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Statistics in the elementary schoolBetty Beck a , Lorraine Denby & James M. Landwehr ba Education Development Center , Newton, Massachusettsb Bell Laboratories , Murray Hill, New JerseyPublished online: 22 Jun 2010.

To cite this article: Betty Beck , Lorraine Denby & James M. Landwehr (1976) Statistics in the elementary school,Communications in Statistics - Theory and Methods, 5:10, 883-894, DOI: 10.1080/03610927608827406

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Br%ty Eeck

B e l l Labora tor ies W r a y t i i l l , Hew J e r s e y

This paper &izc~sies c:le r o l e of s t a t i s t i c s ~ i t h i n an

ince rd i sc ip i<na ra p r o g r & ~ on r e a l ~ ; rob lem sclv'icg i n rl;.:i!ent.?ry

sc:?c,ols (t'rirough grade 9). !$!e f i r s t desc r ibe some gene ra l

f e a t u r e s i ~ f t h e ZSNES ( u n i f i e d Science and Mathematics f o r

Elementary ~ c h o o l s ) curriculum and some s i t u a t i o n s where t h e

a p p l i c a t i o n o f s t a t i s t i c a l p r i n c i p l e s and techniques can e n t e r

t h e program. Then we present our i deas concerning t,he kinds of

s t a t i s t i c a l methods t h a t a r e appropr i a t e i n t h i s environment, a n d

we d i scuss t h e use of t h i s m a t e r i a l with both elementary schooi

s tuden t s and elementary schooi t eache r s .

1. TIE USMES PRCGTIAV

USMES i s based on the hypotheses t h a t r e a l problem solv ing

i s an important s k i l l t o be learned and t h a t many math, science,

s o c i a l sc ience , and language a r t s s k i l l s may be learned more

Copyright O 1976 by Ma:cel Dekkrr, Inc. All Rights Reserved. Netther this work nor any part may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopy:ng, microfilming, and recording, or by any information storage and retrieval system, .x:!.h.ol?! pemissinr? in writing from the publisher. D

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884 BECK, DENBY, AM) LANDWEHR

quickly and e a s i l y wi thin the context of s tudent inves t iga t ions

of r e a l problems. Problem solving, a s exemplified by USMES,

irnpiies s s t y l e of education which involves s tudents i!; i n v e s t i -

g a t i n g and soiving r e a l p r o h i m s . I t provides the br idge between . - . . the abctrac:ti3ils 02 bhe sciioai c;jrricu;--- 9 iuii arid tile world of the

s tuden t . Frcb!.erns a re pre.;ente<i i n tile form of' chai lsngss t h a t

a r e i n t e r e s t i n g t o chi ldren because they a re both r e a l and

p r a c t i c a l . The problem i s r e d i n s e v e r e i r e s p e c t s : (1) a solu-

t i o n i s needed and not p resen t ly known, a t l e a s t Tor the p a r t i c -

u l a r case in question; ( 2 ) t h e s tudents a re involved i n ccjmpLete

s i t i i a t ions ; < i t . i ~ a i l t h e accompanying var iab les and complexities;

( 3 ) the problem app l ies t o some a s ~ e c t of st.1adev.t. life i n the

school rJr commltnity; and (4) the problem i s such t,hat t h e work by 1.1.- students cui lead t o sorw ixp~ovement i n the sftu;t3on. Thid

ejiyectatjorl or u s e f u l accotirpiishment provides t h e motivation f o r

ch i ld ren t o c u r r y out the comprehensive i n x s t i g - t i s n s needed t o

f i n d siiine soli.it.in t.o the chel lenge. 8,- ,uenty-r_ine iu!ii:s i:a-:e beei: develsped; ea& ihTit fociisee on a

problem t h a t . has been t ack led by s tuden t s i n grades one 'chro:@

e i g h t (where s tuden t s a re 1 3 or 14 years o ld ) and includes

appropr ia te background and resource m a t e r i a l f o r s tuden t s and

t eachers . More u n i t s a r e cur ren t ly under development. USMES

u n i t s a re now being used i n over 60 school d i s t r i c t s i n 33 s t a t e s .

The p r o j e c t i s fmded by the National Science Fowdat ion through

t h e Education Development Center. Further discuss ion i s given in

Arbet ter , Beck, and Lomon (1375); Lunetta (1974); and i n Mosaic

(1974) - The l e v e l a t which t h e ch i ld ren approach the problems, t h e

i n v e s t i g a t i o n s t h a t they ca r ry out, and the s o l u t i o n s t h a t they

dev i se may vary according t o the age and a b i l i t y of the chi ldren;

However, r e a l problem so lv ing involves them, a t some l e v e l , i n

a l l a spec t s o f the problem-solving process: d e f i n i t i o n of the

problem, determination of the important f a c t o r s i n t h e problem,

observation, da ta c o l l e c t i o n and ana lys i s , measurement, d isous-

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STATISTICS IN THE ELEMENTARY SCHOOL 885

sion, I'orm-dation and trial sf suggested sslu-cions, c l a r i f i c a ~ i o n

of' values, decis icn ~aki~g, and comunications of findings ts . others . yo I+.-- ,-,li t h e p r x e s ~f r e d problem ~ ~ i . ~ r i n g , the

s tudents must encounter, fcrnulate , u d f i n d s o x soiut ivn t o

complete arid r e a l i s t i c probien.~ . 3.e s ~ u d e z i s thexselvss, rat,

~ n e teacher, m i s t malyzs the problem, chacsi. <he -rariab;oies t h a t

s!:wld be inves t iga te i , search out the f a c t s , and judge she

correctness sf 3; h2notheses snd conclusions. Tn r e a i problem

solving a c t i v i t i e s che teacher acLs 2s a coi;rdina:or arid

cal_l.sborator, not as an au thor i t a t ive answer-giver.

Dealing with t h e s e pi-obierns, in a h o s t every csse, involves

the co i l ec t ion and i n t e r p r e t a t i o n of some data. Since the

chi ldren c o l l e c t da ta on real s i tua t ions , they encounter a l l o f the co ,u-g ~ ~ ~ X . L L L ~ , L - - - : ~ A - - t h a t are presezt i n such s i t u a t i o n s . These

complexities ( t h e data c o l l e c t i o r ~ prncedure as wel l as the

andiysis of da ta which i s r e a l ) usually have been eliminated i n

textbook problem even though the s e t t i n g and uordlng of -the

textbock prcblems ~ m d rea l . Finding some so lu t ion t o a r ea l

problem i s a icng-term process, a id the re may be manjr phases of

col lect ion, m a l g s i s , refinement of procedures, and analysis of'

new data . In t h i s process s t a t i s t i c a l analysis may be used i n

making decis ions about f'urther da ta co l l ec t ion as wel l as i n

drawing inferences from the present data . We w i l l now b r i e f l y

descr ibe several of the u n i t s and show s p e c i f i c xzys i n uhich

s t a t i s t i c s i s r e l s t e d t o these prnblems.

Ln c e r t a i n u n i t s chilit-en may a t t e A q t t o design use fu l

objects so t h a t they a re an appropriate s i z e . For example, i n

the Designing f o r Human Proportions uni t , o r challenge, the

chi idren may w a n t t o deternine how high tln make a new t a b l e f o r

the classroom. The height of a t a b l e t h a t each ch i ld p r e f e r s

is measured, and these values are examined. In the Manu- fac tu r ing challenge, s tudents design aprons, mis tbands, or other

items. They need t o know what d i f f e r e n t s i zes t o make and how

many of each s ize t o produce. Thus, t h e chi ldren need t o c o l l e c t

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886 BECK, DENBY, AND LANDWJZHR

c e r t a i n mrssurements an& analyze r e l a t i o n s h i p s between d i f f e r e n t

measurements. Thio i s o f t en done using histograms and s c a t t e r

p l o t s .

In the Pedestrian Crlssirigs ctiallenga the chi idren t q to

f'igurc out ways t o make a cross ing s a f e r . This may involve

c o l l e c t i n g da ta of' the times i t took childrer. fmm a zertalr:

~ q u l a t i m tn :r.?ss an iriiersectio?, a;;d th? gap Ctinieo between

c z r s during a c e r t a i n chser .~a t im per iod , I r i srdel- ti? determine

a gosd timing f a r a :q;i.lk s i g n .r?f a t r a f f i c l i g h t , the citilitrer!

e m p a r e t h e t:;; ;iistritiiticitls. Simiiar, but more complex,

problems arise i n work on the T r a f f i . ~ Flow chsAlenfle, vhero -- i n v e s t l g a t < ~ o m i:!clride ~peer ' , 0 1 cai-s a*; d i f f e r e n t iimes of day,

road conditions, parking on the s t r e e t , end one-way s t r e e t s .

The challenge in the Dice Design u n i t i s t o cons t ruc t f a i r shapes

t o be used by cilfferent nmkeero of ----'=. p L f i g , Chilriren

determine tile f a i r n e s s of a shape by coli .ecting da ta on the

occurrences of a chiiseii s ide in many s e t s u f R c e r t l ~ i ~ r!tlmber of

tosses . i n each of' these problems it might he reasonable t o com-

" a v p ti.:^: e r :nore s e t s of iiiearurements, ailit various g ~ a p h i c a l mid I -- - numerical methodo might be used.

Several u n j t s involve problems where s tudents may decide t o

design and administer a quest ionnaire or opinion survey. In

problems concerning c l a s s r s ~ m ciesign stiideirts rnw want t o f i n d

out which f l l rnt ture ssrongement, which color, o r which classrocm

job s tuden t s p r e f e r j so fhe c l a s s i s surveyed. I n the Sort Drinir -- Desipn u n i t stiiderits attempt t o devise a s o f t dr ink t h a t i s L

appealing t o many children, so the c l a s s may survey the p r e f e r -

ences of the whole ! The chi ldren lea-ii how t o design

quest ions t h a t wi1.l g ive tllem the information they need by t ry ing

o u t survey quest ions in t h e i r own c l a s s before d i s t r i b u t i n g the

quest ionnaire t o the whole school. The question of t h e s i z e of

t h e sample comes up along with the problems of how t o pick the

sample. The ch i ld ren t a l l y t h e r e s u l t s of t h e i r surveys and

then, because they need t o t e l l o thers about t h e i r f indings ,

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STATISTICS IN TWE ELEMENTARY SCHOOL 887

zney usus l ly make b&x graphs t o skow t h e infomiation c l e a r l y .

In almcst ever:j c i a s s t h e problan z r l s e s o f a 3ifI"erent t o t a l

number of aaswers f o r each ques t ion. S,ozetixes the problem

s r i s e s of deci4ing whether o r not the d i f f e rences between che

responses t o two questions a r? l a r g e enougn L O be sigriir'lcan' L ?

x important .

Tr, ?he Wzys ti. L e a n ~ n f t s c l a s s c f txenty-f ive sr t h l r z y --- ch i ld ren i s d iv ided among tkr;rze ar fcur groups t r y i n g di f ferenr ,

methods of l e ~ r n i n g a c ~ r t a i n t o p i c such as speiiiri,: m r k , t h e

x s t r i c syst-ex, -r a new m t h t o p i c . Af te r de termining t h e amount

learned by each group using p re - and posz-T;esZa, t h e childz-i;n t r y

L. ,, decide .,hich methqd i r : most e f f e c t i v e . The sample s i z e f o r

each method i s l i k e l y t o be small, and chi ldren o f t e n d iscover

',hat it i d i f f i c u l t t o G s c e r n which iiiethod i s s c p e r i o r . Indeed,

t h e s tudents may f e e l t h a t t h e da ta ai-e i n s ~ i f f i c i e n t t o reech E!

v a l i d dec i s ion ,

'The following tjio u n i t s can l ead t o t h e stuay of coupl ice ted

r e l a t i o n s h i p s 'net~.~et?n seve rz l ve iables . I n t h e Wca'iher

P red ic t ions challenge, measurements of temperai;ure, barometric

pressure, humidity, and wind d i r e c t i o n a r e c o l l e c t e d . ChilAren

p l o t t h e f i r s t t h r e e with a record of t h e a c t u a l weather on a

three-dimensional. pegboard graph; wind d i r e c t i o n and weather a r e -

shown on a c i r c u l a r s c a t t e r - p l o t . 4 s t h i s information i s co l -

l e c t e d and p l o t t e d over a per iod of time t'ne chi ldren a r e ab le t o

see c o r r e l a t i o n s between weather f a c t o r s &?d t h e a c t u a l weather

and m&e ijiore accura te f o r e c a s t s i In some c i a s s e s p r o b a b i l i t i e s

a r e a l s o ca lcu la t ed and included i n t h e d a i l y f o r e c a s t . I n t h e

Sof t Drink Cesign challenge t h e ch i ld ren may conduct t a s t e t e s t s --- of populzs dr inks and t a l l y t h e r e s u i t s i n a matrix which 3hm3 t h e

number of each d r ink conf'used wi th each other d r ink . Students i n

upper grades might cons t ruc t a three-dimensional r ep resen ta t ion

of conf i s ion d a t a which can be used t o i d e n t i f y var ious f a c t o r s

such as sweetness, c i t r i c f lavor , t a r t n e s s . This d a t a i s combined

with t.he r e s u l t s of a susvey on s tudent preferenccs t o 'determine

t h e ing red ien t s f o r a new dr ink ,

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2 . STYLTISTICAL PETHODS UPROPRUTE FOR THEBE PROBLEK

S e ~ i e r a l met!:ods a r e needed t h a t g ive the grade school

s tuden t s an i n t u i t i v e , l imi ted, xorking iciiodeQe of how t o

examine c e r t a i n f a i r l y simp]-e s e t s nf da ta . The co:;text in w:iich

they a re used i s t h a t of i n t e r p r e t i n g the yeel. s e t s of d a t a t h a t

the s tuden t s have c o l l e c t e d . Gr~pphics l method:, a re q u i t e o f t en

h e l p f u l i n t h a t tiley b e t t e r snow exac t ly what i s going on than

does plugging the values of the s&ser-fations i n t o some "magical"

formula. We f e e l t h a t i t i s important t o choose techniques t h a t

cm h b e i.mdersto~?d USC? xithol;t g e t t i n g bogged down i n theo-

r e t i c a l background m a t e r i a l .

For example, w i n g the u s u a l t3-test; t o compare two samples'

means would r e q u i r e a_n. ~mde_erstm%ng sf n a t niilj; "59 ~ 1 . - I ~ U ~ I C ~ F L ~3

dis t r ib iut ion b!!t ~f d i s t r i b u t i o n s themsehies a-16 an idea of

p r o b a b i l i t y theory. Presenting a l l t h i s necessary background

would take f a r too much time t o fit within the scope of the grac?e

schoai c:~r i~: i l i :m. (ili addi t ion_ f o . ~ much of the data collect,.;d

it would no t be appropr ia te t o assume the normal d i s t r i b u t i n n , )

Moreover, s impl i fying complex ideas could cause more p o t e n t i a l

misunderstanding of da ta than would techniques which a r e f a r more

i n t u i t i v e , .

Because of a need t o f i n d a possib3.e so lu t ion t o a problem,

t h e s tuden t s focus on understanding t h e da ta and asking quest ions

about i n t e r e s t i n g aspects and comparisons of t h e d a t a . The

t e c h i q ~ e s they use should be easy t o do by hand s ince c a l c u l a t o r s

may not be r e a d i l y ava i l ab le t o elementary school ch i ld ren and

t h e i r a r i thmet ic s k i l l s do not enable them t o perform tedious

c a l c u l a t i o n s quickly . Keeping the problems t h a t the s t a t i s t i c a l

methods should d e a l with and t h e above goals i n mind, we feel.

t h a t t h e following techniques should be presented t o t h e ch i ld ren .

Several of t h e u n i t s r e q u i r e t h e s tuden t s t o t ake a survey.

I n order f o r s tuden t s t o be able t o design, execute, and i n t e r p r e t

t h e resul ts of a survey they must f i r s t c a r e f W l y consider what

s p e c i f i c information they want t o learn, and *om whom they want

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S'!XTISTICS IN THE ELEMENTARY SCHOOL

2.1 :inalyzir?g One Set of Iiata

May of t h e problem described in Section 1 requ i re tne use . -

or' one sairrple s t a t i s t i c s t o 9 ' a m a i z e the data . The average i s

one of the usua l wajrs 01' swmarrizing the loca t ion s f a sex of

da ta . Rovever, the medim i s b e t t e r f x the s tudents t o use s ince

it i s not only e a s i e r f o r the chi ldren t o conrpute but a l s o it is

not as a<fected ijy 3 yew obscri;ati;c; as the -e:q is.

The standard deviation i s the usua l way of descrtbing the

v a r i a b i l i t y ir? a srmple of data . However, the iorn i l~ ia f o r the

staqdard deviat ion is hardly i n t u i t i v e without the appropriate

background9 and i t i s t i n e consuming, tedious, =id d i f f i c u l t t o

compute accurately without a ca lcu la to r . The i n t e r q u a r t i l e r m g e

can be ca lcu la ted i n order t o obtain an idea of how var iab le a

sanple i s . The i n t e r q u a r t i l e range i s equivalent t o the range of

tiie mid&& yj$ Gof + h n U.,, a-+- ,,U,, i .e . , it i s the di f ference between tile

75th and 25th pe rcen t i l e s . It has a d i r e c t i n t u i t i v e meaning, it

i s easy t o cmpute, and a few v i l d observations w i l l not g r e a t l y

change i t s value. Finding e i t h e r the i n t e r q u a r t i l e r m g e or the

inedian requires the student t o order the data. The ordered values

can be uc9d f o r constructing several graphical displays, and

sinply examining the ordered values may make some c h a r a c t e r i s t i c s

of the d a t a more obvious.

Along with summarizing the locat ion and sca le of a sample,

graphical d isplays of the sample are he lp~fu l i n f inding s p e c i a l

features of the data, i n determining the shape of the d i s t r i b u t i o n

of the sample, and a l s o i n detect ing outlying values. The h i s t o -

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890 BECK, DENBY, AM) LANDWEHR

g r m and enp i r i c sb ciun1:lstive d i s t r i b u t i o n Punction (ecdf')

g raph ica l ly display the valnes in t h e sample. After p l o t t i n g one

of these graphs, e spec ia l ly the ecdf', tile sample p e r c e n t i l e s

needed t o ca lcu la te the median and in te rqdarb i l e range =e e a s i l y

obta ined.

Through c a r e f u l use of the 2bove techniques Ir?e c ~ y ob ta in . - and r e s e n t nwch i n f a m a t i o n t ? o n w x i ~ : g a szngie s m p l e of j a t a ,

This does not rnem t h a t they a re t h e t o t e 1 a s v e r t c th:: one

sample d a t a m a l y t i c problem, biit ;is f e e l t h s t they a r e a good

elementary s e t of da ta n m t y t i n t so l s .

2.2 Comparing Two S e t s (if Dn_eia

In Section 1 we saw t h a t i n many of t h e u n i t s s tuden t s need

t o compare two s e t s of d a t a . F n r e x ~ ~ l e , the speeds o f cai-s

under Lwo d i f f e r e n t cnnditinns, o r the &?.,07~:t learned by two

d i f f e r e n t groups of s tudents . I n addi t ion t o using some of the -. .- - . - --

one sample methods discussed doove, we suggest using q-q (qiumti le-

q i i x ~ t i l e ) p l o t s fsr cnmparing twc. s e t s of d a t a . 'The q-ij, p i : ~ t

ensb ies one t o eas i ly , quiclciy, and g raph ica l ly compare many

aspec t s of the two d i s t r i b u t i o n s , and it i s easy t o cons t ruc t ,

The q-q p l o t g ives chosen sample p e r c e n t i l e s ( i . e . ,

q u a n t i l e s ) of one samp1.e p l o t ted rlgsinst the corresponding

p e r c e n t i l e s of a second-sample. When the two samples a r e of

t h e same s i z e t h l s i s simply a p l o t of the ordered po in t s of

one sanple aga ins t the ordered po in t s of' the other sample. . .. wnen tile two s m p l e s are of -;?equal s ize , simpie approximations

f o r t h e corresponding sample p e r c e n t i l e s i n t h e l a r g e r

sample can be used. Cornpesisons between the two samples

a r e made by examining t h e conf igurat ion of the p l o t t e d po in t s i n

r e l a t i o n t o the 45' l i n e , or some other s t r a i g h t l i n e . If the

p o i n t s l i e near the l i n e the two s m p l e s e r e e s s e n t i a l l y

equal . Any d i f fe rences fourid between the two samples a r e seen as

dev ia t ions from t h i s l i n e . I f t h e conf igurat ion l i e s p a r a l l e l t o

the 450 l i n e but above o r beion itj we can sey t h a t t h e r e i s a

d i f f e r e n c e between the l o c a t i o n s c;f t h e samples but not i n any

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These techniques a r e the ones t h a t -xe f e e l t h a t grade scliool

ch i ld ren can use i n handling s t a t i s t i c a l h t a . E ~ u a i i y FT~or ta r i t

i s t h e necess i ty t o c a r z f u l l y examine t h e date, t s u x k r s t ~ ~ c : how

and under uha t c i r c ~ m s t s n c e s the numbers were obt a i ~ e c ? ~ t o ask

appropr ia te quest ions of the data, and t o r z d i i z e the l i m i t a t i o n s

of t h e &it n. T h i s s p p r ~ a z h gives xiiat we perceive a s a broad but

Pntuht lve b a s i s upon vhich t o biitli: iirore s o p h i s t i c a t e d ideas .

vnC L C ~ , w e hope, these tecnniques w i l l not r e s u l t i n as ~ u c h misuse

and misunderstanding as i n the rote c a l c u l a t i o n of more compli-

ca ted s t a t i s t i c a l t ecb . iques t h a t r e q u i r e more t h e o r e t i c a l

knowleae for t h e i r apprecia t ion and proper app l ica t ion . For

another discuss ion abcut the type of s t a t i s t i c s appropr ia te f o r

elementary schooi students, see - The Cnr re la t ion of Sleme::twv - Science and Mathematics (1969):

3. PRESENTATION OF TIESE ME",T!ODS - NqmLerous backgi-ound papers have been w r i t t e n a s a r e fe rence

resource f o r the t eachers involved i n the IJSbES p r o g r m . Some of i.~m* ----...- >---A=- -.,, _C1GLJC13 U l i - ~ ~ ~ i b e kiie type or' da ta Chat s tuden t s may c o l l e c t a s

they look f o r so lu t ions t o some of the chal lenges . Other papers

desc r ibe methods f o r c o l l e c t i n g data ; f o r example, t h e r e is a

paper on how t o make measurements accurate ly .

The s t a t i s t i c a l methods m e mainly presented i n t h e following

four papers prepared by personnel a t Be l l Telephone Lebors tor ies .

" A General. S t r a t e g y m d Qne Sample Methods!', Dsiiby and Landwehr

(1975a), summarizes the r o l e of s t a t i s t i c s wi th in the USEES

program. It presen t s s genera l philosophy of experimental des ign

md diita a n a l y s i s . The one sample s t a t i s t i c s mentioned i n

Sect ion 2 are presented here . " A Graphical Method f o r Ccrmparing

Two Samples", Denby and Landwehr (l975b ), summarizes the construc-

t i o n and i n t e r p r e t a t i o n of the q-q p l o t . "Assessing the S i g n i f i -

cance of the Differences Between Two Samples", Denby and Landwehr

(1975c), p resen t s t h e philosophy behind t h e permutation t e s t , its

cons t ruc t ion and i n t e r p r e t a t i o n . "Design of Surveys and Samples",

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STATISTICS IN TKE ELmTA%Y SCHOOL

Devlin arid Freeriy (197Lj7 gives m elementary idea of ..&at, how

and whom t o surrey. It t e l l s about ways t o sample and how t o

i n t e r p r e t the r e s u i z s .

Many elementary s c h o d teachers s e noi; f v n i l i a r with mosr;

o f ?;he s t a ~ i s t i c a l methods m d he generai point of -riew he have

&?scr ibed i n Beccion 2 . The resomce mate r ia l discasseri In the

i z o w p a r z g r ~ g h i s iztended t n p a r t i a l l y f i l l t k i s gap. Some of

t h i s n a t e r i a l i s presented t o teachsrs durlng workshops =d

t r a i n i n g prcgrms, and the papers a re prcvided t o classrocjni

teachers f?r t h e i r reference. This mater ia l can a l s o be taught

i n col lege programs t r a i n i n g elementary school teashers . lekcher;

f~'i:ili= ~ 5 t h t h i s mater id . can present the methods t o the chi ldren

a t an appropriate time when they need t o analyze da ta they have

coller. ted. The backgroud papers not iztended d i r e c t l y f o r

r w w - l o Cards1' ere ava i l eb le far t h e chi ldrent s reading. However, " "---

the c h i l b e n t o use. The t i t l e s are: "HOW t o Show Your Data on a

Bar (3raph;" "How t a 1Use a B a r Graph Histograrr,:' "How t o Find the

).Ie&i&vL ." G t h ~ r e e being ~ r e u a r e d . -

As p ~ r t of the general classroom deveioprnent and t r i a l

implementation o f u n i t s within the USPS program the recept ion of

these ideas by elementary school teachers and students will be

explored. Since much of t h e s t a t i s t i c a l resource mater ia l

prepared by B e l l Laboratories personnel has only recent ly been

developed, the re has not as ye t been much feedback.

We would l i k e t o thank R . Gnanadesikan, C . L. Mallows, and

H. 0 . Pol l& of B e l i Laboratories, and E. Lomon o f Education

Developnent Center, Tor t h e i r i a e a s and conhieiits concerning the

development of t h e s t a t i s t i c a l ma te r ia l discussed here.

BIBLIOGRAPHY

Arbetter, C . C. , Beck. 3. & Loman, E. (1975). Real problem solving i n USMES: In te rd i sc ip l ina ry education and much more. School Sci. Math. 75 (11, 53-64. -

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89 4 BECK, DENBY. AND LANDWm

Denby L. & Landwehr, J. 1.1. (1975a). Examining one and two s e t s o f data - P a t I: A general s t ra tegy and one sample methods. 1JSi?ES Background Paper PS 5, Edwation Development Center, . . Newion, tlasc.

Denby L. & Landwehr, 3. M. (1975b). Examining 'me and two s e t s of data - F s r t TI: '4 gr-aphicai iiieti~oi f o r ccrqming two s a ~ p l e s . ~ lISK% Backgr-iuid fane^ PS 6 , Education Devslopnant Center, Newton, Mass.

Uenbj; L. & Lardwhr , J . F?. :I:1?5c). C:l:ai?iining one and two s e t s of da ta - Part ZII: Assessing +he s i g c i f i c n : ~ ~ ~ of tile differences between two samples. US!.ES Background Pzyer rNj 7, E d u ~ a t i o n Develqment Center, Newton, Mass.

K-8, The Cor re la t i sn of E1ernetitru.y Science and Mathematics, Qe2c1-t 3f t he Cmhrl'lgc Co::ierence cn t h e Correia t ion of Science and Mathematics i n the Schools. Boston, Mass.: Houghton-Mif f l i n Ca. , 1969, 164-83.

Devlin, S . J . & Freeiiy-, A . 2 . jl.974). Design of surveys and r t . - = F - saiipies. uor.us; Background Paper PS 4, Education Development

Center, Nevtcs:, M,zss,

Leihiitrur, E, L. (1959). Test ins S t a t i s t i c a l ipypotneses. New York: dohn Wiley & Sor~s, Inc., 183-1853,

Lunetta, V . ( 197'1 ) . IJSMES --a s tep toward an in tegra ted c u r r i c u h n . Learning 2 ( r ) , 59-60.

Wllk, 94. B. & Gnmadesikan, R . (1968 j , Probabi l i ty p l o t t i n g methods f o r the ana lys i s of d a t a . Biometrika $5, 1-17"

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