Explore the World of Software: Engineering & Science

home *** CD-ROM | disk | FTP | other *** search

/ Explore the World of Soft…e: Engineering & Science / Explore_the_World_of_Software_Engineering_and_Science_HRS_Software_1998.iso / programs / statistc / cltw9404.exe / CLTWIN.DOC < prev next >

Wrap

Text File | 1994-04-15 | 43KB | 906 lines

┌──────────────────────────────────────────────────────────┐ │ ┌──────────────────────────────────────────────────────┐ │ │ │ Central-Limit-Theorem Simulator │ │ │ │ CLTWIN.EXE...Version 9404 │ │ │ ├──────────────────────────────────────────────────────┤ │ │ │ A Quality Assurance Training Tool: │ │ │ │ Statistics Committee of the QA Section of the PMA │ │ │ ├──────────────────────────────────────────────────────┤ │ │ │ Bob Obenchain, CompuServe User [72007,467] │ │ │ │ 5261 Woodfield Drive North, Carmel, IN 46033 │ │ │ └──────────────────────────────────────────────────────┘ │ └──────────────────────────────────────────────────────────┘ The Central-Limit-Theorem Simulator, CLTWIN, is a Microsoft Windows application with several potential uses in computer-based training for "statistical thinking." CLTWIN uses Monte-Carlo simulation techniques with distributions of four different basic shapes to display colorful histograms of the resulting AVERAGES or SAMPLES accumulated into 10, 20 or 40 cells covering the range from -5.0 to +5.0. For example, CLTWIN software can be used in hands-on workshops as well as in self-paced learning to illustrate basic statistical concepts... [1] The statistical distribution of the average of independent random variables tends to become more and more "bell- shaped" as N, the number of observations in the average, increases [Central-Limit-Theorem], and... [2] The variance of an average of N independent, identically distributed random variables is inversely proportional to N. CLTWIN.DOC Table of Contents: ============================= (i) Introduction to the Central-Limit-Theorem & Statistical Concepts. (ii) CLTWIN Component Files and Main-Menu Items. (iii) CLTWIN Learning Points and/or Instructional Objectives. (iv) References and Update History Central-Limit-Theorem Simulator for Windows Page 2 of 21 ============ <<< INTRODUCTION >>> ============ The primary message commonly associated with the "Central Limit Theorem" is this: The distribution of AVERAGES of observations taken from almost any well-behaved distribution will approach a Normal distribution in the limit as the sample size, N, approaches infinity. Technically speaking, when the observations being averaged are independent and identically distributed, a sufficient condition for the Central-Limit-Theorem to hold is that the parent distribution has a positive (but finite) variance. CLTWIN displays histograms of observations from 4 distributions... Shape = 1: Uniform...over the Range from -5 to +5, Shape = 2: Skewed....a Right Triangle with Mode at +5, Shape = 3: Bimodal...2 Triangles with Modes at -5 and +5, Shape = 4: Normal....with mean = 0 and variance = 16. To illustrate Central-Limit-Theorem concepts, the CLTWIN Histograms and Summary Statistics of primary interest are those relating to... ================================================================ AVERAGES of between 2 and 100 values from these 4 distributions. ================================================================ CLTWIN histograms contain 10, 20 or 40 cells covering the range from -5 to +5; the resulting cell-widths are thus 1.00, 0.50 or 0.25 units. CLTWIN illustrates that the distribution of averages may "appear" to be approximately normal for sample sizes as small as only N = 4 or 5. But, for "pathological" parent distributions (such as the skewed and bimodal cases implemented in CLTWIN), a much larger sample size may need to be averaged before the resulting distribution will tend to be both symmetrical and smoothly bell-shaped. Some potential users of CLTWIN may feel a need to be convinced that they can "trust" the insights they will develop using CLTWIN. Any users wishing to read a discussion of technical issues related to pseudo-random number generation, including extensive literature references, are referred to the ASCII documentation file, SSSWIN.DOC, that is provided with the "Six Sigma Simulator" module ...also authored by Bob Obenchain. Central-Limit-Theorem Simulator for Windows Page 3 of 21 =========================================== <<< CLTWIN Component Files and "MainMenu" ITEMS >>> =========================================== Files associated with the Central-Limit-Theorem Simulator are: CLTWIN.EXE ...Main Windows Executable Module CLTWIN.DOC ...This ASCII Documentation File CLTWIN.PAR ...Default Parameter Settings File CLTWIN.SAV ...Simulation Results (Frequencies,Bar Charts) CLT-RAND.C ...C-language source code related to pseudo- random number generation algorithms. CLTWIN.EXE may be copied to the sub-directory of your choice on your personal computer's hard disk. In the installation instructions detailed below, we assume that the Drive:\Path of your choice is C:\WINDOWS. Next, you should then use the "File/New.../Program" menu item from within Microsoft Windows to define file properties such as... Description: CenLimThm Command Line: cltwin.exe Working Directory: c:\windows Once installed in this way, you would then double-click on the ICON named "CenLimThm" to launch CLTWIN. <<< CLTWIN Main Menu ITEMS: >>> Histograms =--------- Distribution of Averages ...histogram showing the shape of the accumulated distribution of AVERAGEs. Distribution Being Sampled ...histogram showing the shape of the "parent" distribution being sampled. Values in Last Sample ...histogram containing only the observed values in the last sample generated. Sample =----- Number of Values Averaged = Sample Size = N ...Dialog EDIT Box ┌──── Shape of Distribution Being Sampled ───────┐ 2 <= ssize <= 100 │ Uniform --- over Range from -5 to +5 │ │ Skewed ---- Right Triangle with Mode at +5 │...Check the shape │ Bimodal --- 2 Triangles with Modes at -5 and +5│ you wish by │ Normal ---- Mean = 0 and Variance = 16 │ clicking on its └────────────────────────────────────────────────┘ Menu Sub-Item. Central-Limit-Theorem Simulator for Windows Page 4 of 21 Key to SHAPES of CLTWIN Parent Distributions... =============================================== Shape = 1: Uniform over the Range from -5 to +5... █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ -5 -4 -3 -2 -1 0 1 2 3 4 5 Shape = 2: Right Triangular with... Maximum Frequency at +5 (right of screen) and Zero Frequency at -5 (left of screen) ▄ ▄ ▄ ▄ █ █ █ █ ▄ ▄ █ █ █ █ █ █ █ █ ▄ ▄ █ █ █ █ █ █ █ █ █ █ █ █ ▄ ▄ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ ▄ ▄ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ ▄ ▄ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ _ ▄ ▄ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ -5 -4 -3 -2 -1 0 1 2 3 4 5 Shape = 3: Bimodal (Pair of Triangles) with Modes at both +5 and -5 and Zero Frequency at 0. ▄ ▄ █ █ ▄ ▄ █ █ █ █ █ █ ▄ ▄ █ █ █ █ █ █ █ █ █ █ ▄ ▄ █ █ █ █ █ █ █ █ █ █ █ █ █ █ ▄ ▄ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ ▄ ▄ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ ▄ ▄ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ ▄ ▄ █ █ █ █ █ █ █ █ █ █ █ █ █ █ -5 -4 -3 -2 -1 0 1 2 3 4 5 Shape = 4: Normal with mean 0 & variance 16 (std.deviation 4)... ▄ █ ▄ ▄ █ █ █ ▄ ▄ █ █ █ █ █ ▄ ▄ █ █ █ █ █ █ █ ▄ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ _ ▄ █ █ █ █ █ █ █ █ █ █ █ █ █ ▄ _ -12 -8 -4 0 4 8 12 │ │ │<── 68% ──>│ │ │ │ │<──────── 95% ────────>│ │ │<───────────── 99.7% ─────────────>│ Central-Limit-Theorem Simulator for Windows Page 5 of 21 Note, however, that only the range from -5 to +5 of this unbounded distribution (i.e. only the middle 79% of the parent distribution) is displayed by CLTWIN.EXE in its "Distribution being Sampled" histogram... ▄ ▄ ▄ █ █ █ ▄ ▄ ▄ ▄ ▄ ▄ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ ▄ ▄ ▄ ▄ ▄ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ ▄ ▄ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ █ -5 -4 -3 -2 -1 0 1 2 3 4 5 Cell Width =--------- One Unit => Maximum ...Use "widest" histogram cell widths. Half Unit => Middle Width ...Use "intermediate" cell widths. Quarter Unit => Minimum ...Use "narrow" histogram cell widths. NOTE: In addition to their primary purpose of determining histogram "Cell Widths", each of these menu sub-items also discards all previously accumulated results (without saving them in the CLTWIN.SAV file) and resets the simulation to zero. Central-Limit-Theorem Simulator for Windows Page 6 of 21 Other =---- Compare Variation Statistics... this item "Pops Up" the dialog box: ┌───────────────────────────────────────────────┐ │ Number of Values Averaged = N = │ │ │ │ Sample Variance of Data = │ │ Sample Variance of Averages = │ │ Ratio of Sample Variances = │ │ Theoretical Var. Reduction = 1/N = │ │ │ │ Sample Std. Error of Data = │ │ Sample Std. Error of Averages = │ │ Ratio of Sample Std. Errors = │ │ Theoretical Std.Err.Reduc.=1/sqrt(N)= │ │ ┌──────────────┐ │ │ │ Continue... │ │ │ └──────────────┘ │ └───────────────────────────────────────────────┘ Samples Between Refreshes 10 <= refresh <= 10,000. Maximum Total Replications 1,000 <= maxcycl <= 2,000,000,000 Initial Seed Value 0 <= iduminit <= 32,767 About CLTWIN... Author/Address/Version information. NOTE: The "Samples Between Refreshes" sub-item on the "Other" menu controls the number of averages that must be generated before histograms will be redrawn. When this "refresh" setting is very small (say, only 10 or 25), histograms get redrawn very quickly and delays in mouse/keyboard response time are minimized. Don't slow mouse response time down (and speed the simulation up) any more than absolutely necessary to avoid annoying "screen flicker." NOTE: The "Initial Seed" menu sub-item controls the starting point of the simulation that will follow the current simulation. Any change you make here does not become the permanent default seed until you select the "Save Parameter Settings" sub-item on the CLTWIN "File" menu. And "iduminit=0" actually means that CLTWIN will use your system's clock to get a DYNANIC seed for each reset rather than use any FIXED seed value (1 to 32767). NOTE: None of the sub-items on the "Other" menu automatically resets the simulation currently being performed to zero. On the other hand, using the "Maximum Total Replications" item to reduce "maxcycl" below the currently accumulated total would have this effect (and would also trigger an append to CLTWIN.SAV). File =--- Save Parameter Settings ...Write parameter values to CLTWIN.PAR file. Write Detailed Output ...Append results to the CLTWIN.SAV file. Exit ...Leave CLTWIN and return to MS Windows. Central-Limit-Theorem Simulator for Windows Page 7 of 21 The six CLTWIN parameter settings written to CLTWIN.PAR are: ┌─────────────────────────────────────────────────────────────────┐ │ Variable Minimum Default Maximum │ │ Name Brief Explanation Value Value Value │ │ ------- ------------------------ ------- ------- ------- │ │ shape Shape of Distribution 1 1 4 │ │ │ │ ssize Observations Averaged in 2 10 1000 │ │ each Simulation Cycle │ │ │ │ refresh Screen update delay Cycles 2 50 500 │ │ │ │ maxcycl Maximum Cycles before 10,000 2billion 2billion │ │ Automatic Termination │ │ │ │ idum Random Number SEED Value 0 0* 30000 │ └─────────────────────────────────────────────────────────────────┘ WARNING: The CLTWIN.SAV file grows in size each time the user clicks on the "Write Detailed Output" menu item. Results are also appended to this file any time a simulation of specified "maxcycl" size is completed and a new simulation starts. WARNING: Using the ASCII text editor or word process of his/her choice, the user should periodically edit the CLTWIN.SAV file to print results or save them to a different filename.ext. It is highly recommended that all out-dated text be removed from the CLTWIN.SAV file during this manual editing process. Of course, the user may also use the DOS "del cltwin.sav" command or the Windows File Manager to simply delete a CLTWIN.SAV file that has become too large and/or outdated. In CLTWIN, results are accumulated using "long" integer values that cannot exceed 2,147,483,647. CLTWIN.EXE automatically stops (and writes out summary results in a file named CLTWIN.SAV) after "maxcycl" replications. On the other hand, that many replications would require about 70 hours on a 33Mhz 486-machine! Of course, you can always stop CLTWIN.EXE whenever you want via the "File/eXit" menu item. ========================================== <<< INSTRUCTIONAL OBJECTIVES / LEARNING POINTS >>> ========================================== [1] Histograms of averages from non-"bell-shaped" distributions [such as CLTWIN shapes 1=uniform, 2=skewed and 3=bimodal] do become more and more "bell-shaped" as the number of observations, N, in the average increases. Central-Limit-Theorem Simulator for Windows Page 8 of 21 ***************************************************************** SYMMETRY... Three of the parent distributions simulated in CLTWIN [shapes 1, 3 & 4] are symmetric about ZERO. Histograms of averages from these distributions should also be symmetric about ZERO ...except, of course, for sampling error that will appear to evaporate as results accumulate over the long run. But CLTWIN shape=2 is the "skewed" Right-Triangular distribution with zero frequency at -5.0, expected value (mean, centroid) at +1.67, and mode frequency (of 0.2) at +5.0. Averages from this distribution will also have their true mean at +1.67 [rather than at 0, the midpoint of the [-5,+5] range for each observation.] When only a very small number of "skewed" shape=2 observations are averaged, "skewness" will also be clearly visible in the sampling distribution of the resulting averages. Central-Limit-Theorem Simulator for Windows Page 9 of 21 Large-Sample Histogram for the Average of 2 skewed values... 5 ▀▀▀ ▀▀▀▀▀▀▀▀▀▀ ▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀ 4▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀ ▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀ ▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀ ▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀ 3▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀ ▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀ ▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀ ▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀ 2▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀ ▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀ ▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀ ▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀ 1▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀ ▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀ ▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀ ▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀ 0▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀ ▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀ ▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀ ▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀ -1▀▀▀▀▀▀▀▀▀▀▀▀▀▀ ▀▀▀▀▀▀▀▀▀▀▀ ▀▀▀▀▀▀▀▀▀ ▀▀▀▀▀▀▀ -2▀▀▀▀▀▀ ▀▀▀▀ ▀▀▀ ▀▀ -3▀ ▀ This skewness of averages will tend to disappear as the number of observations, N, in the average increases. This illustrates the Central-Limit- Theorem concept that even averages from non- symmetric distributions tend to become more and more symmetric as the number of observations in the average increases. ***************************************************************** MODES (Global and Local)... The normal distribution is uni-modal; due to symmetry, this single mode is located at the distribution mean value. The distributions of averages drawn from shapes 1 and 4 will also have these uni-modality and symmetry properties. Central-Limit-Theorem Simulator for Windows Page 10 of 21 Averages of skewed (shape=2) observations also have uni-modal distributions. When N=2, this mode will occur at approximately +2. But this modal frequency will decrease as N increases, ultimately approaching the mean value of +1.67. Distributions of averages of bimodal (shape=3) observations can have several modes, but are always symmetric. When M is even or larger than 8, the averages distribution will have a single, global mode. But when N is odd and less than 8, the averages distribution will have a pair of global modes that are easily "seen" using narrow (0.25-sigma) cell widths. In addition, these distributions may also have pairs of local modes. Central-Limit-Theorem Simulator for Windows Page 11 of 21 Large-Sample Histogram for the Average of 2 bimodal values... 5 ▀▀▀ ▀▀▀▀▀▀▀▀ ▀▀▀▀▀▀▀▀▀▀▀▀ 4▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀ ▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀ ▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀ ▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀ 3▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀ ▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀ ▀▀▀▀▀▀▀▀▀▀▀▀ ▀▀▀▀▀▀▀▀▀ 2▀▀▀▀▀▀▀▀ ▀▀▀▀▀▀▀▀ ▀▀▀▀▀▀▀▀▀▀ ▀▀▀▀▀▀▀▀▀▀▀▀▀ 1▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀ ▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀ ▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀ ▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀ 0▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀ ▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀ ▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀ ▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀ -1▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀ ▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀ ▀▀▀▀▀▀▀▀▀▀▀▀▀ ▀▀▀▀▀▀▀▀▀▀ -2▀▀▀▀▀▀▀▀ ▀▀▀▀▀▀▀▀ ▀▀▀▀▀▀▀▀▀ ▀▀▀▀▀▀▀▀▀▀▀▀▀ -3▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀ ▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀ ▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀ ▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀ -4▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀ ▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀ ▀▀▀▀▀▀▀▀▀▀▀▀▀ ▀▀▀▀▀▀▀▀ -5▀▀▀ Central-Limit-Theorem Simulator for Windows Page 12 of 21 Large-Sample Histogram for the Average of 3 bimodal values... 5 ▀▀ ▀▀▀▀▀ 4▀▀▀▀▀▀▀▀ ▀▀▀▀▀▀▀▀▀▀▀ ▀▀▀▀▀▀▀▀▀▀▀▀▀ ▀▀▀▀▀▀▀▀▀▀▀▀▀ 3▀▀▀▀▀▀▀▀▀▀▀▀ ▀▀▀▀▀▀▀▀▀▀▀ ▀▀▀▀▀▀▀▀▀▀ ▀▀▀▀▀▀▀▀▀▀▀▀▀ 2▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀ ▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀ ▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀ ▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀ 1▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀ ▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀ ▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀ ▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀ 0▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀ ▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀ ▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀ ▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀ -1▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀ ▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀ ▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀ ▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀ -2▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀ ▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀ ▀▀▀▀▀▀▀▀▀▀▀▀▀ ▀▀▀▀▀▀▀▀▀▀▀ -3▀▀▀▀▀▀▀▀▀▀▀ ▀▀▀▀▀▀▀▀▀▀▀▀ ▀▀▀▀▀▀▀▀▀▀▀▀▀ ▀▀▀▀▀▀▀▀▀▀▀▀▀ -4▀▀▀▀▀▀▀▀▀▀▀ ▀▀▀▀▀▀▀▀▀ ▀▀▀▀▀ ▀▀ -5 Central-Limit-Theorem Simulator for Windows Page 13 of 21 Large-Sample Histogram for the Average of 4 bimodal values... ▀ 4▀▀▀ ▀▀▀▀▀ ▀▀▀▀▀▀ ▀▀▀▀▀▀ 3▀▀▀▀▀▀▀ ▀▀▀▀▀▀▀▀ ▀▀▀▀▀▀▀▀▀▀▀ ▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀ 2▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀ ▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀ ▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀ ▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀ 1▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀ ▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀ ▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀ ▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀ 0▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀ ▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀ ▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀ ▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀ -1▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀ ▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀ ▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀ ▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀ -2▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀ ▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀ ▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀ ▀▀▀▀▀▀▀▀▀▀▀ -3▀▀▀▀▀▀▀▀ ▀▀▀▀▀▀ ▀▀▀▀▀▀▀ ▀▀▀▀▀▀ -4▀▀▀▀ ▀▀▀ ▀ ***************************************************************** [2] The variance of an average of N independent and identically distributed random variables is inversely proportional to N. Equivalently, the standard deviation of an average of N independent and identically distributed random variables is inversely proportional to the square root of N. The measure of "spread" (or expected "width") of a bell-shaped distribution that one can most readily discern, visually, from a histogram is the distribution's standard deviation, not its variance. Central-Limit-Theorem Simulator for Windows Page 14 of 21 NOTE: The technical explanation here is that variance is measured in squared units, while standard deviations are measured in the same units as the individual observations. The above phenomenon can be illustrated in CLTWIN by examining the distributions of averages of sizes both N and 4 x N. Ask yourself the question : "Does the distribution of averages of 4 x N observations look exactly one-half as wide as the distribution of averages of N observations?" ***************************************************************** Histograms of observed average values are listed below for runs in which 2 million UNIFORMLY distributed observations (shape=1) were averaged in groups of size N = 2, 4, 8, 16 or 32. The standard deviation of a uniform distribution with range R is R/sqrt(12). Thus the simulated data in every CLTWIN run should yield sample standard deviations close to the square root of 100/12 = 8.33, which is 2.887. Statistical theory also tells us that the estimated standard deviation of the average of N observations should be close to sqrt(8.333/N). <<< SAMPLE SIZE: N = 32 >>> Random Number Seed = -10821 Total Number of Averages = 62,500 Variance of Averages = 0.261 Variance of Data = 8.333 ▀ 1▀▀▀ ▀▀▀▀▀▀▀▀▀▀ ▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀ ▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀ 0▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀ ▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀ ▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀ ▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀ -1▀▀▀▀▀▀▀▀▀▀ ▀▀▀▀ ▀ Central-Limit-Theorem Simulator for Windows Page 15 of 21 <<< SAMPLE SIZE: N = 16 >>> Random Number Seed = -8099 Total Number of Averages = 125,000 Variance of Averages = 0.520 Variance of Data = 8.334 2 ▀ ▀▀▀ ▀▀▀▀▀▀▀ 1▀▀▀▀▀▀▀▀▀▀▀▀ ▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀ ▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀ ▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀ 0▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀ ▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀ ▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀ ▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀ -1▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀ ▀▀▀▀▀▀▀▀▀▀▀▀ ▀▀▀▀▀▀▀ ▀▀▀ -2▀ Central-Limit-Theorem Simulator for Windows Page 16 of 21 <<< SAMPLE SIZE: N = 8 >>> Random Number Seed = -2565 Total Number of Averages = 250,000 Variance of Averages = 1.039 Variance of Data = 8.334 3 ▀ ▀▀ 2▀▀▀▀▀ ▀▀▀▀▀▀▀▀ ▀▀▀▀▀▀▀▀▀▀▀▀ ▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀ 1▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀ ▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀ ▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀ ▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀ 0▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀ ▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀ ▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀ ▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀ -1▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀ ▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀ ▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀ ▀▀▀▀▀▀▀▀▀▀▀▀ -2▀▀▀▀▀▀▀▀ ▀▀▀▀▀ ▀▀ ▀ -3 Central-Limit-Theorem Simulator for Windows Page 17 of 21 <<< SAMPLE SIZE: N = 4 >>> Random Number Seed = -9040 Total Number of Averages = 500,000 Variance of Averages = 2.085 Variance of Data = 8.337 4 ▀ ▀▀ 3▀▀▀▀ ▀▀▀▀▀▀ ▀▀▀▀▀▀▀▀▀ ▀▀▀▀▀▀▀▀▀▀▀▀ 2▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀ ▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀ ▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀ ▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀ 1▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀ ▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀ ▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀ ▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀ 0▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀ ▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀ ▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀ ▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀ -1▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀ ▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀ ▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀ ▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀ -2▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀ ▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀ ▀▀▀▀▀▀▀▀▀▀▀▀ ▀▀▀▀▀▀▀▀▀ -3▀▀▀▀▀▀ ▀▀▀▀ ▀▀ ▀ -4▀ Central-Limit-Theorem Simulator for Windows Page 18 of 21 <<< SAMPLE SIZE: N = 2 >>> Random Number Seed = -31647 Total Number of Averages = 1,000,000 Variance of Averages = 4.167 Variance of Data = 8.334 5 ▀ ▀▀▀ ▀▀▀▀▀ 4▀▀▀▀▀▀▀ ▀▀▀▀▀▀▀▀▀ ▀▀▀▀▀▀▀▀▀▀▀ ▀▀▀▀▀▀▀▀▀▀▀▀▀ 3▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀ ▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀ ▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀ ▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀ 2▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀ ▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀ ▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀ ▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀ 1▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀ ▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀ ▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀ ▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀ 0▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀ ▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀ ▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀ ▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀ -1▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀ ▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀ ▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀ ▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀ -2▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀ ▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀ ▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀ ▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀ -3▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀ ▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀ ▀▀▀▀▀▀▀▀▀▀▀▀▀ ▀▀▀▀▀▀▀▀▀▀▀ -4▀▀▀▀▀▀▀▀▀ ▀▀▀▀▀▀▀ ▀▀▀▀▀ ▀▀▀ -5▀ Central-Limit-Theorem Simulator for Windows Page 19 of 21 Note that the histogram for averages of N=8 observations tends to have a "width" that is exactly half that of the histogram for averages of N=2 and twice that of the histogram for averages of N=32. After all, 2, 8, and 32 differ by a multiplicative factor of 4, and the square root of 4 is 2. Similarly, the histogram for N=16 is one-half as wide as that for N=4. Observation: ------------ All of the above histograms were of the same height. This is an unfortunate limitation of CLTWIN in the sense that the total screen area covered by histograms appears to decrease as N increases. Ideally, this total area would always be the same; after all, this area always represents a total probability mass of one unit. As the histograms become more narrow (decreasing variance), they should also become taller! Unfortunately, this strategy would have made the histogram for averages of 2 observations VERY short...after all, this is the averages histogram of maximum "width" (i.e. maximum standard deviation and variance.) ***************************************************************** [3] To make numerical (rather than visual) comparisons of variances and standard deviations between the distributions of individual values and their averages, use the "Compare Variation Statistics" item on the CLTWIN "Other" menu. If you do try out this option, you will note that... the variance of the population and the variance of the average are not usually estimated to differ by EXACTLY 1 over N, and the standard deviation of the population and the standard deviation of the average are not usually estimated to differ by EXACTLY 1 over root N. Why is this? And, since they differ, which estimate is the more accurate estimate? ***************************************************************** The CLTWIN estimate of the standard deviation (or variance) of the population being sampled uses ALL of the observations that were generated by the pseudo-random algorithm. The CLTWIN estimate of the standard deviation (or variance) of the AVERAGE is based solely on those observed averages, discarding all information about the individual values observed within the individual samples of size N. Central-Limit-Theorem Simulator for Windows Page 20 of 21 As a result, the CLTWIN estimate of the standard deviation (or variance) of the population is expected to be more accurate (numerically closer to its true value) than the corresponding estimate based solely on sample averages. [Technically, the degrees-of-freedom for the population estimate are essentially (N-1) times larger than the degrees-of-freedom for the estimate based only on sample averages.] In actual statistical practice, one usually HAS to use the standard deviation estimated from all of the observed data to estimate the standard deviation of the average. After all, one has frequently observed only one such average! Thus standard statistical practice is to use the overall estimate, which is expected to be better in the long run, anyway. ***************************************************************** [4] CLTWIN can display histograms of cumulative counts observed in cells of width 1.00, 0.50 or 0.25 units. What are the trade-offs associated with choice of cell width? ***************************************************************** ...Use of narrow cells in CLTWIN will lead to more accurate depictions of the parent distribution being sampled as well as the implied distribution of averages. But the user of CLTWIN will need to be relatively patient in this narrow-cell case; large numbers of replications may be needed to give narrow- cell histograms a "smooth" appearance. ...Use of wide cells in CLTWIN limits the resolution of histogram displays, but these wide-cell displays tend to stabilize much more quickly than narrow-cell histograms do. ***************************************************************** ========== <<< REFERENCES >>> ========== L'Ecuyer, P. (1988). "Efficient and Portable Combined Random Number Generators." Communications of the ACM 31, 742-749,774. Heyde, C. C. (1982). "Limit Theorem, Central." Encyclopedia of Statistical Sciences, (N.L.Johnson, S.Kotz & C.B.Read, editors). Vol.4, 651-655. New York: John Wiley and Sons. Hoaglin, D. C. (1982). "Generation of Random Variables." Encyclopedia of Statistical Sciences, (N.L.Johnson, S.Kotz & C.B.Read, editors). Vol.3, 376-382. New York: John Wiley and Sons. Central-Limit-Theorem Simulator for Windows Page 21 of 21 CLT Update History: =================== 9208 ...Original Version. 9210 ...First upload to CompuServe...Sample only from Uniform Dist. 9211 ...Add three more SHAPES of Distributions for sampling: 2=Skewed, 3=Bimodal, and 4=Normal. 9404 ...First Microsoft Windows version. ...Major revision of this documentation file.