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 15411871. 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 OneName 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) 

15411599  6.8  5.2  2.0 
16001649  7.4  5.9  1.9 
16501699  6.9  6.8  1.7 
17001749  8.8  7.8  2.2 
17501799  12.1  9.0  2.9 
18001849  21.9  13.8  4.7 
18501880  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:
“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.”
(Source: Alan Bardsley How many Smiths are there? in Journal of OneName Studies, April 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.
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.