related questions, typically using one native distribution model. In this manuscript, native (distribution or process) refers to the underlying population characteristics.
We have extensive CLT pedagogical experience based on graduate and undergraduate teaching, interacting with students (and teaching assistants) and evaluating students’ performance in various probability and statistics classes. In our endeavors, we have used a variety of classical (e.g., mathematical formulations), hands-on activities (e.g., beads, sums, Quincunx) and technological approaches (e.g., applets, demonstrations). Our prior efforts have identified the following instructional challenges in teaching the concept of the CLT using purely classical and physical hands-on activities. Some of these challenges may be addressed by employing modern IT-based technologies, like interactive applets and computer activities:
Traditional CLT teaching techniques (symbolic mathematics and physical demonstrations) are typically restricted in terms of time and space (e.g., shown once in class) and may have the limitations of involving one native process, studying one population parameter and restricting the scope of the inference (e.g., sample-size constraints).
Modern IT-based blended instruction approaches address many of these CLT teaching challenges by utilizing the Internet and the available computational power. For example, a Java CLT applet may be evoked repeatedly under different initial conditions (choosing sample-sizes and number of experiments, native process distributions, parameters of interest, etc.). Such tests may be performed from remote locations (e.g., classroom, library, home), and may provide enhanced interactive features (e.g., fitting Normal model to sampling distribution) demonstrated in different experimental modes (e.g., intense computational vs. visual animated sampling). Such features are especially useful for active, visual and deductive learners. Furthermore, interactive demonstrations are thought to significantly enhance the learning process for some student populations (Demir, 2006; Laakso, Salakoski, Grandell, Qiu, Korhonen, & Malmi, 2005; Masters, Madhyastha, & Shakouri, 2005).
Students in probability and statistics classes are generally expected to master difficult concepts that ultimately lead to understanding the basis of data variation, modeling and analysis. For many students relying on procedural manipulations and recipes is natural, perhaps because of their prior experiences with (deterministic) Newtonian sciences. Various statistics-education researchers have experimented with technology to explore novel exploratory data-analysis techniques that emphasize making sense of data via data manipulation, visualization and simulation. Such investigators refer to statistical literacy as the process of acquiring and utilizing intuition for discovering and interpreting trends, proposing solutions and counterexamples to basic problems in probability, as well as understanding statistical data modeling and analysis (Ben-Zvi & Garfield, 2004).
Because the concepts of distribution, variation, probability, randomness, modeling and estimation are so ubiquitously used and entangled (Wild, 2006), instructors frequently forget that these notions should be defined, explained and demonstrated in (most) undergraduate probability and statistics classes. Various sampling and simulation applets and demonstrations are quite useful for this purpose.
The UCLA Statistics Online Computational Resource (SOCR) is a national center for statistical education and computing. The SOCR goals are to develop, engineer, test, validate and disseminate new interactive tools and educational materials. Specifically, SOCR designs and implements Java demonstration applets, web-based course materials and interactive aids for IT-based instruction and statistical computing (Dinov, 2006b; Leslie, 2003). Various types of users (e.g., instructors, students and researchers) may find the SOCR resources useful. The SOCR Motto, “It’s Online, Therefore It Exists!", implies that all of these resources are freely available on the Internet (www.SOCR.ucla.edu).
There are four major components within the SOCR resources: computational libraries, interactive applets, hands-on activities and instructional plans. The SOCR libraries are typically used for statistical computing by external programs (Dinov, 2006a). The interactive SOCR applets are subdivided into Distributions, Experiments, Analyses, Games, Modeler and Charts. The hands-on activities are dynamic Wiki pages (SOCRWiki, 2006) that include a variety of specific instances of demonstrations of the SOCR applets. The SOCR instructional plans include lecture notes, documentation, tutorials and guidelines about statistics education. We have decided to develop and disseminate SOCR educational resources using Wiki pages, as the Wiki infrastructure greatly facilitates building community-based, version-controlled and dynamic materials.
The goals of the SOCR CLT activity are as follows:
The SOCR CLT applet is designed as a meta-experiment (part of SOCR Experiments, integrating functionality from SOCR Distributions, Charts and Modeler). In this applet, we allow the most flexibility in terms of choosing the native process distribution, population parameters of interest, sample-sizes, number of experiments, animated and computational sampling of data. Figure 1 illustrates the main components of the applet interface. This applet may be accessed directly online by going to the SOCR Experiments (http://www.socr.ucla.edu/htmls/SOCR_Experiments.html) and selecting Sampling Distribution CLT Experiment from the drop-down list of experiments (top-left). There are tool-tips included for every widget (active graphical component) in this applet. Applet tool-tips (pop-up information fields describing interface features) are activated by bringing the mouse over a component within the applet window. The most important components of this applet are boxed in blue rectangles on Figure 1 and explained below:
o the main action buttons in the CLT experiment;
o button for refreshing the summary statistics table after completion of a sampling experiment;
o two groups of fields allowing control over the type and sample-size for each of the (two possible) parameters of interest;
o progress monitor (for large sampling tasks);
o a button to go directly to the SOCR CLT Wiki Activity (see below);
o tab-panel selection (Graphical Histogram and Numerical Summaries (default) and Distribution selection);
o four distribution graphs (native population, last sample and the sampling distributions of the two parameters of interest chosen by the user).
In addition to other SOCR libraries, this applet utilizes ideas, designs and functionality of the Rice Virtual Laboratory of Statistics (RVLS) and the University of Alabama Virtual Laboratories in Probability and Statistics (VirtualLabs).
The generality and usability of the SOCR CLT applet comes from the fact
that the user has full control over of each of the features listed above and
because the applet allows a very large number of possibilities to test and
observe the power of the central limit theorem. In particular, all of the CLT
activity goals presented in section I.f,
above, can be addressed using this applet. There is a second panel (called Distributions)
in this applet (shown on the bottom of Figure
1), which allows the selection of the native process we wish to sample
from. In the CLT Applet, the user may manually generate a
distribution or choose one from the 50+ families of SOCR Distributions
(including commonly used distributions like the Binomial and
The SOCR CLT activity is available online (http://wiki.stat.ucla.edu/socr/index.php/SOCR_EduMaterials_Activities_GeneralCentralLimitTheorem) and accessible from any internet-connected computer with a Java-enabled web browser. The activity includes dynamic links to web resources on the CLT, interactive CLT demonstrations, and relevant SOCR resources.
Following the outline of its goals, the CLT experimental settings are described online in the second paragraph of the SOCR CLT Wiki activity. Each of the four experiments suggested in the activity addresses different feature sets of the applet and demonstrates separate characteristics of the CLT.
The first experiment depicts the effects of small sample-sizes on the sampling distributions of the sample average and the sample variance, Figure 2. The arrows in this figure point to the most important features in this experiment (left-to-right and top-to-bottom):
o button is for taking one sample from the selected distribution;
o button is for recomputing the sample and sampling distribution summary statistics in the tables to the right;
o two drop-down lists allow the user to select the sample-sizes for one or two sampling distributions;
o second graph panel shows the distribution of the last sample;
o third graph panel represents the sampling distribution of the first parameter (mean, in this case);
o the bottom graph panel shows the sampling distribution of the second parameter (variance); and
o the data tables on the right show the summary statistics for all 4 distributions (native, sample, sampling (mean) and sampling (variance)).
The user observes graphically the process of random sampling from a
distribution (in this case
This experiment draws parallels between the sampling distribution statistics (rows 3 and 4) and their analogues from the native population (on the top row). For example, the mean of the multiple sample-averages is about the same as the mean of the native population. On the other hand, the standard deviation of the sample-averages is about , where σ is the standard deviation of the original native process.
The third experiment involves sampling from an arbitrary SOCR Distribution, Figure 4. This demonstrates that the central limit theorem is valid for numerous families of distributions. This experiment may be used to empirically validate that the sample average is a unique data statistics that has invariant limiting of its sampling distribution. In addition, the convergence of the sampling distribution to a Normal may be validated, relative to the chosen sample-size.
The fourth experiment in this activity illustrates how to generate and sample from a new distribution not included in the list of SOCR Distributions, Figure 5. The user may draw the shape of a hypothetical distribution by clicking and dragging the mouse in the top graphing canvas on the applet. This functionality allows exploring the assumptions and convergence guaranteed by the CLT for continuous and discontinuous, symmetric and asymmetric, unimodal and multi-modal, leptokurtic and mesokurtic, and other types of distributions or processes. Also, we explore the differences between two data-driven estimates for the population centrality – the sample mean and the sample median.
There are a variety of ossible applications of these resources in a general classroom setting. Instructors should consider fine-tuning these materials to their specific course curricula. In the past, we have experimented in a number of ways with these materials. A step-by-step of illustration of one possible strategy for using these resources in a classroom is described below. We first choose a distribution that we want to start with (typically Normal or constructed by clicking and dragging the mouse in the top graphing panel). Then the instructor begins an interactive discussion with the students, which involves applet usage, CLT properties, examples and counterexamples, student feedback and computer simulations:
Such questions to be discussed are:
The instructor builds complexity as he or she is satisfied with students’ responses, attentiveness and comprehension. A homework assignment that reinforces these CLT principles is appropriate in many cases and may improve knowledge retention.
There are many practical examples using CLT to solve real-life problems. Here are some instances, which use CLT to approximate probabilities of interests and can be solved using SOCR resources.
The last quantity is obtained using the SOCR Normal Distributions applet, without using continuity correction. Using continuity correction the approximation improves, P(T<300 | Arguably, the CLT-based calculation is less intense and more appealing to students and trainees, compared to computing the exact probability.
Denote Notice that in this case the exact distribution of is (generally) not exponential, even though the density may be computed in closed form (Khuong & Kong, 2006). If we use the CLT, however, we can approximate the probability of interest
as we know that the mean and the standard deviation of and the standard deviation of Therefore, , using the CLT approximation and the SOCR Normal Distributions calculator, Figure 6.
The exact probability of interest may be computed by using the Binomial Distribution. If the total casino return in 100 games is denoted by , then the expected casino return in 100 games is $5.26 and P(T>28)=P(X>k), where and k is the integer solution of the following dollar amount equation: k × $1 - (100-k) × $35 = $28, k=98. Therefore, P(T ≥ 28)=P(X ≥ 98)=0.508326. The last probability represents the exact solution and is computed using the SOCR Binomial Distribution applet (http://www.socr.ucla.edu/htmls/SOCR_Distributions.html). This exact calculation is numerically intractable for large sample-sizes (n>200), even though approximations exist. Again, one could use the CLT to find a very good approximation to this type of probabilities. For example, in the case above (n=100), we can estimate the probability of interest by
Notice that this calculation is sample-size independent, and hence widely applicable, whereas the former exact probability calculation is limited for small n. What caused the large discrepancy between the exact and approximate values of the probability of interest: P(T ≥ 28)= 0.508326 and P(T ≥ 28) ≈ 0.347? This is an example where the usual rule of "30 measurements" breaks, because of the skewed underlying Binomial distribution. Such limitations of the CLT even for large sample-sizes have been previously observed and reported for severely skewed distributions (Freedman, Pisani, & Purves, 199). Here one would need much larger sample to get a reasonably good approximation using CLT. For example, if n=1,000 and we are looking for P(T>100), then k=975, the exact probability is P(T>100)=P(X>975)=0.4493287, and the CLT approximation is much closer:
Finally, we show how one can use the SOCR CLT applet alone to completely empirically estimate the probability of interest, P(T>28), for n=100. Figure 8 demonstrates how one can manually construct the native probability mass function for the random variable Y (casino payoff of one roulette game). A simple linear transformation is needed to convert the values of Y to W , so that the range of the Y variable [-35 : 1] may be mapped to the default range of the CLT Applet native distribution [0 : 32], W. Now, recalling the definitions above (for the n=100 case) we have that , see Figure 8. The last equality is obtained by noticing that W will have approximately Normal (μ = 31.23332, σ2=0.488118952) distribution, with empirical mean and standard deviation obtained from row 3 in the table.
A key feature of the SOCR CLT activity is that it demonstrates the fact that, most of the time, the native process distribution is unknown to the user. If one assumes that a curtain hides away the native population, one may only observe samples randomly drawn from the unknown process. A good analogy of this is an experiment of observing shadows (samples) of an object (process of interest) projected under different light directions. One may want to answer questions about the object (e.g., its center) in terms of the observed shadow projections (e.g., sample-driven estimates of centrality). Then the CLT allows us to quantify how biased and dispersed are these sample-driven estimates of the object-specific parameter of interest (e.g., its center-of-mass). In the case that the question of interest is the object’s center-of-mass, the CLT implies that averaging the x- and y-coordinates of points in the 2D shadow projections yields unbiased estimate of the object center-of-mass, and this estimate becomes tighter as the number of projections increase.
There are two unique characteristics of the SOCR CLT applet and activity. The first one is the large number of families of distributions that are incorporated in the CLT applet. This feature may be used to clearly justify the quasi-independence of the CLT result from the distribution of the native process. The second distinct aspect of this activity is the broad spectrum in which the CLT can be demonstrated. An instructor may easily wrap his or her own interpretation of the CLT, tie it to their course curriculum within the SOCR Wiki framework and quickly make this available to their students and the community. For example, one instructor may want to stress the effects of the sample-sizes, whereas another may emphasize the properties of the chosen sample-statistics. And still another may focus just on the random sampling, histogram plotting, counterexamples or sample-size reasoning, all within the same SOCR CLT applet-activity framework.
The SOCR resource has been continuously developing other similar hands-on activities. All of these activities are paired with one or more SOCR applets and typically illustrate one possible approach for demonstrating a probability or statistics concept via a SOCR distribution, experiment, analysis, graphing or modeling applet. Most general topics covered in lower and upper division probability theory or statistical inference courses already have available SOCR Wiki activities (http://wiki.stat.ucla.edu/socr/index.php/SOCR_EduMaterials). For example, there are two other SOCR CLT activities, which involve the Binomial and Gamma distributions. There are many other instances of SOCR activities covering the distributions, analyses, experiments, graphs and modeler applets. Finally, there are also some SOCR applets and activities, which may be used in more advanced undergraduate and graduate level classes, e.g., law of large numbers, mixture modeling, expectation maximization, Fourier and wavelet signal representation, etc.
These new SOCR CLT resources were developed with support from NSF DUE grants 0716055 and 0442992, and from NIH Roadmap for Medical Research, NCBC Grant U54 RR021813. We are also indebted to the anonymous reviewers that contributed significantly to improve the presentation of the manuscript. We also thank Annie Chu, Jenny Cui, Priscilla Chui, Rahul Gidwani, Ariana Anderson and Robert Gould for their improvement ideas and help with validation.
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