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Vote counting

Vote counting is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Vote counting rather than just read about it. In short: Vote counting is the process of counting votes in an election. It can be done manually or by machines.

Vote counting — main illustration
Vote counting — illustration

Key takeaways

  • Vote counting belongs to mathematics; place it in that map before memorising details.
  • Learn the definition first, then one example that makes the definition concrete.
  • Connect Vote counting to a quantity you can measure, compute or draw — that is where exam questions come from.
  • Reproduce the core statement of Vote counting from memory before moving on to harder problems.

Reference excerpt

Vote counting is the process of counting votes in an election. It can be done manually or by machines. In the United States, the compilation of election returns and validation of the outcome that forms the basis of the official results is called canvassing. Counts are simplest in elections where just one seat is being filled and only a few candidates are on the ballot, and these are often counted manually. In elections where many seats are being filled on the same ballot, counts are often done by computers to give quick results. Tallies done at distant locations must be carried or transmitted accurately to the central election office. Manual counts are usually accurate within one percent. Computers are at least that accurate, except when they have undiscovered bugs, broken sensors scanning the ballots, paper misfeeds, or hacks. Officials keep election computers off the internet to minimize hacking, but the manufacturers are on the internet. They and their annual updates are still subject to hacking, like any computers. Further, voting machines are in public locations on election day, and often the night before, so they are vulnerable. Paper ballots and computer files of results are stored until they are tallied, so they need secure storage, which is hard. The election computers themselves are stored for years, and briefly tested before each election. Despite the challenges to the U.S. voting process integrity in recent years, including claims by Republican Party members of error or voter fraud in 2020 and 2021, a robust examination of the voting process in U.S. states, including Arizona (where claims were most strenuous), found no basis in truth for those claims. The absence of error and fraud is partially attributable to the inherent checks and balances in the voting process itself, which are, as with democracy, built into the system to reduce their likelihood.

Manual counting

Manual counting, also known as hand-counting, requires a physical ballot that represents voter intent. The physical ballots are taken out of ballot boxes and/or envelopes, read and interpreted; then results are tallied. Manual counting may be used for election audits and recounts in areas where automated counting systems are used.

Manual methods One method of manual counting is to sort ballots in piles by candidate, and count the number of ballots in each pile. If there is more than one contest on the same sheet of paper, the sorting and counting are repeated for each contest. This method has been used in Burkina Faso, Russia, Sweden, United States (Minnesota), and Zimbabwe. A variant is to read aloud the choice on each ballot while putting it into its pile, so observers can tally initially, and check by counting the piles. This method has been used in Ghana, Indonesia, and Mozambique. These first two methods do not preserve the original order of the ballots, which can interfere with matching them to tallies or digital images taken earlier. Another approach is for one official to read all the votes on a ballot aloud, to one or more other staff, who tally the counts for each candidate. The reader and talliers read and tally all contests, before going on to the next ballot. A variant is to project the ballots where multiple people can see them to tally. Another approach is for three or more people to look at and tally ballots independently; if a majority (Arizona) or all (Germany) agree on their tallies after a certain number of ballots, that result is accepted; otherwise they re-tally. A variant of all approaches is to scan all the ballots and release a file of the images, so anyone can count them. Parties and citizens can count these images by hand or by software. The file gives them evidence to resolve discrepancies.

The fact that different parties and citizens count with independent systems protects against errors from bugs and hacks. A checksum for the file identifies true copies. Election machines which scan ballots typically create such image files automatically, though those images can be hacked or be subject to bugs if the election machine is hacked or has bugs. Independent scanners can also create image files. Copies of ballots are known to be available for release in many parts of the United States. The press obtained copies of many ballots in the 2000 Presidential election in Florida to recount after the Supreme Court halted official recounts. Different methods resulted in different winners.

When manual counts happen The tallying may be done at night at the end of the last day of voting, as in Britain, Canada, France, Germany, and Spain, or the next day, or 1–2 weeks later in the US, after provisional ballots have been adjudicated. If counting is not done immediately, or if courts accept challenges which can require re-examination of ballots, the ballots need to be securely stored, which is problematic. Australia federal elections count ballots at least twice, at the polling place and, starting Monday night after election day, at counting centres.

Errors in manual counts Hand counting has been found to be slower and more prone to error than other counting methods. Repeated tests have found that the tedious and repetitive nature of hand counting leads to a loss of focus and accuracy over time. A 2023 test in Mohave County, Arizona used 850 ballots, averaging 36 contests each, that had been machine-counted many times. The hand count used seven experienced poll workers: one reader with two watchers, and two talliers with two watchers. The results included 46 errors not noticed by the counting team, including:

… excerpt ends here. Continue reading the full article.

Illustrations

Vote counting: Counting ballots, Ouagadougou, 2015
Counting ballots, Ouagadougou, 2015
Vote counting: Time for hand counts: Minutes per vote counted
Time for hand counts: Minutes per vote counted
Vote counting: 2010 Iowa Republican Primary ballot
2010 Iowa Republican Primary ballot
Vote counting: Ballot marking device
Ballot marking device
Vote counting: Scanner marked multiple candidates with black line
Scanner marked multiple candidates with black line

Worked examples

Example 1 — a first encounter with Vote counting

Start with the simplest possible case. Write down what Vote counting claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Vote counting before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Vote counting ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Vote counting

In research
Vote counting appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry
Engineering practice reuses Vote counting in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom
Vote counting is common in secondary-school and first-year university syllabi. It links to neighbouring topics Elections, Elections in the United States, Statistical methods, so understanding it makes those chapters shorter.
In everyday life
Look for Vote counting outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.
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How to study Vote counting in 20 minutes

  1. Read the reference excerpt below once, without taking notes.
  2. Close the page and write down what Vote counting means in your own words.
  3. Compare your version with the excerpt and mark what you missed.
  4. Work through the three examples above with pen and paper.
  5. Explain Vote counting out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Vote counting in simple terms?

Vote counting is the process of counting votes in an election. It can be done manually or by machines.

Why does Vote counting matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Vote counting?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Vote counting.

Tags

  • Elections
  • Elections in the United States
  • Statistical methods
  • Vote counting
  • Voting

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