Estimating the total size of a surname

Can one estimate how many bearers of a name there have existed, since parish records began? The seminal work on local demography is Wrigley and Schofield's monumental The Population History of England 1541-1871. This tome rests on the analysis of the registers of 404 English parishes to produce estimates of the like of population growth, fertility, age at marriage etc. Graham Fidler in How big is your One-Name Study condensed some of the essential figures into the following table, which reveals the estimated number of births, marriages and deaths for each 50 years since 1541.

Year Crude Total Births (millions) Crude Total Deaths (millions) Crude Total Marriages (millions)
1541-1599 6.8 5.2 2.0
1600-1649 7.4 5.9 1.9
1650-1699 6.9 6.8 1.7
1700-1749 8.8 7.8 2.2
1750-1799 12.1 9.0 2.9
1800-1849 21.9 13.8 4.7
1850-1880 22.4 13.8 5.2
Total 86.2 62.4 20.8


Graham found that in the current telephone directories there are 1,625 Fidlers, equivalent to 101 entries per million. (Article written before the appearance of the 1881 Census index, whose figures would have been a preferable baseline.)

Using the above table, suggests that there have been 8 700 Fidler births since 1541, and as the work of Martin Ecclestone indicates, the IGI should contain on average 50% of these.

In my own case (and Dance being a predominantly English surname) there were 2 515 Dances in England and Wales in 1881. The relevant national population was 26 046 142, resulting in a ratio of 96.56 per million. Multiplying by the factor of 86.2 results in 8 323 total births since 1541. Contrast this with the result by the Bardsley method below.

Provisos:
assumes that the present rate has been consistent in the past, and
applies only to a name that is predominantly distributed in the English counties

The Wrigley and Schofield figures were further refined by Alan Bardsley. He attempted to both provide a longer time frame by adding GRO data post 1871, and to provide factors for the estimation of a name population in any decade. How the factors are derived is not explained in detail, so I will quote verbatim from his article (1):

"The basic information needed for each year is the annual population and birth rate. From these a multiplication factor can be calculated which gives the relationship between the births in any one year and the total for all time. Similarly a factor for the total alive in any one year and the total for all time can be derived.

The former can be used with the annual birth registers and the latter with the census records...

To find the total number of births from 1541 to 1996 multiply the number (of births) for any one year by the factor given for Births. Similarly, if you know the total number of individuals in any one year, multiply by the factor given for Totals to give a total of individuals from 1541 to 1996."

Year Births
factor
Total
factor
   Year Births
factor
Total
factor
1541 1901 64.6  1771 789 27.8
1551 1502 59.5  1781 716 25.4
1561 1602 60.1  1791 602 23.1
1571 1644 54.8  1801 606 20.5
1581 1472 49.7  1811 449 18.0
1591 1551 45.9  1821 372 15.2
1601 1388 43.5  1831 373 13.1
1611 1185 40.5  1841 319 11.5
1621 1107 38.2  1851 280 10.2
1631 1322 36.6  1861 253 9.1
1641 1092 35.2  1871 229 8.0
1651 1189 34.2  1881 207 7.0
1661 1227 34.8  1891 203 6.3
1671 1303 36.0  1901 196 5.6
1681 1207 36.3  1911 207 5.1
1691 1131 36.3  1921 212 4.8
1701 1035 35.4  1931 289 4.6
1711 1185 34.2  1941 301 4.4
1721 1059 33.5  1951 267 4.2
1731 935 34.0  1961 225 4.0
1741 1028 32.1  1971 239 3.7
1751 907 31.0  1981 286 3.7
1761 837 29.1  1991 271 3.6

Alan Bardsley continues:

"For example..the number of Bardsleys in 1851, from a census count, is about 2,500 and from the multiplier of 10.2 for the population in that year I would conclude that there have been about 25,000 Bardsleys in the UK since 1541 and from the 1991 ratios working backwards that there should be currently about 92 births a year and a current population of 6,900."

In my own case, there were 2,515 Dance's enumerated in 1881 in England and Wales. So by the above table, multiplying by a factor of 7.00 leads to a total of 17,605 live births since 1541.

Alan does warn that "there will be large errors for individual years when you try to apply the calculations to small groups" and later that "significant differences are bound to occur when the number of births per year is small, say less than ten per annum. The vagaries of procreation and pestilence play havoc with the statistics. The greater the number of years that births can be counted over the more accurate will be the estimate."

I know from the 1996 Electoral Rolls that there are currently some 4,500 Dances alive in the UK. Multiplying by the latest listed factor, results in a figure of 16 250. Which fits in with the range.

(Source: Alan Bardsley How many Smiths are there? in Journal of One-Name Studies, April 1996.

Provisos:

The Wrigley and Scofield data is based on English counties: the RG figures are for England and Wales. This may cause some slight discrepancy.