How did women count? A note ongender-specific age heaping differencesin the sixteenth to nineteenth centuries1

By PETER FÖLDVÁRI,* BAS VAN LEEUWEN,** andJIELI VAN LEEUWEN-LI

The role of human capital in economic growth is now largely uncontested. Oneindicator of human capital frequently used for the pre-1900 period is age heaping,which has been increasingly used to measure gender-specific differences. In this note,we find that in some historical samples, married women heap significantly less thanunmarried women. This is still true after correcting for possible selection effects. Apossible explanation is that a percentage of women adapted their ages to that of theirhusbands, hence biasing the Whipple index.We find the same effect to a lesser extentfor men. Since this bias differs over time and across countries, a consistent compari-son of female age heaping should be made by focusing on unmarried women.

The role of human capital in economic growth is largely uncontested.2

Recently, research in this field has started to focus on the effect of genderinequality in education on economic growth, often finding a negative relation.3

However, for the pre-1900 period, data on human capital are scarce, so researchershave reverted to proxies. In the last few decades, age heaping has began to fulfilthat role.4 Age heaping describes the tendency of people with a lower level ofhuman capital/numeracy to round off their ages to multiples of five (such as 25, 30,35, and so on).5 This is expressed in the form of the so-called Whipple index,defined as the number of people ending their ages in 0 or 5 divided by 1/5 of thetotal sample, multiplied by 100. Consequently, a Whipple index of 100 indicatesno heaping, while a Whipple index larger than 100 indicates a certain degree ofheaping. Clearly, a higher level of age heaping indicates less human capital or lessstrict numeracy. Hence, most studies find a strong negative relationship betweenliteracy and the Whipple index.6 Within the general trend of an increase in the

1 We would like to thank three anonymous referees for their extensive comments on an earlier version of thisarticle.

2 See, for example, Schultz, ‘Investment’; Becker, Human capital; Lucas, ‘Mechanics’; Romer, ‘Technologicalchange’.

3 For example, Barro and Lee, ‘Sources’; D. Dollar and R. Gatti, ‘Gender inequality, income, and growth: aregood times good for women?’,World Bank Policy Research Report on Gender and Development working paperser. no. 1 (1999); Klasen, ‘Low schooling for girls’.

4 Mokyr, Why Ireland starved; Crayen and Baten, ‘New evidence’; A’Hearn, Baten, and Crayen, ‘Quantifyingquantitative literacy’; Clark, Farewell to alms.

5 Although the heaping at multiples of five dominates, some authors have found other heaping patterns as well.For example, de Moor and van Zanden, ‘Uit fouten kun je leren’, pp. 63–5, andThomas, ‘Numeracy’, pp. 125–7,show that heaping on other numbers also happened, such as multiples of 12 and even numbers.

6 Nagi, Stockwell, and Snavley, ‘Digit preference’; Crayen and Baten, ‘Global trends’.

Economic History Review, 65, 1 (2012), pp. 304–313

© Economic History Society 2011. Published by Blackwell Publishing, 9600 Garsington Road, Oxford OX4 2DQ, UK and 350 MainStreet, Malden, MA 02148, USA.

number of age heaping studies in recent years, gender-specific age heaping studieshave also become increasingly common in the literature.7

Many studies look at gender-specific human capital in the medieval and earlymodern periods. Those that focus on literacy generally find that women are wellbehind men (up to 20 percentage points).8 Yet, as recent research has shown, thegender difference in age heaping is small.9 Sometimes, women even have lower ageheaping (and hence higher numeracy) than men. This result seems strange, sinceit not only deviates from the picture suggested by other measures of human capital,but also fails to find support in other material on numeracy. Indeed, as Thomasobserved for early modern England: ‘[w]omen lagged behind in numeracy,perhaps even more than they did in literacy’,10 while the Revd Andrew Urquhartfound in a statistical survey of Portpatrick in 1832 that women were only slightlybehind men in reading, but much further behind in simple arithmetic.11

In this note, we argue that age heaping can be a misleading measure ofnumeracy if its interrelatedness with marriage is not taken into account. Ageheaping differences between the two genders are much lower in the case of marriedcouples than in the case of the non-married population, even after correcting forpossible differences in their age structure and selection effects (where data avail-ability makes it possible).

There are basically three possible explanations for finding lower age heaping formarried than for unmarried women. The first one is selection, that is, a possibletendency that spouses were selected on the basis of their abilities and skills, amongothers numeracy. The second one is learning within marriage, namely, that wiveslearned basic arithmetic skills from their spouses, making it possible for them toreport their age with greater accuracy. If these explanations are valid, the ageheaping statistics of married women reflect the truth and can be trusted.We argue,however, that these two factors alone, even though there are certainly numerousexamples of both, cannot offer an adequate explanation for the observed lowerlevel of age heaping among married women in this time period.

What remains as a third explanation is that wives (and to a lesser extenthusbands)12 adjusted their reported ages to that of their spouses, causing anunderestimation of age heaping for women.13 As a result, data on non-marriedpeople should be used when estimating gender differences in numeracy.

7 de Moor and van Zanden, ‘Uit fouten kun je leren’; Manzel and Baten, ‘Gender equality’; Manzel, Baten, andStolz, ‘Convergence’.

8 For example, Kuijpers, ‘Lezen en schrijven’, p. 498; Reis, ‘Economic growth’, p. 202.9 de Moor and van Zanden, ‘Uit fouten kun je leren’, p. 71.

10 Thomas, ‘Numeracy’, p. 113.11 Tranter, ‘Reverend Andrew Urquhart’, pp. 55–6.12 In general, women have a higher age heaping that is associated with a lower level of numeracy. See, for

example, de Moor and van Zanden, ‘Uit fouten kun je leren’, p. 71. Hence, if any adaptation takes place, womenadapt to the age of their husbands and much less vice versa. Indeed, we show later in this note that the age heapingbetween married and unmarried men is not statistically different, while married women heap significantly lessthan unmarried women.

13 For example, de Moor and van Zanden, ibid., pp. 71, 75, also found that age heaping among women wasabout equal to that of men, even though literacy (as calculated by the number of people able to sign their names)was considerably lower for women. They approached this problem by suggesting that women deliberatelyexaggerated their level of illiteracy so as not to outshine their husbands who could not write.We look at this froman opposite perspective and argue that, because their information is partly based on marriage certificates, it islikely that women adapted their ages to those of their husbands, hence downward biasing age heaping.

HOW DID WOMEN COUNT? 305

© Economic History Society 2011 Economic History Review, 65, 1 (2012)

I

A possible indication for age adjustment is apparent in figure 1, which plots thereported age differences between spouses in the Norwich Census of the Poor from1570.When there is a tendency to report ages divisible by five, we expect peaks at0, 5, and 10, which is evident in figure 1. More surprising, however, is the tendencyto prefer even numbers as differences. Even though this is by no means a decisiveproof of age adjustment within marriage, since the real distribution of the agedifferences of the spouses is unknown, it does indicate that there might have beenpreferences in age differences within marriage as well. If a considerable share ofwives reported their ages as being two or four years different from their husbands,then the wives of age-heaping husbands would probably be counted as accuratereporters.

In order to prove our suggestion, we used surveys from different periods stretch-ing over 400 years.14 We tested two hypotheses: first, that the gender difference inage heaping is smaller for married individuals than for the rest of the population,and second, that marriage significantly improves the age heaping of women but notof men.We tested for possible heaping at even numbers and at multiples of 12, andeven though such secondary patterns were seen in the earliest censuses, they didnot affect the finding that married women heap less than unmarried women.

14 Most censuses are in one way or the other answered by the head of household.This implies that it is unclearif women reported their own ages. However, in the literature the ages 23–62 are specifically chosen in order toomit children who might not report their own age.The same argument might apply for women. Furthermore, deMoor and van Zanden, ibid., p. 66, argue that in the 1796 census for Flanders it is sometimes indicated whetherthe head of household reported the ages. Using these data, they show that reporting by the head of householddoes not lead to higher age heaping.

20

40

60

80

100

120

0

Difference in reported age (husband-wife)−20 −18 −16 −14 −12 −10 −8 −6 −4 −2 0 2 4 6 8 10 12 14 16 18 20

No.

of

obse

rvat

ions

Figure 1. Differences of the reported age of husbands and wives, Norwich Census of thePoor, 1570Note: Differences larger than 20 years are not reported due to the low number of observations.Source: Pound, ed., Norwich Census.

306 FÖLDVÁRI, VAN LEEUWEN, AND VAN LEEUWEN-LI

© Economic History Society 2011 Economic History Review, 65, 1 (2012)

Furthermore, estimates were also carried out including only people over the age of32 in order to avoid other possible non-linearities in heaping; however, this did notmake any significant difference in the results either. Therefore, we focused onmultiples of five, since in general those heaping patterns dominate.

One of the first censuses we used to test this hypothesis was from the Catasto ofTuscany in 1427, a detailed recording of the population ofTuscany.15 We also useddata from the Norwich Census of the Poor for 1570 and the census of Britain in1851.16 For an empirical proof, we applied independent sample t-tests (withunequal variances assumed) to find out if age heaping depends on marital statusand if the relationship of the two genders is affected by marriage (see tables 1–3).17

These estimates show a clear gap in age heaping between married and unmar-ried women, but no statistically significant difference for men. In order to makesure that the difference was not caused by the different age structure of thesubsamples, we cross-checked our results with a logit regression analysis, captur-ing the effect of age on age heaping directly (see table 4).

Once the effect of age on age heaping was captured, we found no effect ofmarriage on the age heaping of women in the Norwich sample; we did, however,find an effect in the 1851 population census of Britain and the 1880 census of theUS. In the latter cases, there seemed to be a marriage effect even for men, albeitof a lower magnitude. Since the quality of the population censuses of Britain andthe US is incomparably better than that of Norwich, which in addition onlyfocuses on the poor, we can safely argue that, even after taking into account thepossible impact of age, we still find that marriage is negatively correlated with ageheaping.

Since there is some evidence for selection effects in marriage, one may argue thatonly the richer women married; hence, heaping is lower among married women.However, the US 1880 census also provided indicators of economic status(occscore, median income in 1950 per occupation, projected on the occupations inthe 1880 census) and social status (sei, a linear combination of income andeducation). These selection effects, however, had only a small effect on the mar-riage coefficient in the regressions. Hence, these socio-economic selection effectscannot explain the lower heaping of women within marriage.

The final question is whether the observed pattern can be adequately explainedby learning or selection prior to marriage. According to the literature, numeracylearning in marriage was infrequent. Many authors have argued that basicnumeracy was absorbed during early childhood or when in contact with theexchange economy in the early modern period.18 Other authors have claimed that

15 Herlihy and Klapisch-Zuber, Les Toscans, pp. 656–63.16 Pound, ed., Norwich Census; Minnesota Population Center, North Atlantic Population Project.We also carried

out the tests for Florence in 1427, England in 1881, and the US in 1880, but due to limitations of the length ofresearch notes, these are not reported. These samples lead to the same conclusions, however.

17 The non-normality of the age heaping should not be a problem since the samples are quite large, and thet-statistics are asymptotically normal, as suggested by the Central Limit Theorem. We cross-checked our t-testresults by a non-parametric Mann-Whitney U-test (no assumption required on the distribution of the population)and obtained the same results, so the tests are valid and reliable.

18 Spufford, ‘First steps in literacy’, pp. 414–15; Cohen, Calculating people, p. 118; Emigh, ‘Numeracy orenumeration?’, p. 676.

HOW DID WOMEN COUNT? 307

© Economic History Society 2011 Economic History Review, 65, 1 (2012)

Tab

le1.

Age

heap

ing

and

gend

erby

mar

itals

tatu

s,Tu

scan

y,14

27,a

gegr

oup

23–6

2

No.

ofob

serv

atio

nsA

vera

geag

esa

mpl

e

Age

heap

ing,

fem

ale

(1)

Age

heap

ing,

mal

e(2

)

Age

heap

ing,

non-

mar

ried

(3)

Age

heap

ing,

mar

ried

(4)

t-st

atis

ticof

diffe

renc

e(1

)-(2

)

Pva

lue

ofH

1:(1

)<(2

)

Pva

lue

ofH

1:(1

)�(2

)

Pva

lue

ofH

1:(1

)>(2

)

Tot

alpo

pula

tion

94,6

6741

.630

6.8

282.

814

.51.

000

0.00

00.

000

Mar

ried

popu

lati

on84

,519

40.7

293.

728

2.7

6.24

1.00

00.

000

0.00

0N

on-m

arri

edpo

pula

tion

10,1

4848

.637

4.4

285.

313

.61.

000

0.00

00.

000

Tot

alpo

pula

tion

94,6

6741

.635

7.5

288.

328

.71.

000

0.00

00.

000

Men

43,9

2442

.528

5.3

282.

70.

420.

662

0.67

60.

338

Wom

en50

,743

40.8

374.

429

3.7

28.9

1.00

00.

000

0.00

0

Sou

rce:

Her

lihy

and

Kla

pisc

h-Z

uber

,L

esTo

scan

s,pp

.65

6–63

.

Tab

le2.

Age

heap

ing

and

gend

erby

mar

itals

tatu

s,N

orw

ich,

1570

,age

grou

p23

–62

No.

ofob

serv

atio

nsA

vera

geag

esa

mpl

e

Age

heap

ing,

fem

ale

(1)

Age

heap

ing,

mal

e(2

)

Age

heap

ing,

non-

mar

ried

(3)

Age

heap

ing,

mar

ried

(4)

t-st

atis

ticof

diffe

renc

e(1

)-(2

)

Pva

lue

ofH

1:(1

)<(2

)

Pva

lue

ofH

1:(1

)�(2

)

Pva

lue

ofH

1:(1

)>(2

)

Tot

alpo

pula

tion

1,06

542

.128

2.1

296.

1-0

.91

0.18

20.

364

0.81

8M

arri

edpo

pula

tion

856

41.2

263.

029

9.0

-2.1

30.

017

0.03

40.

983

Non

-mar

ried

popu

lati

on20

945

.232

7.1

238.

11.

520.

929

0.14

10.

071

Tot

alpo

pula

tion

1,06

542

.131

8.2

280.

42.

020.

978

0.04

40.

022

Men

434

41.8

238.

129

9.0

-1.0

70.

149

0.29

80.

851

Wom

en63

042

.532

8.9

263.

03.

130.

999

0.00

20.

001

Sou

rce:

Min

neso

taP

opul

atio

nC

ente

r,N

orth

Atla

ntic

Popu

latio

nP

roje

ct;

Sch

ürer

and

Woo

llard

,‘N

atio

nal

sam

ple’

.

308 FÖLDVÁRI, VAN LEEUWEN, AND VAN LEEUWEN-LI

© Economic History Society 2011 Economic History Review, 65, 1 (2012)

Tab

le3.

Age

heap

ing

and

gend

erby

mar

itals

tatu

s,E

ngla

ndan

dW

ales

,185

1,ag

egr

oup

23–6

2

No.

ofob

serv

atio

nsA

vera

geag

esa

mpl

e

Age

heap

ing,

fem

ale

(1)

Age

heap

ing,

mal

e(2

)

Age

heap

ing,

non-

mar

ried

(3)

Age

heap

ing,

mar

ried

(4)

t-st

atis

ticof

diffe

renc

e(1

)-(2

)

Pva

lue

ofH

1:(1

)<(2

)

Pva

lue

ofH

1:(1

)�(2

)

Pva

lue

ofH

1:(1

)>(2

)

Tot

alpo

pula

tion

172,

107

38.3

127.

112

5.6

1.43

0.92

40.

153

0.07

6M

arri

edpo

pula

tion

106,

364

39.6

122.

112

4.8

-2.0

40.

021

0.04

10.

980

Non

-mar

ried

popu

lati

on65

,743

36.3

134.

512

7.2

4.25

1.00

00.

000

0.00

0T

otal

popu

lati

on17

2,10

738

.313

1.2

123.

47.

191.

000

0.00

00.

000

Men

82,5

3938

.312

7.2

124.

81.

510.

934

0.13

20.

066

Wom

en89

,568

38.3

134.

512

2.1

8.33

1.00

00.

000

0.00

0

Sou

rce:

Min

neso

taP

opul

atio

nC

ente

r,N

orth

Atla

ntic

Popu

latio

nP

roje

ct.

Tab

le4.

Log

itre

gres

sion

resu

ltsof

the

age

heap

ing

dum

my,

age

grou

p23

–62

Con

stan

tA

geO

ccsc

ore

Sei

Mar

riag

eP

seud

oR

2E

ffect

ofm

arri

age

atm

ean

N

Nor

wic

h,15

70,

men

-1.2

48(-

2.28

)0.

0283

(3.0

8)—

—0.

466

(1.1

3)0.

019

0.12

443

7

Nor

wic

h,15

70,

wom

en-1

.762

(-4.

45)

0.05

03(6

.34)

——

-0.1

66(-

0.89

)0.

053

-0.0

5863

6

Bri

tain

,18

51,

men

-1.4

98(-

51.2

9)0.

0119

(16.

21)

——

-0.0

841

(-4.

88)

0.00

3-0

.016

82,5

39

Bri

tain

,18

51,

wom

en-1

.402

(-49

.71)

0.01

07(1

5.67

)—

—-0

.148

(-9.

47)

0.00

3-0

.028

89,5

68

US

,18

80,

men

-1.3

83(-

277.

9)0.

0144

(112

.7)

——

-0.1

13(-

35.4

2)0.

0039

-0.0

232,

679,

522

US

,18

80,

wom

en-1

.217

(-21

9.2)

0.01

37(1

08.6

)—

—-0

.259

(-84

.3)

0.06

3-0

.053

2,51

7,95

3

US

,18

80,

men

-1.4

19(-

251.

8)0.

0146

(114

.2)

0.01

08(4

5.8)

-0.0

09(-

67.4

5)-0

.112

(-35

.2)

0.00

55-0

.023

2,67

9,52

2

US

,18

80,

wom

en-1

.308

(-22

3.6)

0.01

44(1

13.1

)0.

039

(82.

2)-0

.017

(-60

.8)

-0.2

08(-

63.3

)0.

086

-0.0

432,

517,

953

Not

es:

t-st

atis

tics

are

repo

rted

inpa

rent

hese

s.T

hem

argi

nal

effe

ctof

the

mar

riag

edu

mm

yre

flect

sa

high

erpr

obab

ility

ofag

ehe

apin

gin

the

case

ofm

arri

edpe

rson

s,w

hen

eval

uate

dfo

ran

aver

age

indi

vidu

al.

For

the

US

in18

80,

we

used

ara

ndom

lydr

awn

sam

ple

(siz

e25

%),

beca

use

the

orig

inal

data

set

was

too

larg

eto

hand

leus

ing

our

soft

war

e.S

ourc

e:D

ata

for

Nor

wic

han

dB

rita

inas

inta

bs.1

and

2.T

heda

tafo

rth

eU

Sw

ere

take

nfr

omR

uggl

es,T

rent

Ale

xand

er,G

enad

ek,G

oeke

n,S

chro

eder

,and

Sob

ek,I

nteg

rate

dpu

blic

use

mic

roda

tase

ries

.Alt

houg

hth

eN

AP

Ppr

ovid

edth

esa

me

info

rmat

ion,

the

sei

(soc

ial

econ

omic

indi

cato

r)an

doc

csco

re(o

ccup

atio

nal

scor

e)va

riab

les

wer

epr

ovid

edby

Rug

gles

etal

.

HOW DID WOMEN COUNT? 309

© Economic History Society 2011 Economic History Review, 65, 1 (2012)

illiterate women were less in contact with a commercialized economy than men;19

the same applies to married women.20

Unfortunately, there is no direct way to test the validity of the possible expla-nations, but we can look for indirect proof.There are many widows in our samples.If they increased their numeracy during their marriages or were selected becauseof their better skills, then even after the deaths of their husbands, their numericalskills would remain superior to those of unmarried women. If, however, womenadapted their ages to those of their husbands, one would expect that, after theirhusbands died, they would soon revert to rounding their ages to -5 and -0. Wetested this proposition on the 1851 British and the 1880 US census data. Theresults are reported in table 5.

Our results suggest that even after capturing the effect of age on age heaping (theaverage age difference between widows and the rest of the female population in thesamples ranging between 8 and 13 years), widows display significantly higher ageheaping than married women in both samples.Their age heaping is statistically notdifferent from that of the unmarried women in the British census, but slightlyhigher than that of the unmarried women in the US. This is also true whenincluding variables capturing socio-economic status. It seems therefore thatwidows are more likely to heap than their married counterparts, which is a strongindication against selection based on numeracy or learning in marriage.

II

In this note we addressed the issue that the gender difference in age heaping ismuch smaller than in other measures of human capital.This is strange, since thereis very little independent additional evidence to support this finding. For severaldatasets stretching over four centuries, we found that married women heap sig-nificantly less than unmarried women; the age heaping difference among marriedand non-married men is generally not significant. There are three possible expla-nations: firstly, that more numerate spouses were preferred (socio-economicstatus); secondly, that women improved their numeracy during marriage; orthirdly, that married women (and to a lesser extent married men) had the tendencyto adapt their ages to those of their husbands. Since men, on average, had lowerage heaping, this reduced the observed age heaping among married women aswell.

Plotting the number of spouses by age difference in marriage for the 1570Norwich Census of the Poor revealed a remarkable number of couples with aneven number of years age difference. It seems indeed very unlikely that womenonly married men who were an even number of years older/younger. This cannotbe explained by either learning or the preference for more numerate spouses. As amore formal test, we used data on widows. If actual learning in marriage had takenplace, or if more numerate women were preferred as spouses, widows on averageshould have a lower level of age heaping than unmarried women. On the other

19 J. Reis, ‘How helpful is age heaping? An exploration using the Portuguese Census of 1940’, paper presentedat the European Social Science History Conference, Lisbon (2008) (http://www2.iisg.nl/esshc/programme.asp?selyear=9&pap=6569), p. 22.

20 Horrell, Meredith, and Oxley, ‘Measuring misery’, p. 96.

310 FÖLDVÁRI, VAN LEEUWEN, AND VAN LEEUWEN-LI

© Economic History Society 2011 Economic History Review, 65, 1 (2012)

Tab

le5.

Log

itre

gres

sion

resu

ltsof

the

age

heap

ing

dum

my

for

Bri

tain

,185

1,an

dth

eU

S,18

80,a

gegr

oup

23–6

2

Con

stan

tA

geO

ccsc

ore

Sei

Wid

owP

seud

oR

2M

argi

nale

ffect

ofw

idow

dum

my

atm

ean

N

Bri

tain

,18

51W

idow

san

dm

arri

edw

omen

-1.3

31(-

36.3

7)0.

005

(5.7

4)—

—0.

239

(8.9

5)0.

003

0.04

6861

,815

Wid

ows

and

unm

arri

edw

omen

-1.6

41(-

39.7

8)0.

0172

(15.

24)

——

-0.0

35(-

1.10

)0.

003

-0.0

068

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HOW DID WOMEN COUNT? 311

© Economic History Society 2011 Economic History Review, 65, 1 (2012)

hand, if the lower level of age heaping is caused by women adjusting their age tomatch that of a spouse, we would expect widows to revert to heaping on -0 and -5again after their husbands passed away. We find that age heaping of widows, evenafter correction for the age structure and selection effects, does not significantlydiffer from that of unmarried women.

Unfortunately, the third possibility—that women (and to a lesser extent men)adapt their ages to that of their husband (wife)—cannot be tested directly given thedual distribution (an age distribution of both men and women). However, afteranalysing the other two arguments, we concluded that the age adaptation bymarried women to that of their husband may have been a factor in the unexpect-edly low overall level of age heaping among women. This implies that, in order tocalculate actual age heaping among women reliably, it is preferable to use data onnon-married women.

*University of Debrecen, Utrecht University**Free University, Utrecht University, University ofWarwick

Date submitted 24 July 2009Revised version submitted 27 July 2010Accepted 14 September 2010

DOI: 10.1111/j.1468-0289.2010.00582.x

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