DEMOGRAPHY© Volume 21, Number 3 August 1984

AVERAGING POPULATION DENSITY

John CraigOffice of Population Censuses and Surveys, London WC2B 6JP, United Kingdom

Abstract-Population density is a commonly quoted statistic. Almost nogeneral descriptive summary of the population of an area is completewithout a density listing, table or map. As each such density statistic is anaverage, it is worth considering what kind of average is being used. Thisarticle analyzes this and illustrates the effect of some alternative calcula­tions using population density data for Great Britain; the findings, howev­er, are of general validity.

The population density (d) of an areaas conventionally calculated, is an areaweighted mean of the densities of anyand every way in which the original areamay be spatially subdivided. For, fromthe definition of density

where A; and d, are the physical area anddensity of the ith subdivision of the origi­nal area; d, is itself a mean density ofsubdivisions of the ith area.

These conventional densities will bereferred to as "crude" densities. This isby analogy with crude birth and deathrates. These are also simple but do notallow for the distribution of the popula­tion by age, just as crude densities do notallow for distribution by area. The inde­pendence of the crude density from theactual distribution of population is veryconvenient. Nevertheless it is necessaryto consider the rationale of using areaweights, what standardizing for spatialdistribution would involve, what themost basic subset .. i' would be andindeed whether d, itself should be trans­formed. An earlier article (Stairs, 1977)listed some possibilities on a fairlypragmatic basis, and I have previouslycalculated a particular weighted meandensity (Craig 1975; 1980) without spe­cifically justifying the measure chosen.

d = 'i.A;d/iA; (1)

Deficiencies of Area WeightedDensities

Area weights are obviously appropri­ate with a land use approach when thebasic question is how many units of areaare inhabited at various densities. But ifthe prime interest is in people and thedensity at which they live, then it seemswrong to use A; as the weight and desir­able to use some kind of populationweighted average. If this propositionseems heretical, the analogy with thestandardization of crude birth or deathrates-or more generally with weightedindices as opposed to "unweighted"ones-provides reassurance. Anotheranalogy is that if densities are mappedconventionally then the different densi­ties are equivalent to height in a thirddimension (and the population of eachareal unit is equivalent to volume). Apopulation weighted density is the aver­age height about which all the columnsmapped would "balance," whereas thecrude density is equivalent to the heightthat would result if the total population(volume) were spread evenly over thewhole area. The former seems a reason­able average; the latter involves the as­sumption of a highly unrealistic popula­tion distribution. As we shall see thevalue of the two averages differs greatly.

All this may be laboring the obvious,

405

and log dGM = !, (Pi. log d;)/!' Pi (4)

Population Weighted Averages­Algebra

Arithmetically, the population weight­ed arithmetic mean density is simplest

The expression!' (Pi. log d;) in equation(4) is of interest because it is closelyrelated to an index of inequality which intum is associated with the ideas of "in­formation gain" and "entropy" (for ex­ample, Chapman, 1977). The geometric

(3)

(2)

dGM = II d."

where Pi and d, are the population anddensity of the ith subdivision. However,there are good reasons for transformingd.. Thus it seems very plausible that, fora given physical size of area, the differ­ence between a population of 500 andI ,500 is not the same as the differencebetween 10,000 and 11 ,000 persons­even though the absolute increase indensity is the same in both cases. Whatis relevant to the people concerned isthat in the former case there are threetimes as many of them while in the lattercase the increase is only 10 percent.Therefore it is the relative density differ­ence that matters rather than the abso­lute one. (There is an analogy here withphysical noise and social physics. Thehuman ear detects noise changes loga­rithmically and density is, in a sense, anindicator of the amount of human"noise"-in the sense of movement orinteraction-in an area, so there is anappealing symetry.) Using a logarithmictransformation gives the geometric meanas the appropriate average. We have

406 DEMOGRAPHY, volume 21, number 3, August 1984

but it is important to be able to empha- important thing is not the choice of ansize to the layperson that a population optimum set of areal units but an aware-weighted average is a sensible and desir- ness that a choice exists. We shall seeable measure. Of course a population later the effect of the choice of unit onweighted average is not independent of the mean density."i" and there are many possible spatialunit subsets-which is where the analo­gy with crude birth and death rates orother index numbers is inadequate. Thisis not a particularly welcome complica­tion but that is no reason for not consid­ering it. Indeed, it adds to the need tothink about what the fundamental unit ofdensity actually is.

The Basic Unit to Which DensityRefers

The basic unit appears to be an area,but if this unit contains several people itis still an average and should be subdi­vided until it contains, at most, oneperson. In effect, this would be the "per­sonal space" of an individual and wouldbe very small, with most of the countrybeing empty space. Accordingly, a popu­lation weighted average would be unreal­istically high.

One explanation of this anomaly isthat, if the emphasis is on people, thenthe basic unit should be area per personrather than persons per unit of area; thenthe "personal space" is larger. Even so,further thought is necessary. Thus sup­pose we had a series of small, equallysized, isolated villages each some dis­tance from the next village. To draw aline around the external boundary ofeach village ignores the empty spacebetween villages; to allocate the emptyspace to those living on the perimeter ofeach village is too extreme. It would bebetter to give each village a share of the where (Xi is the proportion of the popula-surrounding countryside, so the basic tion living at density d,unit is desirably, and not through neces­sity, that for a group of people. Thelesson is that at one extreme we want toavoid too small a unit which separate the"villages" from their hinterlands; at theother extreme we must avoid too large aunit which in effect combines differenttypes of settlement. At this stage the

Averaging Population Density

mean provides a link between the meandensity and these more sophisticatedmeasures.

It is also relevant that with a hierarchi­cal system of units there is a simpleconnection between the geometricmeans at different levels in the hierar­chy. Suppose we have a "country"which is subdivided into "regions," eachregion into "local authorities" (LAs);each local authority into "wards" and soon. Taking the country's geometric meancalculated first from "local authorities"and second from "wards" we have

407

to use 1971 Census data for Great Brit­ain. Though the areal units are peculiarto Great Britain the patterns they revealwould apply to most hierarchical sys­tems. In addition, it is convenient thatfrequency distributions for the more nu­merous areal units have been tabulated(Craig, 1980) and as an administrativehierarchy of areal units, there is an alter­native independent geometric one of gridsquares as well. The results of the calcu­lations are set out in Table 1 and Figure 1provides a diagrammatic analysis. Themost obvious patterns in Figure 1are: (a)

summation forall regions

for allLAs

IPk • log d; = IPj ' log d,

for allwards

IP; . log d, = IPj • log dj

summation forall LAs

+ 77P, log ( ~; )

+ L L Pk' log (~k)j k J

summation forall wards

(d;)+ I P;' log dk

where dj is the regional density, dk is thelocal authority density and d, the warddensity. Hence for the geometric meanthe effect of any number of hierarchialdisaggregations of an areal unit is easilyanalysed. Other transformations of den­sity, apart from the logarithmic one, areof course possible. These have not beenexplored, partly because the main aim isto draw attention to the possibility ofweighted densities rather than to make adefinitive study; but also because thelogarithmic transformation is a simpleone and has many appealing features.

Population Weighted Averages­Calculations

To illustrate the differences that occurin practice between the geometric meanand the arithmetic mean it is convenient

all the population weighted means farexceed the crude area weighted density.Both the means increase as the units areprogressively subdivided, though notvery rapidly relative to the increase inthe number of areas; (b) as the areal unitsincrease in number (and decrease in size)the rate of increase of both means falls;and (c) both means keep in step in asmuch as the two lines in Figure 1 keep,from the counties onwards, an approxi­mately equal distance apart. The arith­metic mean is about twice the geometricmean.

Some of these features are easy toexplain. With regions as the unit, little ofthe variation in density is revealed. Re­peated subdivision produces areal unitswhich are progressively more sensitiveto the settlement pattern and so averagedensity rises. But another factor is at

408

work, in addition to smaller areal unitsdistinguishing progressively smallerbona-fide settlements. Any (and every)subdivision of an areal unit increases theaverage population weighted density un­less, freakishly, each subdivision has thesame density as the original unit. Ac-

DEMOGRAPHY, volume 21, number 3, August 1984

cordingly, the mean density continues torise even if the units are being subdi­vided into very small-and so less mean­ingful-units. A further practical com­plication is that units which seem reason­able for one size of settlement may beless satisfactory for another; in England

Table l.---Computed Values of Average Density, Great Britain, 1971.

Areal Unit Used

Numberof Areal

Units

Population Weighted Means

Arithmetic Geometric(AM) (GM)

Persons per Hectarea

Ratio ofAM to GM

4.3 3.3 1.3

13.0 5.8 2.220.8 8.8 2.4

26. I 12.8 2.0

33.4 16.2 2.1

21.2 10.4 2.038.8 22.3 1.7

Great Britain(gives crude density)

bAdministrative

Regions

"New" local authoritiesCountiesDistricts

"Old" local authorities

Wards/parishes

Grid SquaresC

10 km sidesI km sides

10

64459

1,765

17,643

2,694152,000

2.4 2.4 1.0

a l hectare = 2.47 acres _ 0.00386 square miles.10 persons per hectare _ 4.05 persons per acre _ 2,591 per square mile.

bEngland and Wales has a different administrative system than Scotland. Bothsystems, however, were reorganized in the mid-seventies in such a way that thenumber of basic areal units was greatly reduced, hence the distinction between"new" local authorities and "old" ones. In addition, the terminology used isthat for England and Wales. Scotland has a different system of local govern­ment and equating the two systems is somewhat arbitrary. But as Scotland'spopulation is only 10 percent of Great Britain's, and as it is a comparison ofmeans that is being made, this is not important. Similarly, in England,Greater London and the London boroughs are a different administrative systemthan the rest of the country; Greater London has been treated as a 'county'and the boroughs as 'districts'.

cGreat Britain has a standard national grid which enables any location to begrid referenced to a high degree of accuracy. This was done in the 1971 Cen­sus and hence populations of the standard lkm and 10km sided squares areavailable. Grid squares can be uninhabited; the figures in this table referto inhabited squares. Refer to the text for a comment on this.

Averaging Population Density 409

,/

1 km sided: gri d squares

.......:: /": /"

A'IIII,

Z Geometric.........,:~~:..---- means

10 km sidedgrid squares

.Old localauthoriti esI

III

LOistricts

Arithmetic _---~~~means

LCounties

50

II>CoII>s,

~ 20

~ 10II>CQI"0

C

'"~1: 5....s:'"QI~

Co.~....'"~ C~':.d.:_~r_e~ _w.:~!!h::d_ ~:n_s~ ~ _

!1. + (for all sets of areal units)2 Great Britain

10i

100 1000i

10,000i

100,000

Number of areal units(log scale)

Figure I.-Population Weighted Mean Density Increase with Subdivision of Areal Unit.

large towns are identified at the districtlevel but smaller towns are not. Theconvexity of the graphs in Figure 1 alsoshows that subdivision has a diminishingeffect. Mathematically, this convexitycan be related to the expansions ofI,(P . log d) as the areal units are repeat­edly subdivided (see equations 5 and 6).

The averages for the two sizes of gridsquare follow the same general pattern.But, as grid squares are not purposively

drawn with regard to settlements, theygive lower averages than would a similarnumber of administrative units. In asense grid squares are less efficient. Aparticular problem with the one kilome­ter-sided squares is that many have zeropopulation and density. The calculationsexcluded these squares since, as no onelived in them, they were irrelevant. Ineffect this means that "empty space be­tween villages," to use the earlier analo-

The Practical Consequences

It is clear that the two weighted densi­ties give values which differ substantiallyfrom the conventional crude density.This would not matter if trends overtime, or relative differences between ar­eas, were little changed; however inpractice the changes can be drastic­time trends may differ not merely in sizebut in direction.

For example, for Great Britain be­tween 1931 and 1971 population, andhence crude density, rose by 20.5 per-

AN EXPONENTIAL MODEL

With the classic spatial model of popu­lation distribution within an urban settle­ment, in which there is an exponentialdecline in density with increasing dis­tance from the center, the average crudedensity of a circular area can be derivedas a function of the central density (de>and the boundary density without refer­ence to, say, the physical size of thesettlement. This is also the case for thepopulation weighted means-as theboundary density tends to zero, dAM

tends to dc/4 and dGM to d.Ie".Data for Great Britain are an aggrega­

tion for many settlements but it can beshown (Craig and Frosztega, 1976) thatat the ward and parish level the actualdistribution of population with respect todensity is similar to the distribution pro­duced by a number of separate settle­ments, each with the same central densi­ty (or, more approximately, with thesame average central density). A fortu­nate consequence is that the algebra isthe same as that for a single settlementand can be applied to the data used inFigure 1.

410 DEMOGRAPHY,volume 21, number 3, August 1984

gy, is being ignored. This is a special Fitting an exponential model to thecase of the proposition that small units population and density data for 18,000will produce artifically low and high den- wards and parishes of Great Britain gavesities and that it is not sensible to subdi- an estimated 1971 central density (Craig,vide indefinitely. (The low value of the 1980; p. 3) of 128 persons per hectare.ratio of the AM: GM is a result of this This, in turn, gives a lower limit of 32 (=problem; ideally the uninhabited squares 128/4) for the arithmetic mean (the calcu­would be combined with nearby lated value in Table 1 is 33.4) and of 17.3square[s].) This would of course reduce (= 128/e2

) for the geometric mean (Tablethe mean densities and thus the slope of 1 shows 16.2). The figures of 32 and 17.3the grid square relationships in Figure 1. are termed lower limits because in prac­But apart from the practical difficulties tice the boundary density never quiteof doing such aggregations for the unin- reaches zero, and the limiting ratio of thehabited squares, the same principle ap- two means in the model is e2/4 (= 1.85)plies to a densely populated one kilome- compared with the ratios of about 2.0 inter square surrounded by sparsely Table 1. The differences between thispopulated ones. In other words, the un- model and Table I are of no great signifi­inhabited squares are just an extreme cance being due to the approximate "fit"case of the problem previously men- of the model. But the model has twotioned-how far it is desirable to subdi- virtues: (a) it provides algebraic toolsvide areas. with which to analyse the two means,

and (b) a fresh insight into the phenome­non that mean density continually risesas the areal units decrease in size. Fromthe model it might appear that the twomeans should tend to fixed constants (del4 and dele2) . In practice this is not so asthe central density is itself not indepen­dent of the areal unit. Any central areacan be subdivided and, since some partswill be more dense than others, centraldensity appears to rise. To avoid suchsubdivision takes us back to the proposi­tion stated earlier that beyond a certainpoint it is misleading to subdivide areasany further.

Averaging Population Density

cent whereas the two population weight­ed means of wards and parish densitiesfell: the geometric weighted mean by22.5 percent and the arithmetic mean byno less than 58.3 percent. The reason forthe differences is that the population ofthe country rose but the densities andpopulations of big cities fell. Thus in1931 there were 3.4 million people inGreat Britain living in wards with a den­sity of 250 persons per hectare or more(and such densities went up to 400 or 500per hectare). In 1971 there were only23,000 people in this density group. As aconsequence of this-and also analogousbut progressively smaller reductions inthe number of people living at warddensities of 100-250 persons per hect­are-in 1971 people lived at a loweraverage ward density than in 1931.Moreover, the influence of the very highdensities, where the biggest populationdeclines took place, are greater for thearithmetic mean than for the geometricmean, so the former declined by muchmore than the latter. Theoretical reasonsfor preferring the geometric mean havealready been given. In this particularexample, a look at the frequency distri­butions also suggests that the geometricmean provides the more meaningfulmeasure, but there may be projects forwhich the properties of the arithmeticmean are appropriate.

It also is the case that other levels ofareal unit shown in Figure 1 would give adifferent measure of density change overtime, with the larger areas showing less­er declines and, ultimately, increases.See Craig (1979)for such results togetherwith a more detailed discussion.

So much for trends over time. Com­parisons between areas, at a given time,will also be different with weighted aver­ages and for much the same reason. Thusa region which consists of several freestanding cities, each of which is sur­rounded by a sparsely populated hinter­land, will have a low crude density; thisis usually nearer the hinterland than the

411

cities since the physical area of the citiesis small. But a population weighted den­sity will be nearer that of the cities,because it is in the cities that most of thepeople live. Because the mix of citypopulation (and area) usually varies fromregion to region, the relative positions ofsuch regions on a weighted density scaleare likely to be different from that of acrude scale, especially when cities ofdifferent densities, and varying amountsof suburbia, are allowed for.

As with the example of the trend overtime, although the weighted densities areclearly often more appropriate than thecrude density, the choice between thearithmetic mean and the geometric meanis less obvious. However, it seems likelythat the geometric mean will usually bepreferable. As is often the case in statis­tical geography, there is no uniquelycorrect set of areal units. Rather thereare different units and frequently it willbe sensible to use several: the differ­ences that result are themselves informa­tive. Obviously the same kind of unitshould be used for each between-areacomparison; but, as ever, internationalcomparisons are especially difficult be­cause each country has its own sets ofareal units and settlement patterns. Inthis case also, it will usually be best tomake comparisons for more than one setof areal units to see if any unambiguousconclusions emerge.

CONCLUSIONS

The conventional crude populationdensity is not a good measure of thedensity at which the population lives.Instead a population weighted meanshould be used and the geometricmean-equivalent to a logarithmic trans­formation-has many advantages; butother transformations are possible.Some empirical and algebraic propertiesof this mean have been described and ithas been argued that there is a pointbeyond which, in deriving average densi­ty, it is wrong to subdivide areas. This

412 DEMOGRAPHY, volume 21, number 3, August 1984

point has not been identified, but fordensities a working hypothesis would beat the ward and parish level. This arbi­trariness is not very satisfactory but is anexample of the area scale-effect whichaffects many kinds of spatial analysis.Generally there is no unique best set ofareal units for spatial analysis, merelydifferent sets; fortunately the averageis not particularly sensitive to smallchanges in the number of units. Whatev­er units are used the population weightedaverage is certainly very different, andfor many purposes better, than the crudedensity figure. It often differs in absolutevalue, in trends over time, and betweenareas, from the crude density. There aretheoretical and empirical reasons for pre­ferring the geometric mean to the arith­metic mean.

ACKNOWLEDGMENT

I would like to thank Mr. M. A. Baxter

and the reviewers for some helpful com­ments on earlier versions of this paper.

REFERENCES

Chapman, G. P. 1971. Human and EnvironmentalSystems: A Geographer's Appraisal. London:Academic Press.

Craig, J. 1975. Population Density and Concentra­tion in Great Britain 1931, 1951 and 1961. Studieson Medical and Population Subjects No 35:Office of Population Censuses and Surveys.London: HMSO.

Craig, J. 1979. Population Density: Changes andPatterns. Population Trends 17:12-16.

Craig, J. 1980. Population Density and Concentra­tion in Great Britain 1951, 1961 and 1971. Studieson Medical and Population Subjects No 42:Office of Population Censuses and Surveys.London: HMSO.

Craig, J. and J. Frosztega 1976. The Distribution ofPopulation in Great Britain by Ward and ParishDensity, 1931, 1951 and 1961. Area 8:187-190.

Stairs, R. A. 1977. The Concept of PopulationDensity: A Suggestion. Demography 14:243­244.

© British Crown Copyright 1984