Reproductions supplied by EDRS are the best that can be made … · 2013-12-16 · Normal...

DOCUMENT RESUME

ED 450 156 TM 032 346

AUTHOR Rollins, Dahl A.TITLE There Are Infinitely Many Univariate Normal Distributions:

Distribution "Bells" Come in Many Appearances.PUB DATE 2001-02-01NOTE 22p.; Paper presented at the Annual Meeting of the Southwest

Educational Research Association (New Orleans, LA, February1-3, 2001).

PUB TYPE Numerical/Quantitative Data (110) Reports Descriptive(141)-- Speeches/Meeting Papers (150)

EDRS PRICE MF01/PC01 Plus Postage.DESCRIPTORS Heuristics; *Scores; *Statistical DistributionsIDENTIFIERS *Normal Distribution; Normality Tests; *Univariate Analysis

ABSTRACTMany univariate statistical methods assume that dependent

variable data have a univariate distribution. Some statistical methods assumethat the error scores are normally distributed. It is clear that the conceptof data normality is an important one in statistics. This paper explainsthat, notwithstanding common misconceptions to the contrary, there areinfinitely many normal distributions, each differing in appearance. Theclassic bell-shaped curve only depicts one single case of the infinitely manyunivariate normal distributions. The bell-shaped curve is the general shapeof all normal univariate distributions, but the bells can have an infiniteassortment of shapes and still be bells. A small heuristic data set involvingscores of 100 people of a variable labeled "x" is used to illustrate thisprinciple. (SLD)

Reproductions supplied by EDRS are the best that can be madefrom the original document.

dahl.wp 1/22/01

Running head: NORMAL DISTRIBUTIONS

There Are Infinitely Many Univariate Normal Distributions:

Distribution "Bells" Come in Many Appearances

Dahl A. Rollins

Texas A&M University 77843-4225

PERMISSION TO REPRODUCE ANDDISSEMINATE THIS MATERIAL HAS

BEEN GRANTED BY

"D. A . aolk t tAs

TO THE EDUCATIONAL RESOURCESINFORMATION CENTER (ERIC)

1

U.S. DEPARTMENT OF EDUCATIONOffice of Educational Research and Improvement

EDUCATIONAL RESOURCES INFORMATIONCENTER (ERIC)

11.1;:is document has been reproduced asreceived from the person or organizationoriginating it.

Minor changes have been made toimprove reproduction quality.

Points of view or opinions stated in thisdocument do not necessarily representofficial OERI position or policy.

BEST COPY AVAILABLE

Paper presented at the annual meeting of the SouthwestEducational Research Association, New Orleans, February 1, 2001.

2

Normal Distributions -2-

Abstract

Many univariate statistical methods assume that dependent variable

data have a univariate normal distribution. And some statistical

methods assume that the error scores are normally distributed.

Clearly, the concept of data normality is an important one in

statistics. The present paper explains, notwithstanding common

misconceptions to the contrary, that there are infinitely many

normal distributions, each differing in appearance.

3


Many univariate statistical methods assume that dependent

variable data have a univariate normal distribution (cf. Hinkle,

Weirsma & Jurs, 1998). And some statistical methods assume that the

error scores are normally distributed (cf. Thompson, 1992).

Clearly, the concept of data normality is an important one in

statistics.

The normal distribution is a model reflecting a particular

mathematical equation (Hinkle, Wiersma & Jurs, 1998). The normal

distribution is not a single distribution, but for every potential

mean and standard deviation there is a different unique normal

distribution (Hinkle, Wiersma & Jurs, 1998).

However, because all normal distributions are symmetrical, the

mean always equals the median of a normal distribution (Vogt,

1993). Furthermore, because all normal distributions are also

unimodal, the mode also equals both the median and the mean of a

given normal distribution.

Abraham Demoivre, a French mathematician, developed the normal

distribution general equation in the eighteenth century (Hinkle,

Wiersma & Jurs, 1998). The normal distribution is also known as the

"Gaussian" distribution, after the European mathematician who

devoted so much of his life to studying this distribution.

The depiction of the normal distribution in textbooks almost

always shows the very famous bell-shaped curve. Many assume this

classic bell-shaped curve is the only shape for the normal

distribution. However, that classic bell-shaped curve only depicts

one single case of the infinitely many univariate normal

4


distributions.

The frequently-depicted normal distribution actually is a

special case of the infinitely many normal distributions. The

depicted "bell-shaped curve" is invariably a picture specifically

of the "standard normal distribution" (i.e., z scores that are

normally distributed) (Hinkle, Wiersma & Jurs, 1998).

To determine if a curve is normal the raw scores must be

converted to z-scores to standardize them. It is impossible to

determine whether or not even a symmetrical and unimodal

distribution is normal simply by looking at it (Bump, 1991). The

bell-shape is the general shape of all normal univariate

distributions, however, bells can have an infinite assortment of

shapes and still all be bells (Burdenski, 2000).

In statistics there are four important statistical estimates

called "moments about the mean." These involve: (a) the mean

itself, (b) the standard deviation, (c) the coefficient of

skewness, and (d) the coefficient of kurtosis.

Skewness indicates whether a distribution or curve is

symmetrical (i.e., mean = median) or asymmetrical (i.e., skewed:

mean # median). Kurtosis refers to how tall or flat the curve is in

relation to how wide or narrow the distribution is.

In symmetrical distributions, if the two halves of the curve

were divided at the median, they would mirror each other. Or

distributions can be skewed positively or negatively. To be

positively skewed or skewed right a few scores (called the "tail")

are to the right of the majority or the body of the scores. To be


negatively skewed or skewed left a few scores (called the "tail")

are to the left of the majority or the body of the scores.

Kurtosis, the last moment, is "the ratio of the average of the

deviation scores raised to the fourth power to the standard

deviation also raised to the fourth power" (Bump, 1991). The normal

curve has a kurtosis value of 3.0 using this formula, which is why

now the common practice is to subtract 3.0 from the kurtosis value

so as to yield a value of zero for kurtosis of a normal curve.

To yield a curve with a positive coefficient of kurtosis,

there has to be a higher concentration of scores around the mean,

resulting in a narrow distribution. In a negative kurtosis curve,

the distribution is broad and there is a low concentration of

scores around the mean (Bump, 1991).

In order for normality to exist, there must be suitable

distribution of height relative to width. A tall, thin distribution

is called "leptokurtic." A wide, flat distribution is called

"platykurtic." As Henson (1999) noted, "Symmetry is a necessary,

but insufficient condition for normality."

The purpose of the present paper is to illustrate in concrete

fashion that there are infinitely many normal distributions. This

will be demonstrated with heuristic data.

Heuristic Examples

Table 1 presents a small heuristic data set involving scores

of 100 people of a variable labelled "x". These data are very close

to normally distributed, as can be seen from the descriptive

statistics presented in Table 2.

6


INSERT TABLES 1 AND 2 ABOUT HERE.

Four Different Normal Distributions Each with Means of 0

Using the SPSS syntax presented in the Appendix, 4 variables

were computed (i.e., "mean0_1," "mean02," "mean03," "mean04"),

each with a mean equal to zero. However, as reported in Table 2,

the standard deviations of the variables differ.

Even though the 4 distributions differ in their

"spreadoutness," all these variables are approximately normally

distributed, as reflected in the coefficients of skewness and

kurtosis reported in Table 2 all being close to zero. This can also

be seen in Figures 1 through 4, which present graphs of the data

distributions with normal curves imposed on the actual data

distributions.

INSERT FIGURES 1 THROUGH 4 ABOUT HERE.

Three Different Normal Distributions, None with Means of 0

Using the SPSS syntax presented in the Appendix, 3 additional

variables were computed (i.e., "mean_1," "mean_2," and "mean 3"),

each with a mean not equal to zero. As reported in Table 2, the

means and the standard deviations of the variables differ.

Again, even though the 3 distributions differ in their

"spreadoutness," all these variables are approximately normally

distributed, as reflected in the coefficients of skewness and

kurtosis reported in Table 2 all being close to zero. This can also

be seen in Figures 5 through 7, which present graphs of the data

7


distributions with normal curves imposed on the actual data

distributions.

INSERT FIGURES 5 THROUGH 7 ABOUT HERE.

Summary

Although all the graphs look different and the second set of

variables all had different means and standard deviations, the

skewness and kurtosis values were exactly the same for all 7

variables. All these variables were representative of normal

curves. In summary, it is possible that univariate normal curve

distributions may look very different.

It is impossible to judge whether even symmetrical, unimodal

data are normally distributed, unless coefficients of skewness and

kurtosis are computed. Clearly, while the "standard normal

distribution" has a classical bell shape, bells come in many widths

and tallnesses, and so do normal distributions.

8


References

Bump, W. (1991, January). The normal curve takes many forms:

A review of skewness and kurtosis. Paper presented at the annual

meeting of the Southwest Educational Research Association, San

Antonio. (ERIC Document Reproduction Service No. ED 342 790)

Burdenski, Jr., T.K. (2000, April). Evaluating univariate,

bivariate and multivariate normality using graphical procedures.

Paper presented at the annual meeting of the American Educational

Research Association, New Orleans.

Henson, R.K. (1999). Multivariate normality: What is it and

how is it assessed?. In B. Thompson (Ed.), Advances in social

science methodology (Vol. 5, pp. 193-212). Stamford, CT: JAI Press.

Hinkle, D.E., Wiersma, W., & Jurs, S.G. (1998). Applied

statistics for the behavioral sciences (4th ed.). Boston: Houghton

Mifflin.

Thompson, B. (1992, April). Interpreting regression results:

beta weights and structure coefficients are both important. Paper

presented at the annual meeting of the American Educational

Research Association, San Francisco. (ERIC Document Reproduction

Service No. ED 344 897)

Vogt, P.W. (1993). Dictionary of statistics and methodology.

Newbury Park, CA: Sage Publications.

9


Table 1Heuristic Data (n = 100)

from Thompson (2001)

ID x1 2.42 2.83 3.04 3.25 3.36 3.47 3.58 3.69 3.6

10 3.711 3.712 3.813 3.814 3.915 3.916 4.017 4.018 4.119 4.120 4.121 4.222 4.223 4.224 4.325 4.326 4.327 4.428 4.429 4.430 4.531 4.532 4.533 4.634 4.635 4.636 4.637 4.738 4.739 4.740 4.741 4.842 4.843 4.844 4.845 4.946 4.947 4.948 4.949 5.050 5.051 5.052 5.053 5.154 5.155 5.156 5.1

10


57 5.258 5.259 5.260 5.261 5.362 5.363 5.364 5.365 5.466 5.467 5.468 5.469 5.570 5.571 5.572 5.673 5.674 5.675 5.776 5.777 5.778 5.879 5.880 5.881 5.982 5.983 5.984 6.085 6.086 6.187 6.188 6.289 6.290 6.391 6.392 6.493 6.494 6.595 6.696 6.797 6.898 7.099 7.2

100 7.6

Copyright, Bruce Thompson, 2001. All Rights Reserved.

11


Table 2Descriptive Statistics

Variable

Statistics

Mean SDCoef. ofSkewness

Coef. ofKurtosis

x 5.00 1.005 .000 -.090mean01 0.00 1.005 .000 -.090amean0_2 0.00 0.502 .000 -.090bmean0_3 0.00 2.010 .000 -.090cmean0 4 0.00 5.025 .000 -.090dmean 1 2.50 0.502 .000 -.090'mean 10.00 2.010 .000 -.090f_2mean _3 25.00 5.025 .000 -.090g

a"mean0 1" = "x" - 5.= "meant) 1" * .5

c= "mean() 1" * 2.= "mean0=1" * 5.

e= "x" * .5= "x" * 2.

g= "x" * 5.

12


Table 3Scores (n = 100) for Three Variables

Each with Means = .0

ID mean° 1 mean° 2 mean° 3 mean0 4

1 -2.60 -1.30 -5.20 -13.002 -2.20 -1.10 -4.40 -11.003 -2.00 -1.00 -4.00 -10.004 -1.80 -.90 -3.60 -9.005 -1.70 -.85 -3.40 -8.506 -1.60 -.80 -3.20 -8.007 -1.50 -.75 -3.00 -7.508 -1.40 -.70 -2.80 -7.009 -1.40 -.70 -2.80 -7.00

10 -1.30 -.65 -2.60 -6.5011 -1.30 -.65 -2.60 -6.5012 -1.20 -.60 -2.40 -6.0013 -1.20 -.60 -2.40 -6.0014 -1.10 -.55 -2.20 -5.5015 -1.10 -.55 -2.20 -5.5016 -1.00 -.50 -2.00 -5.0017 -1.00 -.50 -2.00 -5.0018 -.90 -.45 -1.80 -4.5019 -.90 -.45 -1.80 -4.5020 -.90 -.45 -1.80 -4.5021 -.80 -.40 -1.60 -4.0022 -.80 -.40 -1.60 -4.0023 -.80 -.40 -1.60 -4.0024 -.70 -.35 -1.40 -3.5025 -.70 -.35 -1.40 -3.5026 -.70 -.35 -1.40 -3.5027 -.60 -.30 -1.20 -3.0028 -.60 -.30 -1.20 -3.0029 -.60 -.30 -1.20 -3.0030 -.50 -.25 -1.00 -2.5031 -.50 -.25 -1.00 -2.5032 -.50 -.25 -1.00 -2.5033 -.40 -.20 -.80 -2.0034 -.40 -.20 -.80 -2.0035 -.40 -.20 -.80 -2.0036 -.40 -.20 -.80 -2.0037 -.30 -.15 -.60 -1.5038 -.30 -.15 -.60 -1.5039 -.30 -.15 -.60 -1.5040 -.30 -.15 -.60 -1.5041 -.20 -.10 -.40 -1.0042 -.20 -.10 -.40 -1.0043 -.20 -.10 -.40 -1.0044 -.20 -.10 -.40 -1.0045 -.10 -.05 -.20 -.5046 -.10 -.05 -.20 -.5047 -.10 -.05 -.20 -.5048 -.10 -.05 -.20 -.5049 .00 .00 .00 .0050 .00 .00 .00 .0051 .00 .00 .00 .0052 .00 .00 .00 .0053 .10 .05 .20 .5054 .10 .05 .20 .50

13

55 .10 .05 .20 .5056 .10 .05 .20 .5057 .20 .10 .40 1.0058 .20 .10 .40 1.0059 .20 .10 .40 1.0060 .20 .10 .40 1.0061 .30 .15 .60 1.5062 .30 .15 .60 1.5063 .30 .15 .60 1.5064 .30 .15 .60 1.5065 .40 .20 .80 2.0066 .40 .20 .80 2.0067 .40 .20 .80 2.0068 .40 .20 .80 2.0069 .50 .25 1.00 2.5070 .50 .25 1.00 2.5071 .50 .25 1.00 2.5072 .60 .30 1.20 3.0073 .60 .30 1.20 3.0074 .60 .30 1.20 3.0075 .70 .35 1.40 3.5076 .70 .35 1.40 3.5077 .70 .35 1.40 3.5078 .80 .40 1.60 4.0079 .80 .40 1.60 4.0080 .80 .40 1.60 4.0081 .90 .45 1.80 4.5082 .90 .45 1.80 4.5083 .90 .45 1.80 4.5084 1.00 .50 2.00 5.0085 1.00 .50 2.00 5.0086 1.10 .55 2.20 5.5087 1.10 .55 2.20 5.5088 1.20 .60 2.40 6.0089 1.20 .60 2.40 6.0090 1.30 .65 2.60 6.5091 1.30 .65 2.60 6.5092 1.40 .70 2.80 7.0093 1.40 .70 2.80 7.0094 1.50 .75 3.00 7.5095 1.60 .80 3.20 8.0096 1.70 .85 3.40 8.5097 1.80 .90 3.60 9.0098 2.00 1.00 4.00 10.0099 2.20 1.10 4.40 11.00100 2.60 1.30 5.20 13.00

14



Table 4Scores (n = 100) for Three Variables

Each with Means # .0

ID mean 1 mean 2 mean 3

1 1.20 4.80 12.002 1.40 5.60 14.003 1.50 6.00 15.004 1.60 6.40 16.005 1.65 6.60 16.506 1.70 6.80 17.007 1.75 7.00 17.508 1.80 7.20 18.009 1.80 7.20 18.00

10 1.85 7.40 18.5011 1.85 7.40 18.5012 1.90 7.60 19.0013 1.90 7.60 19.0014 1.95 7.80 19.5015 1.95 7.80 19.5016 2.00 8.00 20.0017 2.00 8.00 20.0018 2.05 8.20 20.5019 2.05 8.20 20.5020 2.05 8.20 20.5021 2.10 8.40 21.0022 2.10 8.40 21.0023 2.10 8.40 21.0024 2.15 8.60 21.5025 2.15 8.60 21.5026 2.15 8.60 21.5027 2.20 8.80 22.0028 2.20 8.80 22.0029 2.20 8.80 22.0030 2.25 9.00 22.5031 2.25 9.00 22.5032 2.25 9.00 22.5033 2.30 9.20 23.0034 2.30 9.20 23.0035 2.30 9.20 23.0036 2.30 9.20 23.0037 2.35 9.40 23.5038 2.35 9.40 23.5039 2.35 9.40 23.5040 2.35 9.40 23.5041 2.40 9.60 24.0042 2.40 9.60 24.0043 2.40 9.60 24.0044 2.40 9.60 24.0045 2.45 9.80 24.5046 2.45 9.80 24.5047 2.45 9.80 24.5048 2.45 9.80 24.5049 2.50 10.00 25.0050 2.50 10.00 25.0051 2.50 10.00 25.0052 2.50 10.00 25.0053 2.55 10.20 25.5054 2.55 10.20 25.50

15

55 2.55 10.20 25.5056 2.55 10.20 25.5057 2.60 10.40 26.0058 2.60 10.40 26.0059 2.60 10.40 26.0060 2.60 10.40 26.0061 2.65 10.60 26.5062 2.65 10.60 26.5063 2.65 10.60 26.5064 2.65 10.60 26.5065 2.70 10.80 27.0066 2.70 10.80 27.0067 2.70 10.80 27.0068 2.70 10.80 27.0069 2.75 11.00 27.5070 2.75 11.00 27.5071 2.75 11.00 27.5072 2.80 11.20 28.0073 2.80 11.20 28.0074 2.80 11.20 28.0075 2.85 11.40 28.5076 2.85 11.40 28.5077 2.85 11.40 28.5078 2.90 11.60 29.0079 2.90 11.60 29.0080 2.90 11.60 29.0081 2.95 11.80 29.5082 2.95 11.80 29.5083 2.95 11.80 29.5084 3.00 12.00 30.0085 3.00 12.00 30.0086 3.05 12.20 30.5087 3.05 12.20 30.5088 3.10 12.40 31.0089 3.10 12.40 31.0090 3.15 12.60 31.5091 3.15 12.60 31.5092 3.20 12.80 32.0093 3.20 12.80 32.0094 3.25 13.00 32.5095 3.30 13.20 33.0096 3.35 13.40 33.5097 3.40 13.60 34.0098 3.50 14.00 35.0099 3.60 14.40 36.00100 3.80 15.20 38.00

16


30

20

10

0

-2.50 -1.50 -.50 .50 1.50 2.50

-2.00 -1.00 0.00 1.00 2.00

MEAN0_1

Graph

30

20-

10

0

-1.25 -.75 -.25 .25 .75 1.25

-1.00 -.50 0.00 .50 1.00

MEANO_2

Std. Dev = 1.00

Mean = 0.00

N = 100.00

Std. Dev = .50

Mean = 0.00

N = 100.00

17

Graph

-5.0 -3.0 -1.0

-4.0 -2.0

MEANO_3

1.0 3.0

0.0 2.0 4.0

5.0

Std. Dev = 2.01

Mean = 0.0

N = 100.00

18

0

-14.0 -10.0 -6.0 -2.0 2.0 6.0 10.0 14.0

-12.0 -8.0 -4.0 0.0 4.0 8.0 12.0

MEAN0_4

Graph

30

20

10

0

1.25 1.75 2.25 2.75 3.25 3.75

1.50 2.00 2.50 3.00 3.50

MEAN_l

Std. Dev = 5.02

Mean = 0.0

N = 100.00

Std. Dev = .50

Mean = 2.50

N = 100.00

19

Graph

14

12

10

8

6

4

2

0

0" 6'.0 .00(9

00.0 0 0 0 0

Std. Dev = 2.01

Mean = 10.00

N = 100.00

/ 10 / v tis.00 .00 .00 .00 .00 .00

20

20

10

0

12.0 16.0 20.0 24.0 28.0 32.0 36.0

14.0 18.0 22.0 26.0 30.0 34.0 38.0

MEAN_3

Std. Dev = 5.02

Mean = 25.0

N = 100.00

21

..'


APPENDIXSPSS Syntax to Implement the Reported Analyses

SET BLANKS=SYSMIS UNDEFINED=WARN printback=listing.TITLE 'Bruce Thompson^s normal data **************' .

DATA LISTFILE='a:normal.dta' FIXED RECORDS=1 TABLE/1 id 1-3 x 4-8 .

list variables=all/cases=999 .

subtitle '1 show several normal with same Mean ***' .

execute .

compute mean01 = x - 5. .

compute mean° 2 = mean° 1 * .5 .

compute mean03 = mean0-1 * 2. .

compute mean04 = mean0-1 * 5. .

list variables =id mean0:1 to mean° 4/cases=999 .

descriptives variables=x mean° 1 to mean° 4/_ _statistics=all .

subtitle '2 show several normal with *dif* Mean ***' .

execute .

compute mean _l = x * .5 .

compute mean_2 = x * 2. .

compute mean_3 = x * 5. .

list variables=id mean _l to mean 3 /cases =999 .

descriptives variables=x mean_l to mean_3/statistics=all .

GRAPH /HISTOGRAM(NORMAL)=mean0 1 .

GRAPH /HISTOGRAM(NORMAL)=mean02 .

GRAPH /HISTOGRAM(NORMAL)=mean0:3 .

GRAPH /HISTOGRAM(NORMAL)=mean0_4 .

GRAPH /HISTOGRAM(NORMAL)=mean_l .



22

US. DEPARTMENT OF EDUCATIONOffice of Educational Rosman:it and intorovarnant IOERI)

Educational Resources information Cantor (ERIC)

REPRODUCTION RELEASE(Specific Document)

I. DOCUMENT IDENTIFICATION:

TM032346

ERIIC

Title:THERE ARE INFINITELY MANY UNIVARIATE NORMAL DISTRIBUTIONS:DISTRIBUTION "BELLS" COME IN MANY APPEARANCES

AulnalsE DAHL A. ROLLINSCorporate Source: Publication Date:

2/1/01

U. REPRODUCTION RELEASE

In order to ossemrnate as wooer', as oosstote mere ana signsticant materials of interest to me ectueationat community.documentsannounced In me mammy abstract mutat of me ERIC system. Resources in Education (RIEL are usually mule evadable to usersin trucrcMcne. reproduced paper copy. ono etectronicrooticat meats. and sold tnrougn the ERIC Document Reproduction Service(EORS) or curler ERIC vendors. Credit is ()wen to the source of eacn cocument. ono. it reproduction reiease is granted. one ofthe hollowing notices is affixed to me document.

It permission is granted to reproduce the identified csocument. please CHECK ONE of the following options aria sign me reteasebelow.

0 Sampte sticker to be affixed to document Sample sticker to be alibied to document 0 1=1

Check herePermittingmicrofiche(4-x 6" Wm.paper =ay.electrumand optical mediareproduction

"PERMISSION TO REPRODUCE THISMATERIAL HAS BEEN GRANTED BY

DAHL A. ROLLINS

To THE EDUCATIONAL RESOURCESINFORMATION CENTER (ERIC):'

Lovell

"PERMISSION TO REPRODUCE THISMATERIAL IN OTHER THAN PAPER

_COPY HAS BEEN GRANTED BY

clory

TO THE EDUCATIONAL RESOURCES

INFORMATION CENTER (ERIC)."

tarsi 2

or here

Permittingteproduchonin outer tnanPaint coley.

Sign Here, PleaseDocuments will be processed as molested orcanoed reoroduction ouauty permits. It permission to reproduce is granted, but

nertner box is cnecxea. idocuments will be processed at taws 1.

"1 hereof grant to the Educational Resources information Center (ERIC) nonexctusive permission to reoroouce tae document assnouted above. Reproduction from me ERIC microfiche or etectronscrooticat meats by persons titer man ERIC eneadYeaa and Itssystem contractors mauves oerrntssion from tne cooyngnt notder. Exception is made tor non-orath reorocluethon by Mune* end otherservice agencies to satisfy informatton needs of educators in response to discrete amines."

rtund 05-4.==.ettruen me:DAHL

aA.ROLLINS

---1

RES ASSOCIATEOvaninum:

TEXAS A&M UNIVERSITYleWmorm Number:momm

TAMU DEPT EDUC PSYCCOLLEGE STATION, TX 77843-4225 paw

979/845-1335

1/22/01

Reproductions supplied by EDRS are the best that can be made … · 2013-12-16 · Normal...

Documents

Transcript of Reproductions supplied by EDRS are the best that can be made … · 2013-12-16 · Normal...