SEX DIFFERENCES AND THE STATISTICS OF CROWD FLUIDS

b y AnMciny Palmer and Robert Bailey‘

Univcrsily oj S,ussex, England

In a recent paper L. F. Henderson suggested that the movement of people in a crowd represented an analogous system to that of a collectionof gasmolecules. Specifically he sug- gested that the classical Maxwell-Boltzman theory of the molecular system might also describe the distribution of velocities found in the movement of people. Henderson’s em- pirical results produced good agreement with such a theory except for deviations near the modes of the velocity distributions. These he attributed to sexual inhomogeneity in the sampled population but was unable to test this.

In this paper further empirical results are presented which confirm Henderson’s sexual inhomogeneity conjecture. An analysis of variance demonstrates both sex and age as being highly significant independent factors, and in addition establishes the normality of the residuals.

I t is shown that the observed velocities of people’s movements obey a normal distribu- tion, and as the Maxwell-Boltzman distribution is equivalent to the normal distribution in the one-dimensional situations described, there is no need to invoke the statistics of molecu- lar systems to describe those of the movements of people.

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IT HAS BEEN suggested by Henderson (1971) that the movements of people in a crowd may parallel those of a collection of gas molecules and hence, with certain assumptions, may be described by those statistical equations which have been so suc- cessful in the classical description of mo- lecular system, i.e. the Maxwell-Boltzmann statistics.

These behavior assumptions were : (1) The number of persons per unit area were small enough for the behavioral system to be treated as a crowd gas and not as a crowd liquid, i.e., each individual was able to move at some chosen velocity independent statis- tically of the velocities of other individuals on a two-dimensional surface. ( 2 ) This velocity was independent of the individual’s position. (3) The system was in equilibrium and its average properties could be obtained from consideration of stationary ergodic processes.

Henderson also assumed that the above conditions applied irrespective of whether or not the crowd particles were caused by social bonds which resulted in the movement of pairs or larger groups.

Ko ng . ‘Now a t Headquarters Land Forces, Hong

In his analysis Henderson used the Max- well-Bolzmann function in the form

(1) P(V,) = (diVvs/dV,)iV-1 = (2*)-’”(vr .m .s .I-’

* exp [ ( - 22 ( V=’/ (vr .m . 8 .)‘)I where P(Vx) is the probability density func- tion for a single fluctuating velocity compo- nent V , about its mean and v ~ . , , , . ~ . is the root- mean-square of the speed.

In order to take account of any mean translational motion, e.g., caused by move- ment along a path, Eq. (1) may be rewritten (2) P(V’,) = (dNV’,/dVb)N-’

= (2*)-1’2(vr.m .8 .)-l exp [ ( - 45) ( V’,’/ (vr .m .I .)‘)I

where V’, is the normalized velocity. Henderson measured the velocity com-

ponents of pedestrians and found that a plot of their distribution function was in good agreement with that given by Eq. ( 2 ) , except for pronounced bimodal effect. Simi- lar results were also observed in data meas- uring the velocity of students in a campus area, and also of school children in a play- ground where there was no translational mo- tion. Henderson suggested the bimodal effect may be a result of the data sampled from

223

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224 ANTHONY PALMER AND ROBERT BAILEY

two populations that should have been differentiated on a sex basis. Unfortunately his data did not permit male-female distinc- tions and further investigation of this issue was not possible.

The present paper reports further observa- tions, verifies Henderson's conjecture show- ing the anomaly is a result of sex differences, and that the velocity distributions of both male and female populations separately agree with that predicted by Eq. (2). It is also suggested, however, that it is not neces- sary at this stage to invoke a comparison between a molecular system human walking speeds as no additional advantage is gained by treating Eq. (2) as other than the familiar normal distribution.

VELOCITY MEASUREMENTS

Measurements were made of the walking speed of pedestrians using the footpath of a well-used road in South Kensington, Lon- don. The area chosen was free of obvious constraints that would possibly impose velocity changes on pedestrians using that particular section.

The speed was obtained by noting the time a person took to cover a specified length, 45 m. Measurements were confined to people walking alone, prime particles, and in the same direction. The sex of each person, the time of day and an estimatr of the age group were recorded. The latter was subjective and the classes were rather arbitrary (10-30 years, 31-50 years, over 50 years). These age records and those of the time of day were regarded as additional data to determine if such factors would account for observed variation in the data.

RESULTS

Fig. 1 compares thc empirical distribution for the total sample of speeds, balanced with respect to sex but not yet separated, and the Maxwell-Boltzmann distribution with the same mean and variance. The empirical dis- tribution is very similar to that reported by Henderson, especially in the persistent bimodal effect. Fig. 2 shows the separated, single-sexed distributions taken from the same data. Both are unimodal, the respective modes corresponding exactly to the two peaks of the bimodal combined distribution.

I . " 1.1 2.0 SYFF" " ( O i S )

FIG. 1. Distributions of walking speeds: Males and females not separated.

This would suggest the anomaly displayed by Henderson's data was a result of sex differences.

If the separated distributions are assumed to be normal because of the large samples in- volved, the population variances can be equated to the sample estimates of these variances and the difference between popu- lations means formally tested for significance.

As might be expected, this difference was significant at the 1 percent level, and given the robustness of this test, the result strongly supports Henderson's conjecture.

A closer analysis, however, reveals sig- nificant lack of fit t o the normal distribution in all three cases, suggesting that factors other than sex may be significant in account- ing for the variation in velocity.

The goodness of fit was measured by the classical x2 test and also by tests for de- parture from normality based on higher moments, skewness and kurtosis. These re- sults are summarized in Table 1.

It seemed from the results of these tests that a satisfactory way to handle the data would bt. to subject it to a systematic analy-

Behavioral Science, Volume 20. 1975

SEX DIFFERENCES: STATISTICS OF CROWD FLUIDS 225

FIG. 2. Distributions of walking speeds separated for males and females.

sis of variance wherth the main effects are taken as

(1) sex, a t two levels, male and female, ( 2 ) age subjectively estimated, a t three

levds, 10-30,31-50, >50 years, (3) time of day, a t seven levels correspond-

ing to hourly intervals from 11 a.m. to 6 p.m. The analysis was performed in two stages,

a 2 x 3 experiment on sex and age alone with 70 replications followed by a 2 X 3 X 7 experiment on all three factors with five replications. Although the data were bal- anced with respect to sex alone, they were not evenly balanced with respect to all three effects. Random sampling was used to effect this balance.

In the first analysis tho sex and age main effects were shown to be highly significant a t the 1 percent lcvel while their interaction was nonsignificant. Further, a plot of the residuals against time of day showed no discernable trcnd, suggesting that time of day was also a nonsignificant factor. Moreover the residuals themselves closely fitted a normal distribu-

TABLE 1 TESTS FOR GOODNESS OF FIT OF WALKING SPEEDS

TO THE NORMAL DISTRIBUTION.

x* Goodness of De arture from

bil l1 br

Fit gonnality Sample 'i?2le

x' a.f . % sig. skewness kurtosis

nialea L 1625 47.4 20 0.06'' -0.158' 3.333.

rnalea 814 44.9 16 0.02.. -0.323. 3.594. femalea 811 34.2 18 1.2. -0.253. 3.251

females

** Cormponds to significant lack of fit at the 1 percent level. Corresponds to significant lack of fit at the 6 percent level.

TABLE 2 2 X 3 ANALYSIS OF VARIANCE O N SEX AND AGE.

Source of Sum of Mean F Variation '.f. Squares Square Statistic

BBX 1 1.7828 1.7828 47.21.. linear age 1 1.8198 1.8196 48.18'* quadratic age 1 0.0014 0.0014 0.04 interaction 2 1.1632 0.0816 2.16 error (residual) 414 15.6339 0.0378 total 419 19.4009

Goodneaa of fit, normality of residuals: x* = 17.70 d.f. = 18 - 2 = 16; & sig. = 34.

tion thus validating the tests of significance on the main effects and their interaction, and demonstrating that the underlying distribu- tion of walking speeds is normal. Thc results are given in Table 2.

The second analysis servrd to support the first, clearly showing that the main cffects of sex and age are highly significant. The re- sults are given in Table 3. These results are summarized in Fig. 3 where the sample means of the six sex-age combinations are plotted together with their exact 95 percent confidence interval for the population means. Simple linear regression lines are fitted to the data.

It may be objected that since age is sub- jectively catagorized its significance is open to doubt. Further experiments were per- formed on the data to check this. In one case the contents of the age group 31-50 years were randomly sampled into the 10-30 and the >50agegroup,and2 X 2and2 X 2 X 7 analyses of variance were perfornicd. In the other case the contents of age group 3-30 years were omitted and similar analyses of variance were performed. In all caws srx and

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226 ANTHONY PALMER AND ROBERT BAILEY

age were shown to be highly significant, all other main effects and interactions being nonsignificant. These analyses show un- equivocably, that the underlying distribu- tion of walking speeds is normal, and that sex and age are highly significant noninter- acting linear factors in accounting for ob- t j e ~ e d variation in these speeds. Moreover, tune of day is shown to be nonsignificant in explaining this variation.

Use of the Maxwell-Boltzmann distribution The main purpose of the measurements

and results presented here was to show that sex differences may be assumed to account for the anomalies present in Henderson's data. Henderson, in emphasizing the Max-

TABLE 3 2 x 3 X 7 ANALYSIS OF VARIANCE ON SEX, AGE,

AND TIME OF DAY.

Source of Sumof Mean F Variation '.f. Squares Square statistic

W X 1 1.3516 1.3515 40.34" We 2 0.7%7 0.3979 11.88.. time of day 6 0.1110 0.0185 0.55 B 8 r x age 2 0.0444 0.0222 0.66 age X timeof day 12 0.5875 0.0490 1.46 B ~ X X timeof day 6 0.3003 0.OM)o 1.49 aex X age X time 121 0.4032 0.0336 1.00

error (residual) 168 5.6290 0.0335 total 209 9.2227

of day

1 . 7

1 . h

1 . 5

A . B 1.4 - 0 x v)

1.2

1 . 7 $ PLU3:S - 1.5637 - .OUO6x T

1 . 2

-1 0 1 x

10-30 30-5U > 50

AGES

well-Boltzmann distribution suggests there is sufficient correspondence between the movements of people and those of gas mole- cules to permit the distribution which has so successfully described the latter t o be ap- plied to the former. However, at this stage the empirical data may not justify such an approach. The equations used by Henderson Eq. (1) and (2) are, as stated, the Maxwell- Boltzmann distributions for one dimension. In this case the velocity is the same as the speed and Eq. (1) is equivalent to the well- known normal distribution, more usually written

(3) (dN,,/dV,)N-' = t ( P )

= (27r)-1'ysv)-1

exp ((--%)(v - PY/sY)z)

where v is the speed, pv is the mean speed and S, is the standard deviation of the speed.

In the general three-dimensional case, Maxwell's law for the normalized velocity distribution is given by

(4) P ( V , V J J ~ V ~ V ~ ~ V . = ( 4 2 r k ~ ) 3 / 2

-1 0 1

10-30 30-50 > 50

FIG. 3. Exact 95 percent confidence intervals for the means of all age-sex combinations. Corresponding sample sires, number of replications, are given in brackets.

Behavioral Science, Volume 20, 1975

SEX DIFFERENCES: STATISTICS OF CROWD FLUIDS 227

where m is the mass of a molecule, T is the absolute temperature, and k is Boltzmann’s constant. This has the form of the trivariate normal distribution in the special case where the three variates are independent so that their covariances are zero.

However the corresponding distribution for the speed in the three-dimensional case is given by

( 5 ) f(v)czv = ( m / 2 T ~ ~ ) 3 / 2 . exp (-mV2/2KT)4?rV2dV.

This does not have the form of the univariate normal distribution because of the presence of the V 2 term.

Thus, although the normal distribution differs from the Maxwell-Boltzmann equa- tion for the three-dimensional case, it is identical for the one-dimensional case con- sidered here.

CONCLUSION

It has been demonstrated that when sex and age factors are taken into account the underlying distribution of one-dimensional walking speeds is normal. In this case the

further invocation of the Maxwell-Boltz- mann distribution is unnecessary. In order to develop theoretical comparisons between such movements and those of gas molecules, the assumptions of one dimensionality will have to be removed.

In this case Henderson’s measurements of children’s speed in two dimensions in a school playground, albeit undifferentiated with respect to sex, provide an interesting comparison with the two-dimensional case of Eq. (5) . Without such measurements with their attendant difficulties, the distri- bution of pedestrian speeds can be described by the familiar normal distribution.

The success of the sex and age factors in explaining the observed variation and bi- modal effects could be used as evidence for more fundamental physiological dis- criminators.

REFERENCE

Henderson, L. F. The statistics of crowd fluids. Nature, 1971, 229, 381-383.

(Manuscript received December 21, 1973; revised

March 5, 1975)

Behavioral Science, Volume 20, 1975