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Calculator Statistics

Author(s): Alan GrahamSource: Mathematics in School, Vol. 32, No. 5 (Nov., 2003), pp. 16-17Published by: The Mathematical AssociationStable URL: http://www.jstor.org/stable/30215621 .

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Ccut3Ccut3by Alan Graham

Many teachers find the graphics calculatorto be a powerfulaid in the teaching of graphs and graphing. Unfortunately,the name 'graphicscalculator'gives no hint of the potentialbenefits that this exciting mathematical micro-world canoffer in teaching and learning topics beyond graphs. A goodexample where its potential remains largely untapped isstatistics.

This article spells out three key features of the graphicscalculator that can make the job of statistics teachers andlearners easier as well as muchmore enjoyable.The examples(and cartoons) have all been taken from the book 'CalculatorStatistics'.The book has been written specifically for theTexasInstruments TI-83 family of calculatorsbut the ideascould easily be adaptedfor other makes and models that hadthe appropriatestatistical facilities.Settle Down PleaseInitial lessons on probability often start with tossing coinsand rolling dice. Although potentially useful, this can easilydescend into noisy chaos. An even bigger worry about suchlessons is the mistaken beliefs that they can foster instudents. For example, they may lend credence to the 'Lawof Small Numbers', which is the (mistaken) belief that allsmall samples accurately reflect the characteristics of thepopulation from which they were taken. As a result, studentscan easily come away from such a lesson with incorrect orincomplete concepts that can get in the way of laterlearning.For example, after, say, thirty rolls of a die, you may hearremarkssuch as: "I told you six is hard to get!"

I preferr ( m yown ;

If these areeua((v Ulke(vN(eat r0yb - 0 i e s .e s o

a93"Litg

or, afterthirty tosses of a coin: "But they don't look equallylikely!" and so on.

By exploiting the calculator's randInt command, ten orthirty or one hundred 'virtual coins' can be tossed in a triceand the mean number of heads calculated. Each time is pressed, a new sequence of values is generatedand the proportion of heads immediately calculated anddisplayed on the home screen. The speed and ease ofgenerating these results allows the students to give their fullattention to the following two important awarenesses.That:* whichever sample size is chosen, there is always somedegree of variation, ...* ... but with a sample size as large as 100, there is a clearsettling-down effect.

mean~randInt

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Eachtime SPREAD is executed,twoboxplotsaredisplayed,corresponding to the two sample sizes. And students willobservethat,on each occasion, the samebasic patternresults- the spread of sample means for n = 10 is alwaysconsiderably greater than the spread for samples wheren= 100.

FormattingThinkingThe composerIgor Stravinsky expressed the view that, "themore constraintsone imposes, the more one freesone's self".He believed that creativity required boundaries withinwhich it maybe expressed and indeed argued that, "thearbitrarinessof the constraint serves only to obtain precisionof execution".Studentsof statistics often reporta corresponding inabilityto expressthemselves creativelywithout the aid of structure!When it comes to performing some of the more advancedstatistical calculations such as finding a confidence intervalor carryingout a test of significance, they are often unable tomake a start.The graphicscalculatorprovides a structuredmicro-worldwithin which students can take that first creative step.Concepts are encapsulated in symbols that are clearlydefined and available for further examination and

exploration.For example, it is a great source of confusion instatistics that there are two meanings of the word 'sigma'.Capital sigma, I, refers to 'the sum of', while lower-casesigma,(, means the standarddeviation.This distinction mayseem trivial to the expert statistician but for the novice it isa boon to be able to access these measures directly on thecalculatorand work out exactly what they represent.So, littlesigmaisstandard eviation;big sigmameans

'sumof'!

2 6 . 6 7 8 1 2 2 5 4

As well as reflecting standard mathematical notation,themenu structure of the graphics calculator also reflectsstandard methods.Each menu provides a set of choices thatstudents must take when carrying out a particularprocedure.Eachsetting representsa key decision to be madewhen carryingout the procedure and, in a sense, the entiremenu encompassesa neat summaryof what the procedureisall about. I would suggest that when students work througha calculator menu choosing appropriate settings from thescreen, they aresubconsciously absorbingthe main elementsthat constitute the topic in question - in fact, the calculatoris helping to 'formattheir thinking'. Here aretwo examples,the first fairly simple and the second more advanced.

The STATPLOT screen opposite is the menu thatstudents must 'set' when plotting a statistical graph (in thiscase lists L3 and L4 are to be plotted as a scatterplot.Afterswitching the plot On (line 1), they are faced with a choiceof six possible types of plot. Having chosen the second ofthese (the scatterplot), the screen automatically alters andthey are immediately presented with not one but two lists

(Xlist and Ylist). These stages help to form a subliminalawareness in the student's mind that this particular type ofrepresentationis suitable only for bi-variate data.i lotS MtUO gvp .., 3)

X1ist0--3Vlist:L4Mark:Z-TestInPt Data IlMI

a:5.77:28.2n: 350C:F.4fflx'TCacul-ate Drjaw

The second more sophisticated example shows the choicesto be made when carryingout a statistical test of significance(the Z-test). Working systematically through the menu lineby line enables students to learn that, when conducting a Z-test, they need to make the following choices; firstlywhetherthe test is to be applied to data (that might be stored in a listor lists) or will be entered in the form of summarystatistics(line 1). Having in this case chosen 'Statistics', they are nowpresented with settings choices that define the Z-test(population mean, population standard deviation, samplemean and sample size). Next, crucially, they need to make achoice as to whether they areusing a two-tailed test (#o0) ora one-tailed test (either < io or > to).

Z-Testz= -1.868235721P=.0617290941x=28.2n=35

Fi " . -6I-:.O6?Finally, they can choose either to see a calculatedsummaryof the test values (the Calculate option) or see the resultsrepresented visually (the Draw option). Both of these finaldisplays are shown above.

CalculatorStatistics- The BookFor many years I have worked with students from 11 yearsupwards using the graphics calculator in a range ofmathematical situations, and have always been convincedthat this machine has much to offer learners of mathematics.But the writing of 'Calculator tatistics'has provided my co-author Barrie Galpin and myself with the opportunity tothink with greater precision about the learning benefits thatareparticularto statistics and how the calculatorcan help tobuild a bridge between the learner and the statistical ideasthey are grapplingwith.

Calculatortatisticss one of threebooks forA level publishedby A+B Books (Tel: 01780 444360), the other two beingCalculator Calculus and Calculator Graphing. For moreinformationabout these and othercalculator-basedclassroomresources published by A+B Books, see our web site(Web:www.AplusB.co.uk)Keywords: Statistics; Graphingcalculators; Simulation.AuthorAlan Graham is a lecturer in mathematics education at the Centre forMathematics Education, The Open University. He is also the 'A' n A+BBooks. 23 Waller Street, Leamington Spa CV32 5UEe-mail: a.t.graham@oopen.ac.uk

Mathematics n School, November 2003 The MAweb site www.m-a.org.uk 17