Approximation and Estimation
Approximation and estimation in mathematics involve finding values that are close to the actual value of a quantity. Approximation typically involves simplifying a value to make calculations easier, while estimation involves making an educated guess based on available information. Both techniques are used to obtain practical and workable solutions in various mathematical problems.
5 Key excerpts on "Approximation and Estimation"
- Fuchang Liu(Author)
- 2017(Publication Date)
- Routledge(Publisher)
...In doing school work, students may often use it to check if an answer obtained through exact calculation is reasonable, as a safeguard against careless errors. Even if calculators are used, sometimes a keystroke may not be pressed hard enough, or a series of operations may be executed in a different way than expected. If the answer obtained through exact calculation is very different from such an estimate, it’s a good indication that some error may have been made and more careful calculations may be necessary. Outside school, estimation is often called for in place of exact calculation because the latter in many situations is unnecessary. An example of this is estimating the amount of a tip in a restaurant, in which case a rough amount is usually sufficient. Neither use of estimation is practical if it takes longer than exact calculation. Imagine that you are taking a formal timed test such as SAT, where you have only a limited time for each section. If it takes you 1 minute to manually calculate a problem, you probably wouldn’t spend 2 minutes estimating the same problem in order to determine if you have made any possible careless errors. Likewise, for a real-world situation such as tipping in a restaurant, half a minute is usually sufficient to calculate the exact tip amount with paper and pencil. It is unimaginable that you would spend 4 minutes coming up with an estimate. It’s clear that in addition to the reasonableness criterion, there should be another criterion to judge the usefulness of an estimate: the speediness factor. Generally, an estimate should not take longer than the corresponding exact calculation. Thus, a formal definition of computational estimation, when all these factors are taken into consideration, can be given as: Computational estimation is the process of arriving at a rough but reasonable answer to an arithmetic problem in a relatively quick manner without resorting to any external calculating devices such as paper and pencil...
- Judith L. Green, Judith Green, Gregory Camilli, Patricia B. Elmore, Patricia Elmore, Judith L. Green, Judith L Green, Gregory Camilli, Gregory Camilli, Patricia B. Elmore, Patricia B Elmore(Authors)
- 2012(Publication Date)
- Routledge(Publisher)
...15 Estimation Juliet Popper Shaffer University of California, Berkeley DOI: 10.4324/9780203874769-17 An estimate can be described as “an educated guess about a quantity that is unknown but concerning which some information is available” (Lehmann 1982). Estimation abounds in our everyday lives. Polls and surveys are conducted constantly and are used to obtain estimates of many quantities: the proportion of high school graduates in a state, the average income and the distribution of incomes in a particular professional field, the average proficiency of students and how it has changed over time. Intervention research results in estimates of other types of quantities: The difference in average length of life of cancer patients getting different medical treatments, the differences in reading proficiency of students given different types of reading instruction. Every aspect of our lives depends on obtaining appropriate estimates of many quantities. Generally, estimates are based on data, consisting of some set of units(e.g. individuals, farms, schools, cameras) with values on quantitative variables of interest (e.g. test scores, crop productivity, neighborhood economic level, image quality). Sometimes we are interested only in describing the data, such as when a teacher obtains test scores on the students in her/his class; in that case, estimation does not come into the picture. The interest in these cases is in the kinds of summaries that give good pictures of the data set; this area is called descriptive statistics or data analysis. Descriptions of numerical quantities usually include some measures of central value and spread. There are many possible measures of each; of the former, the mean and median of the data are often given; of the latter, the variance or standard deviation and the interquartile range are commonly supplied. For some types of data, such as majors of students in a class, proportions are of interest...
eBook - ePubMore Trouble with Maths
A Complete Manual to Identifying and Diagnosing Mathematical Difficulties
- Steve Chinn(Author)
- 2020(Publication Date)
- Routledge(Publisher)
...11 Estimation A key life skill used to develop more confidence with mathematics Estimation is a life skill. There are many occasions when an estimate will suffice, for example, a tip in a restaurant, buying the right size can of paint, cooking, working out how long a journey will take. Estimation requires a different attitude to precise computations and a different way of thinking (see also Chapter 10). Estimation requires a sense of number and value and an appreciation of the place values of key numbers such as thousands, hundreds and tens. For example, 933 could be estimated to the nearest hundred, 900, or the nearest thousand, 1000. The nature of the estimation, close or not so close, over- or under-estimates, adds a further dimension to this skill. Rounding is a similar skill, but targets a specific goal number. Estimation can also make use of the ‘Is it bigger or smaller?’ question. An estimate of 900 for 933 is an estimate that is smaller than the true value. An estimate of 1000 for 933 is an estimate that is bigger than the true value. The question, ‘Is it bigger or smaller?’ is also about sense of number. Choosing whether to make an estimate ‘bigger’ or ‘smaller’ and knowing when that is appropriate is also a life skill. So, estimation can draw in the skill of appraising how far ‘off’ the estimate can be to work, to be appropriate, in the circumstances in which it is being applied. Being able to estimate is a sophisticated skill, involving number sense, place value and the appropriate use of number values in a range of contexts. But, in terms of evaluation, it is more forgiving in that it does not demand exactness. A key underlying concept is, as it often is in number work, place value. For example, estimating 933 to 1000 takes a three-digit number to four digits, it takes a hundreds number to thousands...
eBook - ePubIndividual Differences in Arithmetic
Implications for Psychology, Neuroscience and Education
- Ann Dowker(Author)
- 2019(Publication Date)
- Routledge(Publisher)
...In particular, it will look at the extent to which estimation is related to certain other arithmetical tasks. First, however, we must emphasize that estimation itself is not a single unitary process but is made up of numerous components. In comparing and attempting to integrate different studies, it is important to remember that different aspects of estimation are emphasized in different studies: in particular, educational studies tend to emphasize aspects related to mental calculation, while brain studies tend to emphasize approximation. What is arithmetical estimation? Arithmetical estimation is not a single process. There are many processes that contribute to or could be described as arithmetical estimation. Consider the following examples: In a study by Starkey (1992; see Chapter 4 of this book), children as young as 2 were encouraged to put a number of balls into an opaque box. An adult then either added or subtracted balls from the original quantity. The children were then asked to take the balls out of the box one at a time. The number of times they reached into the box was used to indicate their expectation of the resulting quantity. When small numbers were involved, most 2-year-olds made an ordinally appropriate number of retrieval attempts: fewer than the original quantity if items had been subtracted, and more than the original quantity if items had been added. 4-year-old Jack counted a set of eight counters. Another counter was then added, and Jack was asked how many there were now. He guessed, “6”. Paul, also 4, was passing by and commented, “No, it’s got to be more!” 5-year-old Michelle was asked to guess the answers to some single-digit addition problems. She consistently added 1 to the larger addend each time: 3 + 4 = 5, 2 + 6 = 7 and 8 + 2 = 9. 6-year-old Chloe could calculate or remember the answers to addition problems that added up to 10 or less...
eBook - ePub- Vyacheslav Tuzlukov(Author)
- 2017(Publication Date)
- CRC Press(Publisher)
...These errors are defined both by the quality performance and by conditions under which an estimation process is carried out. In general, the requirement of minimization of estimation error by value does not depend on just one assumption. However, if the criterion of estimation is given, the quality performance is measured based on this criterion. Then the problem of obtaining the optimal estimation is reduced to a definition of solution procedure that minimizes or maximizes the quality performance. In doing so, the parameter estimate must be close, in a certain sense, to the true value of estimated parameter and the optimal estimate must minimize this measure of closeness in accordance with the chosen criterion. In theory, for statistical parameter estimation, two types of estimates are used: the interval estimations based on the definition of confidence interval, and the point estimation, that is, the estimate defined at the point. Employing the interval estimations, we need to indicate the interval, within the limits of which there is the true value of unknown random process parameter with the probability that is not less than the predetermined value. This predetermined probability is called the confidence factor and the indicated interval of possible values of estimated random process parameter is called the confidence interval. The upper and lower bounds of the confidence interval, which are called the confidence limits, and the confidence interval are the functions to be considered during digital signal processing (a discretization) or during analog signal processing (continuous function). of the received realization x (t)...
