a statistics field trip?

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^ INTRODUCTION ^ D URING a recent holiday on the small island of Tiree in the Hebrides, our party, which consisted of two families each of three people, three males and three females, stayed in the Alan Stevenson Centre at Hynish. This centre, which is owned and run by the Hebridean Trust, occupies the former storehouse used during the construction of Skerryvore Lighthouse (built between 1838 and 1843 by Alan Stevenson, the engineer uncle of Robert Louis Stevenson). The centre caters for small school parties at other times of the year and I started thinking about possible statistics projects which could be used to justify a ‘statistics ¢eld trip’ to such an idyllic spot. Tiree, gaelic name Tir-Iodh, the land of corn, is one of many islands o¡ the west coast of Scotland. Information about Tiree can be found on the Web site http://www.scotland-info.com/tiree.htm. The west-coast islands are in three groups, the southernmost being the islands of the Clyde, including Bute; further north lie the Inner Hebrides including Tiree, Skye, Mull, Coll and Lismore, and further north still the Western Isles (formerly known as the Outer Hebrides), including Barra and Lewis. An excellent map showing the position of all the islands can be found on the Caledonian MacBrayne Web site http://www.calmac.co.uk/. Tiree is the outermost of the Inner Hebrides; it is a small island of about 30 square miles and with a population of between 800 and 900. Figure 1 gives a map of Tiree showing the location of Hynish (1). ^ TRAVEL TO TIREE: ^ CORRELATION AND REGRESSION Tiree is reached either by air from Glasgow or on the Caledonian MacBrayne (Calmac) ferry ‘Lord of the Isles’ which sails to Scarinish (2) on the island from Oban on the mainland. At the time of our visit, there were complaints in the local paper, An Tirisdeach, that the costs of the journey to A Statistics Field Trip? KEYWORDS: Teaching; Correlation; Ranks; Hypothesis tests; Poisson. Susan Meacock University of Southampton, England. e-mail: [email protected] Summary Ideas for statistical investigations can be found in all sorts of places, even on holiday. Fig 1. Map of Tiree. Teaching Statistics. Volume 22, Number 1, Spring 2000 . 7

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Page 1: A Statistics Field Trip?

^ INTRODUCTION ^

D URING a recent holiday on the small islandof Tiree in the Hebrides, our party, which

consisted of two families each of three people,three males and three females, stayed in the AlanStevenson Centre at Hynish. This centre, which isowned and run by the Hebridean Trust, occupiesthe former storehouse used during the constructionof Skerryvore Lighthouse (built between 1838 and1843 by Alan Stevenson, the engineer uncle ofRobert Louis Stevenson). The centre caters forsmall school parties at other times of the year andI started thinking about possible statistics projectswhich could be used to justify a `statistics ¢eld trip'to such an idyllic spot.

Tiree, gaelic name Tir-Iodh, the land of corn, isone of many islands o¡ the west coast of Scotland.Information about Tiree can be found on theWeb site http://www.scotland-info.com/tiree.htm.The west-coast islands are in three groups, thesouthernmost being the islands of the Clyde,

including Bute; further north lie the Inner Hebridesincluding Tiree, Skye, Mull, Coll and Lismore,and further north still the Western Isles (formerlyknown as the Outer Hebrides), including Barraand Lewis. An excellent map showing the positionof all the islands can be found on the CaledonianMacBrayne Web site http://www.calmac.co.uk/.Tiree is the outermost of the Inner Hebrides; it is asmall island of about 30 square miles and with apopulation of between 800 and 900. Figure 1gives a map of Tiree showing the location ofHynish (1).

^ TRAVEL TO TIREE: ^CORRELATION AND REGRESSION

Tiree is reached either by air from Glasgow or onthe Caledonian MacBrayne (Calmac) ferry `Lordof the Isles' which sails to Scarinish (2) on theisland from Oban on the mainland. At the time ofour visit, there were complaints in the local paper,An Tirisdeach, that the costs of the journey to

A Statistics Field Trip?

KEYWORDS:Teaching;Correlation;Ranks;Hypothesis tests;Poisson.

Susan MeacockUniversity of Southampton, England.e-mail: [email protected]

SummaryIdeas for statistical investigations can be found inall sorts of places, even on holiday.

Fig 1. Map of Tiree.

Teaching Statistics. Volume 22, Number 1, Spring 2000 . 7

Page 2: A Statistics Field Trip?

Tiree were high compared to the costs of journeysto other islands o¡ the west coast of Scotland. TheCalmac timetable (Caledonian MacBrayne 1996)gives travel costs and journey times to all 23islands which it serves. From this, journey timesand costs (for a single passenger ticket) for the 16islands accessible directly from the mainland wereobtained. (Current information is available on theCalmac Web site.) Journey times varied from 30minutes for Bute and Skye to 5 hours for Barraand costs from »1.95 for Lismore to »15.95 forBarra. At that time Tiree had a journey time of3 hours 55 minutes (via Mull and Coll) at a cost of»9.85. Using a computer statistical package, thecorrelation between cost and time was found tobe 0.949, a highly signi¢cant correlation for asample size of 16. The regression line estimatingcost (in pence) in terms of time (in minutes) wasgiven by:

cost � 68:0� 4:89� time� error

This gives, for a journey time of 3 hours 55minutes (235 minutes), an expected cost of »12.17.In these terms the Tireasdachs appear to have littleto complain about! The plot in ¢gure 2 shows that6 of the islands are paying less than expected with10 paying more. Lewis in the Western Isles appearsto be su¡ering most. Of course, it could well beargued that the ferry to Tiree does not take theshortest route and that a fairer measure would beto use sea distances rather than journey times. Toexpand this project, other costs (passenger returns,vehicles) could be used or a comparison made withair fares where these are available. Anotherexample of the use of correlation and regressioncould include examining the relationship betweenpopulation and area on these west-coast islands.

^ `SCULPTURE IN WATER': ^RANKING METHODS

During our holiday we paid a visit to an exhibitionentitled `Sculpture in Water' at Milton (3), in which10 international artists had created sculptures in aseries of watery areas (known as lochans), mainlyusing material found locally. We saw 11 of the 13sculptures (in a typical Hebridean downpour) andthe 6 members of the party ranked them in orderof their personal appeal. Spearman's rankcorrelation coe¤cients were then obtained for eachpair of rankings. Table 1 gives the values (correctto 3 decimal places) of Spearman's coe¤cient foreach of the pairs. The ¢gures in bold type aresigni¢cantly greater than zero at the 1% level;those in italics are signi¢cant at the 5% but not atthe 1% level. (For 11 objects the 5% critical valueis 0.536 and the 1% critical value is 0.709.)

Figure 3 illustrates all the correlations signi¢cantat the 5% and 1% levels. Four of the participantsclearly agreed fairly well in their orderings, whileone, JB, had negative coe¤cients with three of theother participants.

For each group of three people the averagecorrelation can be obtained. These were 0.75 and0.16 for the two families, 0.48 for the women and0.32 for the men.

It can be shown (Kendall 1943) that for groups ofm individuals ranking n objects, the averagecorrelation lies between ÿ1=�mÿ 1� and �1. Themaximum value of �1 will always be obtainedwhen each of the m individuals award the samerankings. The minimum value of ÿ1=�mÿ 1� isobtained when the sum of the rankings given bythe m rankers is the same for all of the n objectsbeing ranked and is equal to m times the averagerank, �n� 1�=2. Hence if n is even and m odd andassuming no tied ranks, the minimum cannot beachieved. It is an interesting exercise to ¢nd sets ofm rankings with minimum average correlation forFig 2. Regression of cost against time.

TB JB SB SM EM GM

TB 1.000 0.191 0.373 0.546 0.136 0.427JB 1.000 ÿ0.082 ÿ0.118 ÿ0.046 0.473SB 1.000 0.718 0.655 0.564SM 1.000 0.764 0.709EM 1.000 0.782GM 1.000

Table 1. Spearman's rank correlation coe¤cients

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

Page 3: A Statistics Field Trip?

m even and any value of n, which is easy, and form odd and n odd, which is more challenging.

Clearly such ranking experiments could easilybe adapted to other situations and to other groups.Such an experiment allows consideration of rankingmethods and levels of signi¢cance. Considerationof how to obtain an overall ranking of the objectscan also be made.

^ COWRIE SHELLS: DATA ^ANALYSIS AND HYPOTHESIS TESTING

Tiree is famous for, amongst other things, its abun-dant wildlife and this could provide a fruitfulsource of data. The sandy beaches contain manyshells, in particular those of the two species ofcowrie found in the British Isles, the EuropeanCowrie (Trivia monacha) and the Spotted Cowrie(Trivia arctica) (¢gure 4). The European Cowrie isunspotted but the Spotted Cowrie has three spots

which are purplish-brown in colour. The EuropeanCowrie is generally found to be smaller in size thanits spotted relative (McMillan 1977).

Over three days I collected and measured thelengths of 338 cowries (270 unspotted and 68spotted) on 3 beaches, Hynish (1) and Balephuil(4) on the south-west of the island and Vaul (5) onthe north-east. Table 2 summarizes the resultsobtained.

The data can be used for exploratory dataanalysis (stem-and-leaf, box-and-whisker), histo-grams, Normal plots and chi-squared goodness-of-¢t tests. (The Normal ¢t was accepted for boththe spotted and unspotted data.) One-tailedhypothesis tests of the equality of the means of thetwo species were rejected at the 0.1% level orsmaller for the overall data and for each individualbeach. Two-tailed hypothesis tests of the equalityof size of specimens on the di¡erent beaches wererejected at the 5% level or smaller except for

Fig 3. Chart to show signi¢cant Spearman correlation coe¤cients.

Fig 4. Spotted Cowrie Trivia arctica from the top and from the bottom.

Teaching Statistics. Volume 22, Number 1, Spring 2000 . 9

Page 4: A Statistics Field Trip?

Hynish/Balephuil for both spotted and unspottedspecimens.

Independent two-sample tests were used through-out on a statistics package which also tested forequality of variances. Clearly analysis of variancewould have been a preferable method with ad-vanced students.

Statistical tests of the di¡erences between theproportions of spotted/unspotted at the threelocations were made. The chi-squared test on thetable of numbers of the two species on the threebeaches gave a highly signi¢cant value of 26.56with 2 degrees of freedom, thus the hypothesis ofequality of proportions can be rejected. Con¢denceintervals for the proportions on the three beachescould be obtained.

Tests on other types of shells could have been usedinstead. Possible experiments could include similartests on the relative sizes of the two species oftortoiseshell limpet, Acmaea, or on the proportionsof the di¡erent colours of the £at periwinkle,Littorina littoralis, or on the height/diameter ratioof the common limpet, Patella vulgata, whichvaries with habitat (Yonge 1966).

Tiree is also famous for its £ower-¢lled machair(low-lying sandy beach or boggy links a¡ordingsome pasturage) and similar analyses on plantheights in varying locations would have been analternative use of hypothesis testing.

^ MEDICAL EMERGENCIES: ^THE POISSON DISTRIBUTION

Medical emergencies on Tiree are dealt with bysending an air ambulance from Glasgow to theairport (6). An Tirisdeach reported that in the twoyears prior to our visit there were 33 ambulancecalls each year. In the week before our visit therewere 2 call-outs. Assuming that call-outs are

random events with a mean rate of 33/52 perweek, what is the probability distribution, p�x�, ofthe number of call-outs per week and in particularthe probability of two or more? Assuming aPoisson distribution with m � 33=52 � 0:6346, wehave

p�x� � eÿmmx=x! x � 0; 1; 2; . . .

or more conveniently

p�0� � eÿm and p�x� � fm=xgp�xÿ 1�

for x � 1; 2; 3; . . . :

This gives table 3, in which the values of p�x� aregiven correct to 3 decimal places.

Thus in only 13.4% of weeks will there be 2 ormore call-outs and in over half the weeks there willbe no call-outs. Data on the actual numbers ofcall-outs over the two years could be obtained andused to estimate the randomness, or otherwise, ofthe distribution. Possible sources of nonrandom-ness would include seasonal e¡ects due to touristsin the summer and bad weather in the winter. Forrandom data the gaps between call-outs wouldfollow an exponential distribution and this couldalso be tested.

Tiree has its own weather station and is acoastal report station which features on the UKshipping forecast. It is also known as the`sunshine island' because of its good sunshinerecords, principally in May and June. Weatherrecords, which can be obtained from the Website http://www.wunderground.com, are thus an-other possible area of investigation which couldgenerate some interesting hypotheses.

Unspotted Spotted

Day Beach Sample size Mean SD Sample size Mean SD

1 Hynish 168 8.90 1.19 42 10.55 1.312 Balephuil 14 9.50 0.94 16 11.25 1.483 Vaul 88 7.99 1.01 10 9.60 1.35

Total 270 8.64 1.22 68 10.57 1.43

Table 2. Summary of data for cowries

x 0 1 2 3 4 � 5

p�x� 0.530 0.336 0.107 0.023 0.004 0.000

Table 3.

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

Page 5: A Statistics Field Trip?

^ 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 modi¢ed 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

di¡erent 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 and

Fares.Kendall, M.G. (1943). The Advanced Theory

of Statistics Vol I. London: CharlesGri¤n.

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