Durham Research Online

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1 Durham Research Online Deposited in DRO: 25 August 2015 Version of attached le: Published Version Peer-review status of attached le: Unknown Citation for published item: Visvalingam, M. (1976) 'Chi-square as an alternative to ratios for statistical mapping and analysis.', Working Paper. University of Durham, Department of Geography, Census Research Unit, Durham. Further information on publisher's website: Publisher's copyright statement: Additional information: Use policy The full-text may be used and/or reproduced, and given to third parties in any format or medium, without prior permission or charge, for personal research or study, educational, or not-for-prot purposes provided that: a full bibliographic reference is made to the original source a link is made to the metadata record in DRO the full-text is not changed in any way The full-text must not be sold in any format or medium without the formal permission of the copyright holders. Please consult the full DRO policy for further details. Durham University Library, Stockton Road, Durham DH1 3LY, United Kingdom Tel : +44 (0) Fax : +44 (0)

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4 WORKING 8 NOVEMBER mli/;up:~ AS.A:N ALTERNATIVE TO RATIOS FOR STATISTICAL MAPPING AND

5 Introduction 2, 2 An Altel"'U'ltive Measure? 4. Discussion Formulation of of Chi~square Values Unreliable Values for Small 4.4. Events the of a 4.5. Variables for Multivariate Ratios Other Merits of the Model Conclusion References

6 common forms non-statistical work to absolute numerical of numerical measures for such as for variables in in are still very and for mathematical and and multivariate The of } has several takes into account both absolute and relative the norm; can be a a ic to the of variables. it demands an assessment before; rather than the mechanical of Illustrations of the the statistic are, in this confined d.ata set, the one-kilometre square on sex Durham from 1971 The choice of sex made for a of reasons In the on of items of data for each areal data of the various statistical discernible the total each one-kilometre the are not affected reasons of the could be

7 and illustrations this areas a marked of one sex or the one-kilometre square data include the entire basis is a o:f all incidencea can be confined consideration sex in this paper and the conclusions reached have much 2 were listed and the proportion, expressed as the as the ratio o:f the exces2 or deficit of males as : the total These autho.rs were o.f the that these three measures removed the effects of variations in size were thare.fore suitable for inter-area, inter-group and temporal Since numerical or deficit males is the and considered measure these noted that the sex r;at sex was a "more that due to its sll of each measures of derived from should

8 ations two. If values 1 Jneasures ) and 62 and 46 squares wi extreme cases merit any further 1) and very low stnall ial less the Chance variations the of females small can lead to or and that the effect the of the still less ratio and the small He some of the ratios derived from small as census data Ratios have traditions macroand for international to is the case one-kilometre adverse effect the extremes it is sma ll the while meas.ures

9 1.,

10 -5 Kilometre Num.be:r ot square males femsles diff o o * ~23 -s * , * B * n -3. * * not l TABLE 2

11 Measu:res and and 2 illustrates these the and shows that effect cases As ( is line a=l, it skewed the arithmetic differenc es below the value 1 sex ratio an arithmetic identifies inverse female either form for the of national w which are close to the norm, neither a broader range like the the sex ratio is a as extremes which are ; are bound the calculation coefficients and Table 3 cut-offs is to mask areas between the sexes vary Total 10 loo 1,000

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13 the scatter ( were be derived as an inverse ion of ratios would still be ummitable for sizes. ~ the use numerical differences alone is 1 the discrimination differences ions in most cases relative differences in small popula ions. Ana i.ona difference in tion s is to tions ones. alternatives and removed the worst anomalies squares so far makes clear ths a distorted &xists. If ive basis, One r&lat between males and females lacks thus the various t-tests are circumvented la middle-sized so variance more standardised for the range si~e in each subset. However not the delimination 1 to if not will also vary with different ranges of and groups; studies more difficult. with heterosoedast necessary f()r delimitation "small" and in diffe.rent data sets those of a.nd Great for that the selected measure should over the lues suitable internal be

14 Numerical differences in a range of ion constant o:f all one-kilometre squares less than 100 males or fem~~tles

15 sex from that a.n x relative where "" The icance of then assessed at different levels of confidence. If a numerical sexes, and the particular the that Thus for area tenus In r thus 2 2 For lues much 4. 5, 50 1

16 in of difference to -11 rated as a since the while a difference in difference 4) a value statistic size. A i:f the formula :for of males females is inverted derive for a that inversion In this ca.se tolerance + Hence, square root a ( define variat indicate levels are shown :1n 5 This measure automat With all males# (:fo.r not at the w1thocut due to the

17 2 4. Tolerance as a function of tor values ot

18 99 and 95 per cent confidence limits of for one ot freedom

19 the of t between areas ma purpose, where in tive and remain exact and = sgn function Thus ma ive and the same as for show that the spat values a is of at 95 cent confidence level unrelated (see also Table 1 express errors additional and extend the pattern per cent in 1971 Durham had les as aga les (. a difference of and a mascul would of one-kilobatre squares fema than with male Table 5 lf an numbers of males and females are indica the va If the for males and females were 489T + where total number of one-kilometre 7 shows the areas of male and female 99 per cent levels,. of per squares with a and males

20 50 per

21 Distribution map of sex ba,sed males

22 a new based average there are more of females of males, more extreme and and 7 are observable between Peterle e and around and norththe east More detailed variat in sex would be necessary to Pr'oc.es:se,s, and would be of sex tables other variables from and census data. may be measured terms of some issue in use Values for are derived tables or related some Normal lljleo,er;matl'll 1971) sex and norm of per cent males is assesses the of with males 9 per cent did not make a one-kilometre when of of the a and that

23 While convenient the of areas with extreme male in other instances and map data loos ions. This is useful when a derived the basis istioal criteria in a. skewed lues. measure considered unreliable when the eel table is 956 p. The conventional solution to loca ions and ascertain whether a similar spatia ttern results. As ha beer.% ratio.s are also most for small l pattern of sex compos different levels of is be a of the 1971 one-kilometre census da for Great Brita the and and tio measures, The results will a 11 :f cn~ms is of random events distribut 1 is ive can be utilised as another means the of variat in domai11. Variables to multivariate tions because the va 11 dominate of correla lues is to be.undertaken Evans and the will

24 because the are and who are e for the use the :measure p , Variables closed from zero retired as a of work as a of take the a closed ratio form for can be derived from the mascul variables such as per the age further consideration,. discrete without reference to conditions of and should when for the use of which base estimates of standard tolerance evaluated of data set, which is less values of would have data influence the of values were 4 4 the and relative are known. variables a from measures appears be :measure of and lends set

25 forward this paper. At the same time are conducted into the of variables, such as retirement and for multivariate The author members of the Census for their comments the selection of non-sta and Nick Cox for many the process are course for his editorial on the and

26 based W.,, eds, ( Research ( J.,S M an evaluation of of leukaemia