Pre-modern demography is a particularly fraught discipline. M. M. Postan, in the Cambridge Economic History of Europe, makes the point that the data available to work with is so poor "that the estimates inevitably incur the risk of errors on a truly heroic scale", and the Domesday Book is no exception to this. For one, it's a record of tenants, rather than a census, and for another it's incomplete. Four entire counties (Cumberland, Westmorland, Northumberland and Durham) were omitted, along with part of Lancashire, and not all of the towns have populations recorded. We have no break down of household structure, no mention of the itinerant population, and the very real possibility that poorer inhabitants were under-reported. This means that any numbers produced a very much the product of the biases and assumptions of the demography.

To begin with, we can look at your assumptions, particularly the assumption that England's "hundreds" were in hundred hide groupings and that a hide was the amount of land needed to support a family. While we do have some evidence to suggest that large parts of the Midlands were more or less divided up into hundred hide blocks (or in "half hundred" (50 hide) and "double hundred" (200 hide) blocks) by the time of the Domesday Book, elsewhere hundreds could cover as little as 20 hides or as much as 150 hides. Additionally, while hides varied in size - usually ranging between 40 and 120 acres - they were all fairly substantial areas and could have supported several families each, even in their smallest form.

If we were to use your estimate of the number of hundreds and say that there were four families per hide, an average household size of 3.5 would yield an estimate of 1.68 million people, without factoring in the urban population (perhaps 168 000 using the old 10% ratio). The final number is somewhere around 1.85 million, which is definitely towards the higher end of population estimates, but highlights the importance of starting with the right numbers.

So, what are the raw numbers for the Domesday Book? There were 234 800 rural tenants, 28 200 slaves, 5 800 men belonging to other categories (priests, etc) and approximately 20 000 men living in the boroughs. To this can be added approximately 7000 knights, barons and bishops, which brings us to 296 000 men we can find in the document itself. To this need to be added estimates for under reporting (0-25%), estimates for the counties without a recorded population (7-10 000 households) and estimates for unrecorded urban population (c. 10 000 households). Altogether, we're looking at a maximum of 390 000 households.

And this is where things get tricky. How many men were under reported? Did the slaves have families, or were they single? How large was the average household? How many households were there in the four counties not recorded? There are many, many variables that could come into play here, and this is why estimates of population vary so much. J.C. Russell, who pioneered the study of medieval populations in England using the Domesday Book and late 14th century poll taxes, assumed that the Domesday Book had managed to record every household, that each household averaged 3.5 members and made some estimates for the missing counties based on population densities in nearby counties, coming up with a total of 1.1 million people. Andrew Hinde, using a multiplier of 4.5 people per household, the assumption that 5% of households weren't recorded and using more recent estimates for missing boroughs, calculated 1.5 million people. With a 25% omission rate, a household size of 5 and the more recent urban population estimates, a maximum of 2 million people can be achieved.

The question of household size is an important one. The figure of 3.5 comes from J.C. Russell who, although he heavily influenced both the study of populations in medieval England and the medieval world in general, is almost the sole advocate of the figure. He arrived at it via the poll tax of 1377, where he looked at the average number of persons over the age of 14 (the minimum age for paying the tax) in the records. He found an average of 2.3 persons per household, roughly one child over the age of 14 or one servant for every third couple, and then used the life tables he had computed (based on the lives of noblemen) to estimate that a third of the population was under 14. He then attempted to demonstrate the universality of the figure by going through a body of evidence to show that it was common.

This approach has not been without critics. J. Krause pointed out that not only was Russell missing reliable data for infant mortality, but that rural peasantry as a rule don't have the same life spans as the rural elite, and so there would be both a higher childhood morality and a higher proportion of children. Similarly, the work of Ferdinand Lot with the polytyptique of Irminon prior to Russell had already updated the average household size listed in the 1895 printed volume Russell cited, and David Herlihy has since shown that the average household consisted of 5.75 people, although the average of the majority of the rural households comes to 4.9 people.

Other work, such as H. E. Hallam's work with a handful of 13th century censuses from Lincolnshire, provide numbers from about 4.14 (Postan's estimate for the 1377 poll tax) to upwards of 5.9, as with David Herlihy's examination of Tuscany in 1427. In general, though, household sizes of 4.5 or 5 are used in medieval demography. The process continues to be refined, and more use is being made of the evidence from cemeteries to generate life tables, so the multiplied might change in the future.

Further Reading

J. C. Russell's British Medieval Population, for all its many flaws, is nonetheless an important and extensive study of medieval demographics. Andrew Hinde's England's Population: A History Since the Domesday Survey is much less involved and provides the most modern range of estimates for medieval English populations, along with a discussion of the evidence and how to interpret it.

For early criticism of Russell, J. Krause's "THE MEDIEVAL HOUSEHOLD: LARGE OR SMALL?", in The Economic History Review Vol. 9, No. 3 (1957) pp. 420-432, is the most important and most cited, while H.E. Hallam's "Some Thirteenth-Century Censuses", in The Economic History Review, Vol. 10, No. 3 (1958), pp. 340-361, has some alternate figures for rural household sizes in the 13th century.

And, on a distantly related note, a particularly interesting study of medieval demography in Scandinavia is Ole Jørgen Benedictow's The Medieval Demographic System of the Nordic Countries, which combines skeletal evidence from medieval cemeteries, life tables for high mortality populations, scattered pieces of medieval records and early modern demographic evidence to build a fairly comprehensive view of medieval Scandinavian households, along with a reasonable estimate for population.