Ivo D. Dinov, Nicolas Christou and Robert Gould

University of California, Los Angeles

Journal of Statistics Education Volume 17, Number 1 (2009), ww2.amstat.org/publications/jse/v17n1/dinov.html

Copyright © 2009 by Ivo D. Dinov, Nicolas Christou and Robert Gould, 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; Law of
large numbers; Limit theorems; SOCR.

Modern approaches for technology-based blended education utilize a variety of recently developed novel pedagogical, computational and network resources. Such attempts employ technology to deliver integrated, dynamically-linked, interactive-content and heterogeneous learning environments, which may improve student comprehension and information retention. In this paper, we describe one such innovative effort of using technological tools to expose students in probability and statistics courses to the theory, practice and usability of the Law of Large Numbers (LLN). We base our approach on integrating pedagogical instruments with the computational libraries developed by the Statistics Online Computational Resource (www.SOCR.ucla.edu). To achieve this merger we designed a new interactive Java applet and a corresponding demonstration activity that illustrate the concept and the applications of the LLN. The LLN applet and activity have common goals – to provide graphical representation of the LLN principle, build lasting student intuition and present the common misconceptions about the law of large numbers. Both the SOCR LLN applet and activity are freely available online to the community to test, validate and extend (Applet: http://socr.ucla.edu/htmls/exp/Coin_Toss_LLN_Experiment.html, and Activity: http://wiki.stat.ucla.edu/socr/index.php/SOCR_EduMaterials_Activities_LLN).

Contemporary Information Technology (IT) based educational tools are much more than simply collections of static lecture notes, homework assignments posted on one course-specific Internet site and web-based applets. 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 these 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 of learning (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, Christou, & Sanchez, 2008; Dinov, 2006c; Mishra & Koehler, 2006) have demonstrated the power of this new paradigm of technology-based blended instruction. In particular, in statistics education, there are a number of excellent examples where fusing new pedagogical approaches with technological infrastructure has 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 law-of-large-numbers (LLN) applet along with a corresponding hands-on activity.

Suppose we conduct independently the same experiment over and over again. And assume we are interested in the relative frequency of occurrence of one event whose probability to be observed at each experiment is *p*. Then the ratio of the observed sample frequency of that event to the total number of repetitions converges towards *p* as the number of (identical and independent) experiments increases. This is an informal statement of the LLN.

Consider another example where we study the average height of a class of 100 students. Compared to the average height of 3 randomly chosen students from this class, the average height of 10 randomly chosen students is most likely
closer to the *real average* height of all 100 students. This is true because the sample of 10 is a *larger number* than the sample of only 3 and better represents the entire class. At one extreme, a sample of 99 of the 100 students will produce a sample average height almost exactly the same as the average height for all 100 students. On the other extreme, sampling a
single student will be an extremely variant estimate of the overall class average height.

The two most commonly used symbolic versions of the LLN include the *weak*
and* strong* laws of large numbers.

The statement of the *weak law of large numbers* implies that the average of a random sample converges in
probability towards the expected value as the sample size increases. Symbolically,

.

For a given *ε > 0*, this convergence in *probability* is defined by

.

In essence, the weak LLN says that the average of many observations will eventually be within any margin of error of the population mean, provided we can increase the sample size.

As the name suggests, the *strong law of large numbers* implies the weak LLN as it relies on almost sure
(a.s.) convergence of the sample averages to the population mean. Symbolically,

, i.e., .

The strong LLN explains the connection between the population mean (or expected value) and the sample average of independent observations. Motivations and proofs of the weak and strong LLN may be found in (Durrett, 1995; Judd, 1985).

It is generally necessary to draw the parallels between the formal LLN statements (in terms of sample
averages and convergence types) and the frequent interpretations of the LLN (in terms of probabilities of various events). The (strong) LLN implies that the sample proportion converges to the true proportion *almost surely*. One
practical interpretation of this convergence in terms of the SOCR LLN applet is the following: If we repeat the applet simulation a fixed, albeit large, number of times, we will *almost surely* observe a sequence that does *not*
appear to converge. However, almost all sequences will appear to converge – and this behavior would normally be observed when the applet simulation is run. Of course, the probability of observing a non-convergent behavior is trivial when
running the applet in *continuous* mode, without a limit on the sample size.

Suppose we observe the same process independently multiple times. Assume a binarized (dichotomous) function of the outcome of each trial is of interest. For example, *failure* may denote the event that the continuous voltage measure < 0.5V, and the complement, *success*, that voltage ≥ 0.5V. This is the situation in electronic chips, which perform arithmetic
operations by binarizing electric currents to 0 or 1. Researchers are often interested in the event of observing a success at a given trial or the number of successes in an experiment consisting of multiple trials. Let’s denote *p=P(success)*
at each trial. Then, the ratio of the total number of successes in the sample to the number of trials (*n*) is the average

, where

represents the outcome of the *i ^{th}* trial. Thus, the
sample average equals the sample proportion (). The sample proportion (ratio of the observed frequency of that event to the total number of repetitions) estimates the true

There are several attempts to provide interactive aids for LLN instruction and motivation. Among these are the fair-coin applet experiment developed by Sam Baker and the University of South Carolina (http://hspm.sph.sc.edu/COURSES/J716/a01/stat.html); and the applet introduced by Philip Stark at University of California, Berkeley, which allows user control over the probability of success and the number of trials a coin is tossed (http://stat-www.berkeley.edu/~stark/Java/Html/lln.htm). Many other LLN tutorials, applets, activities and demos may be discovered at the CAUSEweb site (http://www.causeweb.org/cwis/SPT--QuickSearch.php?ss=law+of+large).

There are two distinct challenges in teaching the LLN and these are related to the theory and practice of these laws. The theoretical difficulties arise because complete understanding of the LLN fundamentals may require learners’ familiarity with different types of limits and convergence. The practical utilization barriers are centered around the two main LLN empirical misconceptions (Garfield, 1995; Tversky & Kahneman, 1971): (1) In a fair-coin toss experiment, if we observe a long streak of consecutive heads (or tails), then the next flip has a better than 50% probability of landing tails (or heads); (2) In a large number of coin tosses, the number of heads and number of tails become more and more equal.

These challenges may be addressed by employing modern IT-based technologies, like computer applets and interactive activities. Such resources provide ample empirical evidence by allowing multiple repetitions and arbitrary
sample sizes. Applets and activities also expose the scope and the limitations of theoretical concepts by enabling the user to explore the effects of parameter settings (e.g., varying the values of *p= P(Head)*) and to study
the resulting summary statistics (e.g., graphical or tabular outcomes). Interactive graphical applets also address conceptual challenges by enabling hands-on demonstrations of process *limiting* behavior and various types
of convergence.

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). SOCR resources may be utilized by instructors, students and researchers. 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; Dinov et al., 2008). The interactive SOCR applets (top of http://socr.ucla.edu/) are subdivided into Distributions, Functors, Experiments, Analyses, Games, Modeler, Charts and Applications. 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, documentations, tutorials, screencasts and guidelines about statistics education.

The goals of this activity are:

- To illustrate the theoretical meaning and practical implications of the LLN;
- To present the LLN in varieties of situations;
- To provide empirical evidence in support of the LLN-convergence and dispel the common LLN misconceptions.

The SOCR LLN applet is designed as a meta-experiment (integrating functionality from SOCR Experiments and Distributions). In this applet, we provide the flexibility for choosing the number of trials and altering the probability of
the event (in the meta-experiment of observing the frequencies of occurrence of the event in repeated independent trials). Figure 1 illustrates the main
components of the applet interface. This applet may be accessed directly online
at http://socr.ucla.edu/htmls/exp/Coin_Toss_LLN_Experiment.html
(or from the main SOCR Experiments page: http://socr.ucla.edu/htmls/exp). There
are tool-tips included for every widget in this applet. The tool-tips are
pop-up information fields which describe interface features, and are activated
by bringing the mouse over a component within the applet window. The control-toolbar
of this applet, top of Figure 1, is included here as an insert . This toolbar contains the following experiment action control-buttons (left-to-right):
running a single-step trial, running a multi-trial experiment, stopping of a
multi-trial experiment, experiment resetting, frequency of updating the results
table, number of trials to run (*n*), and an information dialog about the
LLN experiment. Note that in the applet, the probability of a Head (*p*)
and the number of experiments (*n*) are selected by the *Probability-of-heads*
slider (defaulted to *p=0.5*) and the *Sample-size/Stop-frequency*
drop-down list (defaulted to "*Stop 10*"), respectively.

In addition to other SOCR libraries, this applet utilizes ideas, designs and functionality from the Rice Virtual Laboratory of Statistics (RVLS) and the University of Alabama Virtual Laboratories in Probability and Statistics (VirtualLabs).

The SOCR LLN activity is available online (http://wiki.stat.ucla.edu/socr/index.php/SOCR_EduMaterials_Activities_LLN) and accessible from any Internet connected computer with Java-enabled web browser. The activity includes dynamic links to web resources on the LLN, interactive LLN demonstrations and relevant SOCR resources.

**2.2.1** The *first LLN experiment* illustrates the statement and validity
of the LLN in the situation of tossing (biased or fair) coins repeatedly, Figure
2. The arrows in Figure 2 point to the applet URL, and the main experimental controls – action-buttons (top), LLN experiment selection from the
drop-down list (left), probability of a Head slider and graph-line (middle), and the headings of the results table (bottom). If H and T denote Heads and Tails, the probabilities of observing a Head or a Tail at each trial are *0
≤ p ≤ 1* and *0 ≤ 1 − p ≤ 1*, respectfully. The sample space of this experiment consists of sequences of H's and T's. For instance, an outcome may be {H,H,T,H,H,T,T,T,....}. If we toss a
coin *n* times, the size of the sample-space is *2 ^{n}*, as the coin tosses are independent. The probability of observing

To start the SOCR LLN Experiment go to http://socr.ucla.edu/htmls/exp/Coin_Toss_LLN_Experiment.html
(note that the *Coin Toss LLN Experiment* is automatically selected in the drop-down list of experiments on the top-left). Now, select number of trials *n=100* (Stop 100) and *p=0.5* (fair-coin). Each time the user runs the applet,
the random samples will be different, and the figures and results will
generally vary. Using the *Step* () or *Run* ()
buttons will perform the experiment one or many times. The proportion of heads in the sample of *n* trials, and the difference between the number of Heads and Tails will evolve over time as shown on the graph and in the results table
below. The statement of the LLN in this experiment reduces to the fact that as the number of experiments increases, the sample proportion of Heads (red curve) will approach the user-preset theoretical (horizontal blue line) value of *p*
(in this case *p=0.5*). Changing the value of *p* and running the
experiment interactively several times provides evidence that the LLN is invariant
with respect to *p*.

In the SOCR LLN activity, the learner is encouraged to explore deeper the LLN properties by fixing a value of *p* and determining the sample-size needed to ensure that the sample-proportion stays within certain limits. One
may also study the behavior of the curve representing the difference between the number of Heads and Tails (red curve) for various *n *and* p* values, or examine the convergence of the sample-proportion to the (preset) theoretical
proportion. This is a demonstration of how the applet may be used to show theoretical concepts like convergence type and limiting behavior. The second misconception of the LLN (section **1.4**) may be empirically dispelled by
exploring the graph of the second variable of interest (the difference between the number of Heads and Tails). For all integers *n*, the independence of the *(n+1)*^{st} outcome from the results of the first *n* trials is also evident by the
random behavior of the outcomes and the unpredictable and noisy shape of the graph of the *normalized* difference between the number of Heads and Tails. In fact, the number of Heads minus Tails difference is so chaotic, unstable and divergent that it can not be even plotted on the same scale as the plot of the Heads-to-Tails ratio.

We have defined an interesting *normalization* of the raw differences between the number of Heads and Tails to demonstrate the stochastic shape of this unstable variable. The normalized difference between the number of Heads and Tails in the graph is defined as follows: First, we let *H _{k}* and

where the maximum *raw difference* between the number of Heads and Tails in the first *k* trials is defined by

.

Since *E((1 &minus p) H _{k} &minus pT_{k}) = 0*, the expectation of the numerator is trivial, we know that the normalized difference will oscillate around

**2.2.2** The *second LLN experiment* uses the Binomial coin applet to
demonstrate that the (one-parameter) empirical and theoretical distributions of
a random variable become more and more similar as the sample-size increases. **Figure
4** shows the Binomial coin experiment (http://socr.ucla.edu/htmls/exp/Binomial_Coin_Experiment.html).
Again, arrows and highlight-boxes identify the applet URL and main components
(top to bottom): action control buttons, variable selection list, graph
comparison between the model and empirical distributions, quantitative results
table, and quantitative comparison between the theoretical and sample empirical
distributions.

The user may select the number of coins (e.g., *n=3*) and probability of heads (e.g., *p=0.5*). Then, the right panel shows in blue color the model distribution (Binomial) of the *Number of Heads* (*X*). By varying the probability (*p*) and/or the number of coins (*n*), we see how these parameters affect the shape of the model distribution. As *p*
increases, the distribution moves to the right and becomes concentrated at the right end (i.e., left-skewed). As the probability of a Head decreases, the distribution becomes right-skewed and centered in the left end of the range of *X*
(*0 ≤ X ≤ n*). The LLN implies that if we were to increase the number of experiments (*N*), say from *10* to *100*,* * and then to* 1,000*, we will get a better fit between the theoretical (Binomial)
and empirical distributions. In particular, we get as a better estimate of the probability of a Head (*p*) by the sampling proportion of Heads (),
cf. section **1.2**. And this convergence is guaranteed for each *p* and each *n* (number of trials within a single experiment). Note that in this applet, the Binomial distribution parameters (*n, p*) are controlled
by *Number-of-coins* (*n*) and *Probability-of-heads* (*p*)
sliders, and the number of experiments we perform is selected by the *Stop-frequency* drop-down list in the control-button toolbar. Figure 5 illustrates this improved match of the theoretical (blue) and empirical (red) distribution
graphs as the number of experiment (*N*) increases.

Similar improvements in the match of the theoretical and empirical distributions may be observed for many other processes modeled by one-parameter distributions using the other SOCR Experiments. For instance, in the *Ball
and Urn Experiment*, where one may study the distribution of *Y*, the number of red balls in a sample of *n* balls (with or without replacement), we see similar LLN effects as the number of experiments (*Stop-frequency*
selection) increases.

Another LLN illustration is based on the SOCR *Poker* experiment where one may be interested in how many trials are needed (on average) until a single pair of cards (same denomination) is observed. We can demonstrate the
LLN by running the Poker experiment 100 times and recording the number of trials (5-card hands) containing a single pair (indicated by a value of *V=1*). Dividing this number by the total number of trials (100) we get a sample proportion
of single-pair trials (),
which will approximate *p*, the theoretical probability of observing a single pair. The same experiment can be repeated with stopping criterion being set to *V=1*, instead of a specific (fixed) number of trials. Then, the
(theoretical) expectation of the number of trials (*n*) needed to observe the first success (single-pair hand) is *1/p*, as this is a Geometric process. This expectation can be approximated by the (empirical) number of
trials ()
to observe the first single-pair hand. In this case, the LLN implies that the
more experiments we perform, the closer the estimates of and to
the theoretical values of their counterparts, *p* and *n*.

There are a number of applications of the LLN (DeHon, 2004; Rabin, 2002; Uhlig, 1996). The following two LLN applications demonstrate *estimation of transcendental numbers* for the two most popular transcendental numbers –*– π* and *e*.

There are a number of equivalent definitions for the natural number *e*
(http://en.wikipedia.org/wiki/E_(mathematical_constant)).
One of these is

.

Using simulation, the number *e* may be estimated by random sampling from a continuous Uniform distribution on (0, 1). Suppose *X _{1},X_{2}, ... ,X_{k}* are drawn from a uniform distribution on (0, 1) and
define

.

It turns out that the expected value ,
(Russell, 1991). Therefore, by the LLN, taking averages of *U _{1}, U_{2}, U_{3}, ... , U_{n}*, where each

as described above, will provide a more accurate estimate of the natural
number *e *(as ).
The SOCR E-Estimate Experiment (http://wiki.stat.ucla.edu/socr/index.php/SOCR_EduMaterials_Activities_Uniform_E_EstimateExperiment)
provides an activity with the complete details of this simulation and the
corresponding *Uniform E-Estimate Experiment* applet http://socr.ucla.edu/htmls/exp/Uniform_E-Estimate_Experiment.html.
The experiment illustrates hands-on this stochastic approximation of *e*
by random sampling. **Figure 6** shows the graphical user interface behind this simulation. The arrows in Figure 6 point to the applet selection
from the drop-down list of applets (left), the results table containing the outcome of 10 simulations (middle) and the (theoretical and empirical) estimate of the bias and precision (MSE) of the approximation of *e* by random
sampling (right). The graph panel in the middle shows the values of the Uniform(0,1) sample, *X _{1},X_{2}, ... ,X_{k}*, used to compute the values of

The SOCR *Buffon’s Needle Experiment* provides a similar approximation to *π, *which represents the ratio of the circumference of a circle to its diameter, or equivalently, the ratio of a circle's area to
the square of its radius in 2D Euclidean space. The LLN provides a foundation for an approximation of *π* using repeated independent virtual drops of needles on a tiled surface by observing if the needle crosses a tile
grid-line. For a tile grid of size 1, the odds of a needle-line intersection
are ,
(Schroeder, 1974). In practice, to estimate *π* from a number of needle drops (*N*), we take the reciprocal of the sample odds-of-intersection. The complete details of this application are also available online at http://wiki.stat.ucla.edu/socr/index.php/SOCR_EduMaterials_Activities_BuffonNeedleExperiment
and http://socr.ucla.edu/htmls/SOCR_Experiments.html.

One specific approach for teaching the concept of the LLN in various classes is to follow
the SOCR LLN activity step-by-step. Instructors would typically begin by presenting one or two motivational examples. Then, as appropriate, the instructor may discuss the formal statement(s) of the LLN, and perform several
simulations using the LLN applet. Students may then be given 5-10 minutes to explore the *hands-on* section of the LLN Wiki activity on their own by following the directions and answering the questions. Instructors must follow
up with a question-and-answer period to ensure all students understand the mechanics, purpose and applications included in the LLN activities. Finally, it is appropriate to encourage an open classroom discussion, utilizing the LLN
applet, explaining the two LLN misconceptions.

These new SOCR LLN resources were developed with support from NSF grants 0716055 and 0442992 and from NIH Roadmap for Medical Research, NCBC Grant U54 RR021813. The authors are indebted to Juana Sanchez for her valuable comments and ideas. JSE editors and referees provided a number of constructive recommendations, revisions and corrections that significantly improved the manuscript.

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Ivo D. Dinov, Ph.D.

Department of Statistics and Center for Computational Biology

University of California, Los Angeles

Los Angeles, CA 90095

Tel. 310-267-5075

Fax: 310-206-5658

Dinov@stat.ucla.edu

Nicolas Christou, Ph.D.

Department of Statistics

University of California, Los Angeles

Los Angeles, CA 90095

Tel. 310-825-8430

Fax: 310-206-5658

NChristo@stat.ucla.edu

Robert Gould, Ph.D.

Department of Statistics

University of California, Los Angeles

Los Angeles, CA 90095

Tel. 310-285-8430

Fax: 310-206-5658

RGould@stat.ucla.edu

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