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^ INTRODUCTION ^

T HIS article is the second of a pair containingexamples where the graphics calculator can

provide students with particularly valuable insightsinto some of the big ideas in statistics. The rstarticle, entitled `Statistical Nuggets with a GraphicsCalculator', appeared in the previous issue of Teach-ingStatistics (volume 21, number 3, pages 70^3).

^ CONCLUSION ^

Biology and geography eld trips are a commonfeature of school and college life these days. I havetried to show that a statistics eld trip could alsobe a possibility and could be used to illustratemany of the main features of statistics syllabuses.

Realistically, I fear that I have probably not suc-ceeded in justifying such an expedition. However, Ihope that some of the ideas could be modied foruse in a `statistics day' somewhat nearer home,with pupils of all secondary ages.

The travel example could easily be adapted to uselocally available bus, coach or train times andfares. I have carried out a similar experiment usingcoach journey times, costs and mileages to 20 UKcities from Birmingham, which worked well.

The ranking experiment could be carried out usingpictures, photographs, music, sweets or any otheritems of interest. Using penny sweets is a cheapoption which has worked nicely with students inthe past.

Shells could be replaced by other creatures foundlocally such as land snails or owers or grasses in

dierent habitats. Finally, data from any of theemergency services could form the basis for thePoisson analysis. For those near the coasts in theUK, life-boat stations usually give records of theirlaunchings and further inland data from re orambulance stations could be used.

Acknowledgements

The author would like to thank the editor and thereferee for many helpful comments. She would alsolike to thank Malcomn Marshall and Israel Vieirafor their help with the illustrations. Finally sheacknowledges the patience of her extended familyin allowing her to collect data while on holiday.

ReferencesCaledonian MacBrayne (1996). Timetables andFares.

Kendall, M.G. (1943). The Advanced Theoryof Statistics Vol I. London: CharlesGrin.

McMillan, N.F. (1977). The Observer's Bookof Seashells of the British Isles. London:Frederick Warne.

Yonge, C.M. (1966). The Sea Shore. London:Collins.

"CC" " " " " " " " " " " " " " " " " "

OMPUTING CORNEROMPUTING CORNER

More Statistical Nuggets with a Graphics Calculator

KEYWORDS:Teaching;Boxplot;Histogram;Graphs.

Alan GrahamThe Open University, Milton Keynes, England.e-mail: [email protected]

SummaryThis article follows on from the one in theprevious issue concerning exploration of thegraphics calculator.

Teaching Statistics. Volume 22, Number 1, Spring 2000 . 11

Too often, when studying statistics, students getso absorbed by the tasks and calculations involvedthat the big idea underpinning these activities failsto register with them. Technology can sometimesremedy this diculty, releasing student time,energy and attention, allowing them to engage withthe concepts rather than being side-tracked by thenumber-crunching.

As with the previous article, all of the examplesgiven here are adapted from Handling Data, one ofve books in the series Calculator Maths which iswritten for the Texas Instruments range ofgraphics calculators, the TI^80, TI^82 and TI^83.The screens shown are copied from the TI^80; thecorresponding TI^82 and TI^83 screens will bevery similar but not always identical to these.Again as before, the adapted student worksheetshave been provided at the end of the article onseparate pages so that they can be photocopiedand used directly with students. Key teachingpoints and solutions/comments are given below.Examples 1 and 2 were in the previous article.

^ EXAMPLE 3: POSITION AND ^SPREAD

Key teaching points

Students can see a batch of data as a list ofnumbers but nd it harder to recognize that,when suitably organized and portrayedgraphically, these numbers form a distributionof some sort. This distribution provides ahandy visual picture of the size and spread ofthe values. This extract gives students theopportunity to explore what happens to thesize and spread of a distribution when certaintransformations are carried out on the entirebatch.

Solution/comments

First, notice the general shape of the boxplot whenthe data in L1 are drawn as one. Note that themaximum value is displayed by pressing

(a) Increasing each data value by a xed amount(20) changes the boxplot by shifting thepositions of the values to the right by 20 units,but the spread remains unchanged.

(b) Here we increase the values by a xed per-centage (50%). This changes the boxplot in adierent way: the positions of the values shiftto the right with the larger values shifting bymore than the smaller ones. As well asincreasing the median, this has the eect ofstretching the boxplot and therefore increasingthe overall spread.

(c) To make a stretch without shifting, enter

L1 1:5 :5MEANL1 ! L6:The values in L1 are all increased by 50% andthen reduced by a xed amount in such a waythat the new mean is equal to the old mean.Now plot L1 and L6 together as boxplots tosee the eect. Notice that the medians alsoapproximately coincide.

^ EXAMPLE 4: BARCHARTS ^AND HISTOGRAMS

Key teaching points

In primary school, pupils are not often requiredto dierentiate between a barchart and a histo-gram. However, later, when they learn aboutthe distinction between discrete and continuousdata, the need for corresponding graphicalanalogues becomes necessary. Unfortunatelythe waters are muddied by the graphicscalculator; themanual talks just of `histograms'and the machines appear only to be able todraw block graphs whose adjacent bars touch.

This extract draws attention to these issuesand shows students how barcharts can becorrectly drawn on their calculator with thebars centered on the values they represent andwith gaps between adjacent bars.

Solution/comments

(i) Two dice

(a) Use the commands shownhere.

(b) The corresponding plot forL3 looks like this. Thescores run from 2 to 12 andthe overall shape is peakedin the middle.

12 . Teaching Statistics. Volume 22, Number 1, Spring 2000

(c) The largest frequency was for a score of 7. Thismight be expected, since there are more waysto get a score of 7 than any other score(6 1; 5 2; 4 3; 3 4; 2 5 and 1 6).

(d) The smallest frequency was for a score of 4.You might have expected the smallest frequencyto correspond to a score of 2 or 12 since thereis only one way to get either of these scores(respectively, 1 1 and 6 6).

(ii) Centering the intervals

(a) Set XMIN 1.5(b) The plot should look like

this. The frequencies are:4, 5, 3, 15, 13, 16, 12, 13, 7,4, 7

^ EXAMPLE 5: GEE WHIZ GRAPHS ^

Key teaching points

Students tend not to realize how arbitrary arethe shape and position of a graph. Merelyadjusting the scales or the minimum andmaximum values on the axes can radicallyalter the impression given by the graph. Theseparameters are contained together on theWindow screen of the graphics calculator andcan provide an entertaining way for studentsto investigate just how much distortion theycan create in otherwise harmless data!

Solution/comments

(a) Sales patterns break intoroughly ve-year trends,rising quickly for the rstve years, then fallingslightly, then rising sharplyagain, and nally rising slowly.

(b)

(c)

(d)

(e)

The examples given here are adapted fromHandling Data, one of ve books in the seriesCalculator Maths by Alan Graham and BarrieGalpin.

The books in the series are:

FoundationsNumberAlgebraShapeHandling Data

Calculator Maths is available from:

AB Books, 15 Top Lodge, Fineshade, NearCorby NN17 3BB, UK.e-mail: [email protected];Web address: http://www.neshade.u-net.com/a b.htm; Tel: 44 (0)1780 444360

The remaining three pages of this article consistof photocopiable worksheets. Readers are per-mitted to make photocopies of these for use withintheir own institutions, but not for any form ofprot, without reference to the Teaching StatisticsTrust or to the authors of the article. The source(Teaching Statistics 22(1), 2000) should be ack-nowledged, and acknowledgement should also bemade that the photocopiable worksheets areadapted from Calculator Maths by Alan Grahamand Barrie Galpin.


^ EXAMPLE 3: POSITION AND SPREAD ^

Clear the data lists (the CLRLIST command is in the STAT menu) and switch o all existing plots bypressing

Enter the numbers below into list L1, press

and draw them as a boxplot. (Note that the TI^83 oers boxplots with or without outliers. If you areusing the TI^83, choose the boxplot option without outliers displayed.)

Suitable WINDOW settings would be XMIN 0, XMAX 100.

47, 52, 64, 55, 42, 49, 55, 49, 52, 54, 53, 4429, 57, 49, 49, 39, 47, 48, 48, 50, 58, 47, 30

You can now investigate what happens to the position and spread of a boxplot when the values are alteredin certain ways.

(a) Increasing each value by a xed amount:

Increase each value in L1 by 20 and store the result in L2.

This can be done from the Home screen by entering L1 20!L2.

Now press

and plot these new values in L2 as a boxplot. How do the position and spread of the values in L2 comparewith those in L1?

(b) Increasing by a xed percentage:

Increase each value in L1 by 50% and store the result in L3.

This can be done from the Home screen by entering L1 1.5!L3.

Press

and plot these new values in L3 as a boxplot. How do the position and spread of the values in L3 comparewith those in L1?

(c) Stretching without shifting:

Could you increase the spread of a batch of data by 50% without changing the mean?

Can you think of a way of doing this using a single calculator command sequence?


^ EXAMPLE 4: BARCHARTS AND HISTOGRAMS ^

(i) Two dice

Clear all statistical lists and switch o all plots by pressing

Reset your calculator's random facility to its factory setting using

0 !RAND. (RAND is in the MATH PRB menu.)

Store 99 dice scores in L1 using the SEQ command in LIST.

Note that this can be done on the TI^83 using the commandRANDINT(1,6,99) !L1.

(a) Store a further 99 dice scores in L2. Then store the sum ofL1 and L2 in L3 by entering L1L2 !L3.

(b) Choosing the Window settings shown here, plot the scores in L3as a `histogram'.

Describe the overall shape of the distribution. Press TRACE anduse3 and" to explore the frequencies of each outcome.

(c) Which outcome actually occurred with the greatest frequency?Which outcome would you expect to occur with the greatestfrequency? Why?

(d) Which outcome actually occurred with the least frequency? Which outcome would you expect to occurwith the least frequency? Why?

(ii) Centering the intervals

(a) Your answer to part (c) above should have revealed that eachbar was not centered on the value it represented. For example,the rst bar (corresponding to the score `2') was not centeredover the value 2 on the horizontal axis.

This is easily xed. Press WINDOW and change the XMINsetting to something suitable. Then press TRACE and checkthat your graph looks like the one shown here.

(b) Here is a sneaky way of ensuring that your barchart is drawncorrectly, so that:

. the bars are centered on the values they represent,

. there are gaps between adjacent bars.

Try this window setting. Then press TRACE and use3 and"to read o the frequencies and write them down.


^ EXAMPLE 5: GEE WHIZ GRAPHS ^

These data show the annual sales, in thousands, of tyres by the `Hot Air' tyre company between 1980and 1998.

Clear the statistical lists and enter these data into L1 and L2. You will nd it quicker to enter the `years',running from 1980 to 1998 using:

SEQ(X,X,1980,1998,1)! L1.

Set a suitable window and plot the data in L1 and L2 as a line graph. From the Graphing screen, useTRACE and" to read o the sales for each year.

(a) In one sentence, describe how company sales have changed over the period 1980 to 1998.

You can now investigate what happens when you adjust the Window settings.

(b) Set YMIN to 25. How does this aect the shape of the graph?

(c) The Hot Air company wants to make their sales gures look impressive. They choose to focus juston the last ve years' data. Adjust the window settings to make the sales over the last ve years lookas favourable as possible. Gee whiz!

(d) A second tyre company, `Smokin' Rubber', redraws the Hot Air sales graph to make their sales lookbad using the slogan `Hot Air down the tubes'. By focusing on the years when sales dropped, do thisfor them with a suitable window setting.

(e) Redraw the section of the graph between 1992 and 1998 to make the increase in sales look tiny.

Year Sales (1000s) Year Sales (1000s)

1980198119821983198419851986198719881989

12161921283332313029

199019911992199319941995199619971998

313742434445464749


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