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┌──────────────────────────────────────────────────────────┐
│ ┌──────────────────────────────────────────────────────┐ │
│ │ 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
▀▀▀
▀▀▀▀▀▀▀▀▀▀
▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀
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▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀
▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀
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
▀▀▀
▀▀▀▀▀▀▀▀
▀▀▀▀▀▀▀▀▀▀▀▀
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▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀
3▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀
▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀
▀▀▀▀▀▀▀▀▀▀▀▀
▀▀▀▀▀▀▀▀▀
2▀▀▀▀▀▀▀▀
▀▀▀▀▀▀▀▀
▀▀▀▀▀▀▀▀▀▀
▀▀▀▀▀▀▀▀▀▀▀▀▀
1▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀
▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀
▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀
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0▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀
▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀
▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀
▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀
-1▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀
▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀
▀▀▀▀▀▀▀▀▀▀▀▀▀
▀▀▀▀▀▀▀▀▀▀
-2▀▀▀▀▀▀▀▀
▀▀▀▀▀▀▀▀
▀▀▀▀▀▀▀▀▀
▀▀▀▀▀▀▀▀▀▀▀▀▀
-3▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀
▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀
▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀
▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀
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Central-Limit-Theorem Simulator for Windows Page 12 of 21
Large-Sample Histogram for the Average of 3 bimodal values...
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Central-Limit-Theorem Simulator for Windows Page 13 of 21
Large-Sample Histogram for the Average of 4 bimodal values...
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*****************************************************************
[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
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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
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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
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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
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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
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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.