Census (mathematics)

Summary: Conducting a valid and reliable census depends on mathematical and statistical methods.

The term “census” comes from the Latin word censere, meaning “to assess.” A census is a systematic collection of data about an entire population of interest. Usually the population is people but historically it has also been done for land, livestock, and trade goods. Sometimes a census is a one-time event, or it may be repeated periodically, like the decennial census in the United States.

There are also two primary philosophies of data collection that can affect the outcome of a census: de jure and de facto. De jure counts people at their usual place of residence, while de facto counts people where they are on the day of the census.

For example, one biblical account of the birth of Jesus involves a census in which individuals were required to return to their town of origin rather than being counted where they lived, as opposed to the U.S. census, which is centered about people’s permanent residences. Archaeological records indicate that many ancient civilizations conducted censuses, the purpose of which was often taxation or military recruitment. The constitutionally stated purpose of the U.S. census is to determine each state’s congressional representation, though it has grown to include additional descriptive and predictive activities. The U.S. Census Bureau is one of the largest employers of mathematicians and statisticians, who not only collect and analyze data but also lead the way in developing new data collection and analysis methods.

Statisticians work internationally as well. For example, in 1949, British statistician Frank Yates was appointed to the United Nations Commission on Statistical Sampling and published Sampling Methods for Censuses and Surveys, which is widely acknowledged to have been influential in establishing sound principles and technical terminology. Overall, mathematical and statistical procedures improve the quality, reliability, and representation of census data, and the methods used by census-takers are constantly evolving.

The History of the Census

The practice of completing a census for an entire population occurred in many ancient civilizations. Records suggest that the Babylonians conducted a census in about 3800 b.c.e., and that Egyptians did so in the second millennium b.c.e. Male Roman citizens had to register for a census every five years and declare both family and property. Elected censors oversaw and coordinated the census process. The censors would then summon every tribe in the country to appear before them so they could record the relevant details. In ancient Rome, the census recorded the names of the family members, along with details of any property or land they owned. This provided the leaders of the country at the time the ability to tax their citizens according to the amount they owned. William the Conqueror carried out a census in Britain in 1086 c.e. for taxation purposes. This census took years to complete and attempted to compile a comprehensive list of all land and property in Norman Britain. Such a comprehensive exercise was previously unheard of in Europe, and it preceded an early example of a modern census by nearly 600 years.

Instructed by King Louis XIV in 1666, Jean Talon, a French colonial administrator, conducted a census in order to expand the colony in New France, North America. Talon used the de jure method and visited many of the colonial settlers personally, compiling data on settlers’ names, age, sex, and occupation. The aim of this census was to help the colony settle by using the statistics to decide how best to develop agriculture, trade, and manufacturing industry. In all, Talon managed to compile details of 3215 inhabitants and paved the way for the development of a number of further censuses in the New World. In Britain, a 1798 paper written by demographer Thomas Malthus discussed the possibility that not knowing the population size and growth rate (demographics), of a country could lead to food shortages and overuse of other resources, resulting in famine and disease as the population is unable to sustain itself. These revolutionary modeling ideas led the British government to pass through parliament the Census Act of 1800. The first modern British census took place in 1801; the process has been repeated decennially since then, except for in 1941 during World War II.

The Modern-Day Census—Data Collection

The U.S. census is required by constitutional law to take place every 10 years and involves sending forms to every residence in the United States and Puerto Rico. The data are then analyzed in order to determine how each state is represented in the U.S. House of Representatives and to provide the correct resource allocation for the current population, that is, how much of the federal fund is given to hospitals, schools, and other public services. Individual responses to the U.S. census are kept confidential for 72 years. In 2020, the US offered the ability to respond to the census completely online for the first time.

A similar process is used in the United Kingdom, although the census details are kept confidential for 100 years. A key difference in the census forms in the United Kingdom (UK) and the United States is that the U.S. form has just 10 questions and is two pages long. The UK census form for 2011 contained 43 questions in a 32-page booklet.

In Canada, a national census is taken every five years. Each household receives a census questionnaire, to either be filled out online or returned in the post.

Practical Problems With Census Taking

A number of problems can arise when attempting to take a census of an entire population. For the data to be useful, the characteristics of the whole population need to be reflected. This requirement means that any non-response could jeopardize the quality of the data. Non-responses can happen, for example, when an address list is not comprehensive, or people fail to fill in their forms fully and return them. There are a number of measures used to prevent this, including following up with nonrespondents in a face-to-face interview, and setting fines for nonrespondents.

A statistical technique called “imputation” was used by the U.S. Census Bureau in its 2000 census to create data using the nearest neighbor “hot deck” method. Where a gap in the counting (for example, an entire household’s data was missing) was identified, data from the geographically closest neighbor were used instead. Where a household had not completed every question fully, the missing data were imputed from a nearest neighbor record where the households are of the same size. Where a respondent within a household gave incomplete data, the characteristics were imputed from the characteristics of other household members. This technique enabled the U.S. Census Bureau to produce a more complete set of data on the U.S. population.

In order to overcome the obstacle of an incomplete address list, a number of different address lists can be combined to get a more complete list—thus ensuring a wider population is reached, and improving the reliability of the data. Alternatively, another statistical method, called “sampling,” can be used to estimate features of the population. A forward-thinking statistical sampling plan was proposed by many mathematicians and statisticians after the 1990 census turned out to be particularly problematic in terms of issues such as undercoverage of certain subpopulations. The U.S. Supreme Court refused to permit sampling to completely substitute for counting. These mathematical methods are used, however, for other types of estimation and to gauge how much undercoverage or other biases might exist.

Analysis of Census Data

A number of mathematical and statistical techniques can be used to draw the most descriptive and predictive information possible from raw census data. For example, to identify resource need, data can be ranked in such a way to identify areas where there are more children, thus enabling the government to plan where to locate schools. Alternatively, areas with a high percentage of elderly people could be identified and provided with more social care. Since the 1990s, census data have become a major resource for both amateur and professional genealogists now that older records are being digitized. Census data are also used to find ways to make future collections efforts better.

Edna Lee Paisano grew up on a Nez Perce Indian Reservation in Sweetwater, Idaho. Talented in both mathematics and science, she attended the University of Washington and earned a graduate degree in social work, studying statistics in the process. In 1976, she was hired by the U.S. Census Bureau to work on issues regarding Americans and Alaskan Natives, and was the Bureau’s first full-time Native American employee.

Using data from both the 1980 census and a survey she developed, Paisano discovered that Native Americans in some locations were undercounted. This was a serious issue, as allocation of federal funds to tribal units is based on census figures. She used statistical methods to improve the accuracy of the census and encouraged others in the Native American community to become educated in mathematics-related fields such as computer science, demography, and statistics. The 1990 census showed a 38% increase in U.S. residents counted as American Indians.

Bibliography

Aly, Gotz, and Karl Heinz Roth. Nazi Census: Identification and Control in the Third Reich. Philadelphia: Temple University Press, 2004.

Anderson, Margo. The American Census: A Social History. New Haven, CT: Yale University Press, 1990.

Kertzner, D. Census and Identity: The Politics of Race, Ethnicity, and Language in National Censuses. New York: Cambridge University Press, 2002.

Wright, Tommy, and Joyce Farmer. A Bibliography of Selected Statistical Methods and Development Related to Census 2000. Washington, DC: U.S. Bureau of the Census, 2000.