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The Useful and Not so Paradoxical Inspection Paradox

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Title: The Useful and Not so Paradoxical Inspection Paradox
Language: English
Authors: James E. Marengo (ORCID 0000-0003-0498-0424), David L. Farnsworth (ORCID 0000-0003-0251-8770)
Source: International Journal of Mathematical Education in Science and Technology. 2025 56(6):1196-1208.
Availability: Taylor & Francis. Available from: Taylor & Francis, Ltd. 530 Walnut Street Suite 850, Philadelphia, PA 19106. Tel: 800-354-1420; Tel: 215-625-8900; Fax: 215-207-0050; Web site: http://www.tandf.co.uk/journals
Peer Reviewed: Y
Page Count: 13
Publication Date: 2025
Document Type: Journal Articles
Reports - Descriptive
Education Level: Higher Education
Postsecondary Education
Descriptors: Statistics, Mathematics Instruction, Student Projects, Undergraduate Students, Student Research, Mathematical Concepts
DOI: 10.1080/0020739X.2024.2327551
ISSN: 0020-739X
1464-5211
Abstract: We offer ways that the inspection paradox can be usefully presented, even in the most elementary statistics courses, and provide guidance about how it can be revisited subsequently in the same course and in succeeding courses. Numerous situations in which the inspection paradox might occur are mentioned, and mathematically simple demonstrations are furnished in order to help convince students of the paradox's veracity. Although remedial actions for the paradox are given, the main point of view is that the two sampling procedures leading to this apparent paradox are simply alternative methods that yield differing, but bona fide, measures and interpretations for the populations from which the data were obtained. Suggestions for student projects and undergraduate research are given.
Abstractor: As Provided
Entry Date: 2025
Accession Number: EJ1474384
Database: ERIC
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  Value: <anid>AN0185445721;imt01jun.25;2025May29.02:12;v2.2.500</anid> <title id="AN0185445721-1">The useful and not so paradoxical inspection paradox </title> <p>We offer ways that the inspection paradox can be usefully presented, even in the most elementary statistics courses, and provide guidance about how it can be revisited subsequently in the same course and in succeeding courses. Numerous situations in which the inspection paradox might occur are mentioned, and mathematically simple demonstrations are furnished in order to help convince students of the paradox's veracity. Although remedial actions for the paradox are given, the main point of view is that the two sampling procedures leading to this apparent paradox are simply alternative methods that yield differing, but bona fide, measures and interpretations for the populations from which the data were obtained. Suggestions for student projects and undergraduate research are given.</p> <p>Keywords: Inspection paradox; bias; sampling method; teaching statistics; length-based sampling; size-based sampling; Primary: 97D40 (Mathematics Education: Teaching methods and classroom techniques); Secondary: 60E15 (Probability: Inequalities, Stochastic orderings); 62B15 (Statistics: Theory of statistical experiments); 62D05 (Statistics: Sampling theory)</p> <hd id="AN0185445721-2">1. Introduction and purpose</hd> <p>In many statistics courses during the first week of each semester in a class discussion, we have used an exercise from an older elementary textbook, partly because it is a good ice breaker. The exercise asks students to comment on what is wrong with the following sampling procedure:</p> <p>To estimate the average length of logs fed into a mill by a constant- speed conveyor belt, measurements are made of the lengths of the logs which pass a given point exactly every five minutes. (Freund, [<reflink idref="bib9" id="ref1">9</reflink>], p. 12)</p> <p>The exercise works well. It does not depend upon any particular discipline, which is preferred when the students have many diverse majors. It is easy to understand the set-up and see it in one's mind's eye. Most students can think of something to contribute to the discussion. It can be used in elementary statistics courses, as well as more advanced courses. It is especially appropriate for quality control, sampling, and stochastic processes courses.</p> <p>Among the students' suggestions have been:</p> <p></p> <ulist> <item> The belt is moving, so the measurements might not be good.</item> <p></p> <item> The point could be between two logs.</item> <p></p> <item> The word 'average' is ambiguous.</item> <p></p> <item> The logs could have different diameters, which might be important to the mill.</item> <p></p> <item> An extremely long log could be measured twice.</item> <p></p> <item> The small logs and the large logs might come in bunches.</item> <p></p> <item> The mechanism that feeds the logs into the mill might change as they are arriving from varying sources, so that the logs could be smaller or larger every five minutes.</item> <p></p> <item> Longer logs take longer to pass by, so they are more likely to be measured.</item> </ulist> <p>We suspect that the next-to-last response may have been the one that the author was specially soliciting, but the final one might have been, as well.</p> <p>Suggestions (<reflink idref="bib1" id="ref2">1</reflink>)–(<reflink idref="bib7" id="ref3">7</reflink>) can be addressed by augmenting or editing the original statement, respectively, by</p> <p></p> <ulist> <item> saying that there is a ruler near the belt, so that sufficiently accurate measurements can be taken,</item> <p></p> <item> saying the logs are tightly end-to-end with no gaps,</item> <p></p> <item> replacing 'average' with 'arithmetic average' or 'mean',</item> <p></p> <item> saying that the logs are of equal diameters,</item> <p></p> <item> saying that no log is so long as to be measured twice,</item> <p></p> <item> saying that the logs are jumbled or mixed together before being feed into the mill, and</item> <p></p> <item> replacing 'exactly every five minutes' with 'at random times'.</item> </ulist> <p>One way to resolve the final suggestion, (<reflink idref="bib8" id="ref4">8</reflink>), is to employ an entirely different sampling method, such as rolling the logs into the mill so that each log is equally likely to be chosen. Suggestion (<reflink idref="bib8" id="ref5">8</reflink>) leads to the inspection paradox. This paradox is discussed in Section 2, including examples in which the paradox affects real sampling situations. Also, in that section, we discuss paradoxes in general and how they have been used in teaching and educational research. Some of the inspection paradox's straightforward mathematics, which we have found helpful for demystifying that paradox, is in Section 3. Section 4 emphasises examples where length-based sampling, which might be considered the 'cause' of the paradox, is the appropriate method. Section 5 discusses some of our experiences with presenting the inspection paradox and students' reactions to it. Concluding comments are in Section 6, which contains recommended ways the paradox can be used in teaching.</p> <hd id="AN0185445721-3">2. The inspection paradox</hd> <p>The <emph>inspection paradox</emph> says:</p> <p>Under certain conditions, a single element that is chosen at random tends to be larger than an ordinary element.</p> <p>In the example in Section 1, longer logs take longer to pass the measuring station, so they are more likely to be measured, which tends to increase estimates of the lengths of the logs entering the mill. The paradox is that, although a random, accurate, and precise process is involved, the estimate is incorrect. Sometimes, this is called the <emph>waiting-time paradox</emph>. In the logs example, the varying waiting times for the logs to pass the measuring site is the cause of an apparently incorrect conclusion. Because the conveyor belt has constant speed, the chance of a log being selected for measurement is proportional to its length. Thus, this is an example of <emph>length-based sampling</emph> or <emph>size-based sampling</emph>. Therefore, the paradox is sometimes called the <emph>length-based sampling paradox</emph> or the <emph>size-based sampling paradox</emph>.</p> <p>A <emph>paradox</emph> is</p> <p>A statement or situation that may be true but seems impossible or difficult to understand because it contains two opposite facts or characteristics. (Cambridge Editors)</p> <p>In the measurement of lengths of logs example, two sampling methods are in play – simple random sampling in which each log would be equally likely to be selected for measurement and length-based sampling with longer logs being more likely to be selected. The inspection paradox is that the arithmetical average yields a number that is too large. Length-based sampling has been employed with its accompanying bias toward the inclusion of longer logs in the sample. Researchers must guard against using the wrong sampling method, which, as we demonstrate, might be either simple random sampling or length-based sampling, depending upon the goal. We examine this duality.</p> <p>The motivating situations often used to illustrate the inspection paradox are waiting times for buses (Feller, [<reflink idref="bib8" id="ref6">8</reflink>]; Masuda & Porter, [<reflink idref="bib26" id="ref7">26</reflink>]; Pal et al., [<reflink idref="bib32" id="ref8">32</reflink>]; Pauls, [<reflink idref="bib36" id="ref9">36</reflink>]; Reuveni, [<reflink idref="bib40" id="ref10">40</reflink>]), time between eruptions of geysers (Rauwolf & Kamps, [<reflink idref="bib39" id="ref11">39</reflink>]), lifetimes of lightbulbs (Ross, [<reflink idref="bib42" id="ref12">42</reflink>], p. 409), lifetimes of batteries (Marengo et al., [<reflink idref="bib25" id="ref13">25</reflink>]; Ross, [<reflink idref="bib42" id="ref14">42</reflink>], p. 447), numbers of students in classes (Hemenway, [<reflink idref="bib13" id="ref15">13</reflink>]; Wagner, [<reflink idref="bib52" id="ref16">52</reflink>]), and number of children in a family or of people in a friendship group (Feld, [<reflink idref="bib7" id="ref17">7</reflink>]; Gelman & Glickman, [<reflink idref="bib11" id="ref18">11</reflink>]; Paulos, [<reflink idref="bib35" id="ref19">35</reflink>]; Ross, [<reflink idref="bib41" id="ref20">41</reflink>]).</p> <p>These are situations in which either simple random sampling or length-based (sized-based) sampling can be properly employed to obtain different outcomes, which could be interpreted as a paradox. In some, the time between events would be the length or size. For example, a bus rider would be more likely to arrive at a bus stop at a longer time period between buses and so experience a longer wait time than the average time between arrivals of buses that the operator of the bus company might believe from sampling routes. The rider's waiting experience is truly measured using length-based sampling, but the operator might sample by routes or trips and obtain a smaller number, which reflects the operator's understanding of the service being delivered.</p> <p>In some of the other examples, the length or size is created by subgroups, such as families and classes. For example, if randomly selected students are asked about the number of students in their classes, probably a larger number will be reported than if random classes were chosen and their numbers of students determined. The length in the length-based sampling is by student; since there are more students in larger classes, more students from larger classes will tend to be included in the sample. The length-based sampling of asking random students yields data that mirrors students' experiences, but sampling by choosing classes reflects the college administration's viewpoint.</p> <p>Bytheway ([<reflink idref="bib2" id="ref21">2</reflink>]) and Jenkins and Tuten ([<reflink idref="bib15" id="ref22">15</reflink>]) present numerous examples in the social sciences literature of published studies that came to apparently incorrect conclusions based on misinterpreting samples that were gathered by length-based sampling.</p> <p>There are many paradoxes and so-called paradoxes in mathematics and the sciences. Quine ([<reflink idref="bib37" id="ref23">37</reflink>]) and others have created classifications of them. In Quine's typology, the inspection paradox is an example of a <emph>veridical paradox</emph>, because under close examination it can be resolved. It does not depend upon performing an incorrect procedure, such as dividing by zero, which would might lead to a <emph>falsification paradox</emph> or fallacy in Quine's typology. Nor is it fundamental or an <emph>antinomy</emph>, i.e. it does not shake the foundations of mathematics, probability, or statistics. Kleiner and Movshovitz-Hadar ([<reflink idref="bib18" id="ref24">18</reflink>]) gave many examples of paradoxes that have been important in the history of mathematics.</p> <p>Paradoxes can be useful tools in educational research. For example, Sriraman ([<reflink idref="bib49" id="ref25">49</reflink>]) reported on the written and verbal reactions of about 120 pre-service mathematics teachers in her classes to an assignment about the barber paradox in which a male barber in town shaves the set of all men in town who do not shave themselves, leaving the unanswerable question: Does the barber shave himself? She found connections between the individual student's reactions and their success in the assignment and the student's beliefs about the nature of mathematics (an ideal or fixed set of ideas, being infallible, a set of formulas to be usefully applied, and so forth) and also their apparent ability or proclivity to think across disciplines. Movshovitz-Hadar and Hadass ([<reflink idref="bib29" id="ref26">29</reflink>]) used paradoxes as significant parts of the curriculum for pre-service mathematics teachers. They reported that it helped the students to clarify their ideas and reasoning in problem solving, to think about validity of proofs and derivations, and to address many other aspects of their learning mathematics. Mamolo and Zazkis ([<reflink idref="bib24" id="ref27">24</reflink>]), Mamolo ([<reflink idref="bib23" id="ref28">23</reflink>]), and Wijeratne and Zazkis ([<reflink idref="bib53" id="ref29">53</reflink>]) looked closely at what their students thought about certain concepts that are parts of some paradoxes, especially when infinity is involved. Sierpinska ([<reflink idref="bib46" id="ref30">46</reflink>]) used falsehoods and contradictions to analyze students' reactions to problem solving and thoughts about the correctness of solutions.</p> <p>Paradoxes can be used as topics to help students learn. Klymchuk and Kachapova ([<reflink idref="bib19" id="ref31">19</reflink>]) used examples of veridicial paradoxes and counterexamples to engage students in thinking about mathematics in what the authors considered a less superficial manner, which might yield incorrect conclusions. Cheong et al. ([<reflink idref="bib4" id="ref32">4</reflink>]) explain how they used computer programmes and simulations to show students how some paradoxes, especially in probability, might be better understood.</p> <p>There is much written about the paradox, but the topic is not routinely addressed in undergraduate courses in mathematics, probability, or statistics with the exception that often textbooks for second courses in undergraduate probability at least mention it. When addressed properly, paradoxes are necessary parts of some courses. The inspection paradox is important in some sampling situations and in renewal theory, as we have pointed out. So, omitting the paradox there is not in in the students' best interest. Because we integrate the paradox into a topic or topics that are already included in the courses, such as sampling and conditional probability, the paradox is presented seamlessly as an interesting potential paradox. Sections 3 and 4 contain an overview of the topic.</p> <hd id="AN0185445721-4">3. Some mathematical development</hd> <p>We use the example of the measurement of the incoming logs as a generic or prototypical situation in Example 1, where just two sizes are imagined. In Example 2, any finite number of sizes is explored. In Example 3, length-based sampling is presented more formally. In Example 4, the specialised setting of consecutive renewals and the waiting times between them is discussed.</p> <p>In a course in which probability and expectation have not been introduced, Example 1 can be presented without the vocabulary of probability, but instead in an ad hoc fashion. For example, the phrases 'fraction of the logs', 'proportion of logs in order to accommodate multiple logs of the same length', and 'weighted mean' can be used. Counts could be used instead of fractions or probabilities.</p> <hd id="AN0185445721-5">Example 1 : Two lengths for the logs</hd> <p>Consider just two lengths <emph>x</emph> and <emph>y</emph> for the logs. The constant fractions of the presence of the lengths are <emph>a</emph> and <emph>b</emph> respectively, with <emph>a</emph> > 0, <emph>b</emph> > 0, and <emph>a</emph> + <emph>b</emph> = 1. The expected value of the length of a log selected randomly is</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi><mo>=</mo><mi>ax</mi><mo>+</mo><mi>by</mi><mo>.</mo></math> </ephtml> </p> <p>Using the sampling method presented in Section 1 with the alterations for (<reflink idref="bib1" id="ref33">1</reflink>)–(<reflink idref="bib7" id="ref34">7</reflink>) implemented, the probabilities of selecting logs of length <emph>x</emph> and <emph>y</emph> are proportional to</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ax</mi></math> </ephtml> and</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>by</mi></math> </ephtml> , respectively. In order that the sum of those probabilities is one, the denominator <emph>ax</emph> + <emph>by</emph> is introduced giving the probability mass function</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mi>p</mi><mi>x</mi></msub></mrow><mo>=</mo><mfrac><mrow><mi>ax</mi></mrow><mrow><mi>ax</mi><mo>+</mo><mi>by</mi></mrow></mfrac><mspace width="thinmathspace" /><mrow><mtext>and</mtext></mrow><mspace width="thinmathspace" /><mrow><msub><mi>p</mi><mi>y</mi></msub></mrow><mo>=</mo><mfrac><mrow><mi>by</mi></mrow><mrow><mi>ax</mi><mo>+</mo><mi>by</mi></mrow></mfrac><mo>.</mo></math> </ephtml> </p> <p>The expected value of the length of a selected log is</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>ϵ</mi></mrow><mo>=</mo><mfrac><mrow><mi>ax</mi></mrow><mrow><mi>ax</mi><mo>+</mo><mi>by</mi></mrow></mfrac><mi>x</mi><mo>+</mo><mfrac><mrow><mi>by</mi></mrow><mrow><mi>ax</mi><mo>+</mo><mi>by</mi></mrow></mfrac><mi>y</mi><mo>=</mo><mfrac><mrow><mo>(</mo><mrow><mi>ax</mi></mrow><mo>)</mo><mi>x</mi><mo>+</mo><mo>(</mo><mrow><mi>by</mi></mrow><mo>)</mo><mi>y</mi></mrow><mrow><mi>ax</mi><mo>+</mo><mi>by</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mi>a</mi><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow><mo>+</mo><mi>b</mi><mrow><msup><mi>y</mi><mn>2</mn></msup></mrow></mrow><mrow><mi>ax</mi><mo>+</mo><mi>by</mi></mrow></mfrac><mo>.</mo></math> </ephtml> </p> <p>This expected value is a weighted mean of the lengths <emph>x</emph> and <emph>y</emph> with weights <emph>ax</emph> and <emph>by</emph>. The difference of the estimates is</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mtext>Difference</mtext></mrow><mo>=</mo><mrow><mo>|</mo></mrow><mrow><mrow><mi>ϵ</mi></mrow><mtext>--</mtext><mi>E</mi></mrow><mrow><mo>|</mo></mrow><mo>=</mo><mrow><mo>|</mo><mrow><mfrac><mrow><mi>a</mi><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow><mo>+</mo><mi>b</mi><mrow><msup><mi>y</mi><mn>2</mn></msup></mrow><mtext>--</mtext><mrow><msup><mrow><mo>(</mo><mrow><mi>ax</mi><mo>+</mo><mi>by</mi></mrow><mo>)</mo></mrow><mn>2</mn></msup></mrow></mrow><mrow><mi>ax</mi><mo>+</mo><mi>by</mi></mrow></mfrac></mrow><mo>|</mo></mrow><mo>.</mo></math> </ephtml> </p> <p>Observe that</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>ϵ</mi></mrow><mo>≥</mo><mi>E</mi></math> </ephtml> by the inequality between the weighted power means of degrees one and two with weights <emph>a</emph> and <emph>b</emph>, whose difference is in the numerator (Hardy et al., [<reflink idref="bib12" id="ref35">12</reflink>], pp. 26–28; Mitrinovic, [<reflink idref="bib27" id="ref36">27</reflink>], pp. 76–77). The inequality is an equality if and only if <emph>x</emph> = <emph>y</emph>, i.e. all logs have the same length. The inequality</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>ϵ</mi></mrow><mo>≥</mo><mi>E</mi></math> </ephtml> is an example of the inspection paradox. For a numerical instance, let <emph>x</emph> = 20 ft, <emph>y</emph> = 40 ft, <emph>a</emph> = 0.75, and <emph>b</emph> = 0.25, so that</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi><mo>=</mo><mo>(</mo><mrow><mn>0.75</mn></mrow><mo>)</mo><mo>(</mo><mrow><mn>20</mn></mrow><mo>)</mo><mo>+</mo><mo>(</mo><mrow><mn>0.25</mn></mrow><mo>)</mo><mo>(</mo><mrow><mn>40</mn></mrow><mo>)</mo><mo>=</mo><mn>25</mn><mspace width="thinmathspace" /><mrow><mtext>ft,</mtext></mrow></math> </ephtml> </p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>ϵ</mi></mrow><mo>=</mo><mfrac><mrow><mo>(</mo><mrow><mn>0.75</mn></mrow><mo>)</mo><mrow><msup><mrow><mn>20</mn></mrow><mn>2</mn></msup></mrow><mo>+</mo><mo>(</mo><mrow><mn>0.25</mn></mrow><mo>)</mo><mrow><msup><mrow><mn>40</mn></mrow><mn>2</mn></msup></mrow></mrow><mrow><mo>(</mo><mrow><mn>0.75</mn></mrow><mo>)</mo><mo>(</mo><mrow><mn>20</mn></mrow><mo>)</mo><mo>+</mo><mo>(</mo><mrow><mn>0.25</mn></mrow><mo>)</mo><mo>(</mo><mrow><mn>40</mn></mrow><mo>)</mo></mrow></mfrac><mo>=</mo><mn>28</mn><mspace width="thinmathspace" /><mrow><mtext>ft</mtext></mrow><mo>,</mo></math> </ephtml> </p> <p>and</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow><mi mathvariant="normal">Difference</mi><mo>=</mo><mn>28</mn></mrow><mo>−</mo><mn>25</mn><mo>=</mo><mn>3</mn><mi>ft</mi></mrow><mrow><mtext>.</mtext></mrow></math> </ephtml> </p> <p>▪</p> <hd id="AN0185445721-6">Example 2: Any finite number of lengths for the logs</hd> <p>Consider an input where there are many logs sharing each of <emph>n</emph> lengths <emph>x<subs>i</subs></emph>, <emph>i</emph> = 1, 2, ... , <emph>n</emph>. The random variable, which is the length of a log, is <emph>X,</emph> and Pr(<emph>X</emph> = <emph>x<subs>i</subs></emph>) = <emph>p<subs>i</subs></emph> with</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mo movablelimits="false">∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></msubsup><mrow><mrow><msub><mi>p</mi><mi>i</mi></msub></mrow></mrow><mo>=</mo><mn>1.</mn></math> </ephtml> The expected or mean length is</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>μ</mi><mo>=</mo><mi>E</mi><mo>(</mo><mi>X</mi><mo>)</mo><mo>=</mo><munderover><mrow><mo movablelimits="false">∑</mo></mrow><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><mo>⁡</mo><mrow><msub><mi>p</mi><mi>i</mi></msub></mrow><mrow><msub><mi>x</mi><mi>i</mi></msub></mrow><mo>,</mo></math> </ephtml> </p> <p>which is positive, and the variance is</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msup><mi>σ</mi><mn>2</mn></msup></mrow><mo>=</mo><mi>E</mi><mo>(</mo><mrow><mrow><msup><mi>X</mi><mn>2</mn></msup></mrow></mrow><mo>)</mo><mtext>--</mtext><mrow><mo>(</mo><mrow><mi>E</mi><mo>(</mo><mi>X</mi><mo>)</mo></mrow><msup><mo>)</mo><mn>2</mn></msup></mrow><mo>=</mo><munderover><mrow><mo movablelimits="false">∑</mo></mrow><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><mo>⁡</mo><mo>(</mo><mrow><mrow><msub><mi>p</mi><mi>i</mi></msub></mrow><msubsup><mi>x</mi><mi>i</mi><mn>2</mn></msubsup></mrow><mo>)</mo><mtext>--</mtext><mrow><msup><mrow><mo>(</mo><mrow><munderover><mrow><mo movablelimits="false">∑</mo></mrow><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><mo>⁡</mo><mrow><msub><mi>p</mi><mi>i</mi></msub></mrow><mrow><msub><mi>x</mi><mi>i</mi></msub></mrow></mrow><mo>)</mo></mrow><mn>2</mn></msup></mrow><mo>.</mo></math> </ephtml> </p> <p>Employing the sampling procedure in Section 1, which has been altered following adjustments (<reflink idref="bib1" id="ref37">1</reflink>)–(<reflink idref="bib7" id="ref38">7</reflink>), gives</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mo movablelimits="true" form="prefix">Pr</mo><mo>(</mo><mrow><mi>X</mi><mo>=</mo><mrow><msub><mi>x</mi><mi>i</mi></msub></mrow></mrow><mo>)</mo><mo>=</mo><mfrac><mrow><mrow><msub><mi>p</mi><mi>i</mi></msub></mrow><mrow><msub><mi>x</mi><mi>i</mi></msub></mrow></mrow><mrow><msubsup><mrow><mo movablelimits="false">∑</mo></mrow><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></msubsup><mo>⁡</mo><mrow><msub><mi>p</mi><mi>j</mi></msub></mrow><mrow><msub><mi>x</mi><mi>j</mi></msub></mrow></mrow></mfrac><mo>=</mo><mfrac><mrow><mrow><msub><mi>p</mi><mi>i</mi></msub></mrow><mrow><msub><mi>x</mi><mi>i</mi></msub></mrow></mrow><mi>μ</mi></mfrac></math> </ephtml> </p> <p>and</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>ϵ</mi></mrow><mo>(</mo><mi>X</mi><mo>)</mo><mo>=</mo><munderover><mrow><mo movablelimits="false">∑</mo></mrow><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><mo>⁡</mo><mrow><mo>(</mo><mrow><mfrac><mrow><mrow><msub><mi>p</mi><mi>i</mi></msub></mrow><mrow><msub><mi>x</mi><mi>i</mi></msub></mrow></mrow><mrow><msubsup><mrow><mo movablelimits="false">∑</mo></mrow><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></msubsup><mo>⁡</mo><mrow><msub><mi>p</mi><mi>j</mi></msub></mrow><mrow><msub><mi>x</mi><mi>j</mi></msub></mrow></mrow></mfrac></mrow><mo>)</mo></mrow><mrow><msub><mi>x</mi><mi>i</mi></msub></mrow><mo>=</mo><mfrac><mrow><msubsup><mrow><mo movablelimits="false">∑</mo></mrow><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></msubsup><mo>⁡</mo><mrow><msub><mi>p</mi><mi>i</mi></msub></mrow><msubsup><mi>x</mi><mi>i</mi><mn>2</mn></msubsup></mrow><mrow><msubsup><mrow><mo movablelimits="false">∑</mo></mrow><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></msubsup><mo>⁡</mo><mrow><msub><mi>p</mi><mi>i</mi></msub></mrow><mrow><msub><mi>x</mi><mi>i</mi></msub></mrow></mrow></mfrac><mo>=</mo><mfrac><mrow><mrow><msup><mi>σ</mi><mn>2</mn></msup></mrow><mo>+</mo><mrow><msup><mi>μ</mi><mn>2</mn></msup></mrow></mrow><mi>μ</mi></mfrac><mo>=</mo><mi>μ</mi><mo>+</mo><mfrac><mrow><mrow><msup><mi>σ</mi><mn>2</mn></msup></mrow></mrow><mi>μ</mi></mfrac><mo>.</mo></math> </ephtml> </p> <p>The expression</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mo movablelimits="false">∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></msubsup><mrow><mrow><msub><mi>p</mi><mi>i</mi></msub></mrow><msubsup><mi>x</mi><mi>i</mi><mn>2</mn></msubsup></mrow><mrow><mo>/</mo></mrow><msubsup><mo movablelimits="false">∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></msubsup><mrow><mrow><msub><mi>p</mi><mi>i</mi></msub></mrow><mrow><msub><mi>x</mi><mi>i</mi></msub></mrow></mrow></math> </ephtml> is a weighted contraharmonic mean of the <emph>x</emph><subs>i</subs>, which is not only greater than the mean</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>μ</mi></math> </ephtml> , but also greater than the root-mean-square mean</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msup><mrow><mo>(</mo><mrow><msubsup><mo movablelimits="false">∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></msubsup><mrow><mrow><msub><mi>p</mi><mi>i</mi></msub></mrow><msubsup><mi>x</mi><mi>i</mi><mn>2</mn></msubsup></mrow></mrow><mo>)</mo></mrow><mrow><mn>1</mn><mrow><mo>/</mo></mrow><mn>2</mn></mrow></msup></mrow></math> </ephtml> (Jenkins & Tuten, [<reflink idref="bib15" id="ref39">15</reflink>]; Mitrinovic, [<reflink idref="bib27" id="ref40">27</reflink>], p. 70). There are two distributions. One has assigned probabilities <emph>p<subs>i</subs></emph>, and the other has probabilities <emph>p<subs>i</subs>x<subs>i</subs>/μ</emph>. The mean of the second distribution is the contraharmonic mean of the first distribution. Then,</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mtext>Difference</mtext></mrow><mo>=</mo><mrow><mo>|</mo></mrow><mrow><mrow><mi>ϵ</mi></mrow><mtext>--</mtext><mi>E</mi></mrow><mrow><mo>|</mo></mrow><mo>=</mo><mrow><mo>|</mo><mrow><mfrac><mrow><msubsup><mrow><mo movablelimits="false">∑</mo></mrow><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></msubsup><mo>⁡</mo><mrow><msub><mi>p</mi><mi>i</mi></msub></mrow><msubsup><mi>x</mi><mi>i</mi><mn>2</mn></msubsup><mtext>--</mtext><mrow><msup><mrow><mrow><mo>(</mo><mrow><msubsup><mrow><mo movablelimits="false">∑</mo></mrow><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></msubsup><mo>⁡</mo><mrow><msub><mi>p</mi><mi>i</mi></msub></mrow><mrow><msub><mi>x</mi><mi>i</mi></msub></mrow></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow></mrow><mrow><msubsup><mrow><mo movablelimits="false">∑</mo></mrow><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></msubsup><mo>⁡</mo><mrow><msub><mi>p</mi><mi>i</mi></msub></mrow><mrow><msub><mi>x</mi><mi>i</mi></msub></mrow></mrow></mfrac></mrow><mo>|</mo></mrow><mo>=</mo><mrow><mo>|</mo><mrow><mfrac><mrow><mi>E</mi><mo>(</mo><mrow><mrow><msup><mi>X</mi><mn>2</mn></msup></mrow></mrow><mo>)</mo><mtext>--</mtext><mrow><msup><mrow><mo>(</mo><mrow><mi>E</mi><mo>(</mo><mi>X</mi><mo>)</mo></mrow><mo>)</mo></mrow><mn>2</mn></msup></mrow></mrow><mrow><mi>E</mi><mo>(</mo><mi>X</mi><mo>)</mo></mrow></mfrac></mrow><mo>|</mo></mrow><mo>=</mo><mfrac><mrow><mrow><msup><mi>σ</mi><mn>2</mn></msup></mrow></mrow><mi>μ</mi></mfrac><mo>.</mo></math> </ephtml> </p> <p>Observe that</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>ϵ</mi></mrow><mo>≥</mo><mi>E</mi></math> </ephtml> . This inequality is an equality if and only if</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msup><mi>σ</mi><mn>2</mn></msup></mrow><mo>=</mo><mn>0</mn><mo>,</mo></math> </ephtml> i.e. all potential measurements <emph>x<subs>i</subs></emph> are identical. Presumably, the mean</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>μ</mi></math> </ephtml> and variance</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msup><mi>σ</mi><mn>2</mn></msup></mrow></math> </ephtml> are unknown, otherwise the measurements might not be taken. However, if estimates of those two parameters can be made, perhaps from historical sample means and variances, then the magnitude of Difference can be assessed. There are rules of thumb for estimating the standard deviation as a multiple of the range (Lord, [<reflink idref="bib22" id="ref41">22</reflink>]; Noether, [<reflink idref="bib31" id="ref42">31</reflink>]; Triola, [<reflink idref="bib51" id="ref43">51</reflink>], p. 100). The range might be well estimated from the physical set-up. The range has the unfavourable feature that it is extremely sensitive to extra-small and extra-large observations, but that effect might be ameliorated in some situations, such as this one in which outsized logs may be rejected before reaching the mill. An approximation of</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msup><mi>σ</mi><mn>2</mn></msup></mrow><mrow><mo>/</mo></mrow><mi>μ</mi></math> </ephtml> might be used to obtain an estimate of <emph>E</emph> from</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>ϵ</mi></mrow></math> </ephtml> . Because</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msup><mi>σ</mi><mn>2</mn></msup></mrow><mrow><mo>/</mo></mrow><mi>μ</mi><mo>=</mo><mo>(</mo><mi>μ</mi><mo>)</mo><mo>(</mo><mrow><mrow><msup><mi>σ</mi><mn>2</mn></msup></mrow><mrow><mo>/</mo></mrow><mrow><msup><mi>μ</mi><mn>2</mn></msup></mrow></mrow><mo>)</mo><mo>,</mo></math> </ephtml> the <emph>coefficient of variation,</emph></p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>σ</mi><mrow><mo>/</mo></mrow><mi>μ</mi></math> </ephtml> , might be directly used here. If the estimate of</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msup><mi>σ</mi><mn>2</mn></msup></mrow><mrow><mo>/</mo></mrow><mi>μ</mi></math> </ephtml> is sufficiently small, this analysis shows that these sampling methods might be safely employed interchangeably. ▪</p> <p>Derivations of the inspection paradox's inequality that the expected value of the length of the selected object or interval is larger than the expected value for an ordinary object or interval is available in many forms.</p> <hd id="AN0185445721-7">Example 3: Continuous lengths for the logs</hd> <p>Consider the distribution of <emph>X</emph>, which might be the length of logs or another application; <emph>X</emph> could be a continuous variable. Take the domain or support of <emph>X</emph> to be a subset of the positive real numbers. The probability mass function or probability density function is <emph>f</emph>(<emph>x</emph>). For simple random sampling,</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>μ</mi><mo>=</mo><mi>E</mi><mo>(</mo><mi>X</mi><mo>)</mo><mo>></mo><mn>0</mn><mrow><mtext> and </mtext></mrow><mrow><msup><mi>σ</mi><mn>2</mn></msup></mrow><mo>=</mo><mi>E</mi><mo>(</mo><mrow><mrow><msup><mi>X</mi><mn>2</mn></msup></mrow></mrow><mo>)</mo><mtext>--</mtext><mrow><mo>(</mo><mrow><mi>E</mi><mo>(</mo><mi>X</mi><mo>)</mo></mrow><msup><mo>)</mo><mn>2</mn></msup></mrow><mo>.</mo></math> </ephtml> </p> <p>For length-based sampling, the probability mass function or probability density function is</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mi>x</mi><mi>μ</mi></mfrac><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>,</mo></math> </ephtml> </p> <p>where <emph>μ</emph> in the denominator assures that the probabilities total 1. The expected value is</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi><mrow><mo>(</mo><mrow><mfrac><mrow><mrow><msup><mi>X</mi><mn>2</mn></msup></mrow></mrow><mi>μ</mi></mfrac></mrow><mo>)</mo></mrow><mo>=</mo><mi>μ</mi><mo>+</mo><mfrac><mrow><mrow><msup><mi>σ</mi><mn>2</mn></msup></mrow></mrow><mi>μ</mi></mfrac><mo>></mo><mi>μ</mi><mo>,</mo></math> </ephtml> </p> <p>which demonstrates the inspection paradox with respect to length-based sampling.</p> <p>For example, consider uniformly distributed <emph>X</emph> with</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mfrac><mn>1</mn><mi>α</mi></mfrac></math> </ephtml> for 0 < <emph>x</emph> < <emph>α</emph> and <emph>f</emph>(<emph>x</emph>) zero elsewhere,</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>μ</mi><mo>=</mo><mfrac><mi>α</mi><mn>2</mn></mfrac><mspace width="thinmathspace" /><mrow><mtext>and</mtext></mrow><mspace width="thinmathspace" /><mrow><msup><mi>σ</mi><mn>2</mn></msup></mrow><mo>=</mo><mfrac><mrow><mrow><msup><mi>α</mi><mn>2</mn></msup></mrow></mrow><mrow><mn>12</mn></mrow></mfrac></math> </ephtml> (Devore, [<reflink idref="bib5" id="ref44">5</reflink>], p. 143). Thus,</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>μ</mi><mo>+</mo><mfrac><mrow><mrow><msup><mi>σ</mi><mn>2</mn></msup></mrow></mrow><mi>μ</mi></mfrac><mo>=</mo><mfrac><mn>2</mn><mn>3</mn></mfrac><mi>α</mi><mo>></mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>α</mi><mo>.</mo></math> </ephtml> Indeed, the length-based measurements have the beta distribution</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mn>2</mn><mi>x</mi></mrow><mrow><mrow><msup><mi>α</mi><mn>2</mn></msup></mrow></mrow></mfrac></math> </ephtml> , whose mean is</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>2</mn><mn>3</mn></mfrac><mi>α</mi></math> </ephtml> (Devore, [<reflink idref="bib5" id="ref45">5</reflink>], p. 167).</p> <p>Similarly, for exponentially distributed <emph>X</emph>,</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mfrac><mn>1</mn><mi>λ</mi></mfrac><mrow><msup><mi>e</mi><mrow><mtext>--</mtext><mfrac><mi>x</mi><mi>λ</mi></mfrac></mrow></msup></mrow></math> </ephtml> with <emph>x</emph> > 0 and <emph>λ</emph> > 0,</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>μ</mi><mo>=</mo><mi>λ</mi></math> </ephtml> , and</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msup><mi>σ</mi><mn>2</mn></msup></mrow><mo>=</mo><mrow><msup><mi>λ</mi><mn>2</mn></msup></mrow></math> </ephtml> (Devore, [<reflink idref="bib5" id="ref46">5</reflink>], p. 157). Thus,</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>μ</mi><mo>+</mo><mfrac><mrow><mrow><msup><mi>σ</mi><mn>2</mn></msup></mrow></mrow><mi>μ</mi></mfrac><mo>=</mo><mn>2</mn><mi>λ</mi></math> </ephtml> . The length-based measurements have the gamma distribution</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mi>x</mi><mrow><mrow><msup><mi>λ</mi><mn>2</mn></msup></mrow></mrow></mfrac><mrow><msup><mi>e</mi><mrow><mtext>--</mtext><mfrac><mi>x</mi><mi>λ</mi></mfrac></mrow></msup></mrow></math> </ephtml> , whose mean is</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mi>λ</mi></math> </ephtml> (Devore, [<reflink idref="bib5" id="ref47">5</reflink>], p. 160). The expected value is twice the expected value using simple random sampling, so that the inspected log or interval is expected to be twice as long as an ordinary uninspected log or interval. ▪</p> <p>The examples in Section 2 that are for the waiting times between buses and geyser eruptions and lifetimes of lightbulbs and batteries possess the additional structure of being sequential in time. In some circumstances, the time of the previous event, and hence the waiting time between events, may be known solely probabilistically. In this setting, the times between events, such as the failures of lightbulbs and arrivals of buses, are called <emph>interarrival times</emph>. The requirement that the sampling be precisely length-based may not hold. An observer arrives and waits until the next event, which follows a probability distribution. The observer makes a probabilistic estimate about the event that occurred before their arrival. The waiting times and lifetimes are random processes in which longer times are more likely to be selected, but the precise form</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mi>x</mi><mi>μ</mi></mfrac><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></math> </ephtml> may not apply exactly. A simple alternative derivation for the inspection paradox for these situations, which are examples of renewal processes, is offered in Example 4.</p> <p>In a <emph>renewal process</emph>, events occur consecutively and independently and their lifetimes have a common probability distribution (Ross, [<reflink idref="bib42" id="ref48">42</reflink>], pp. 409–479; Sigman, [<reflink idref="bib47" id="ref49">47</reflink>]). When a bus leaves the bus stop or a geyser stops erupting, the next waiting time for a bus or an eruption begins. When a lightbulb or battery fails, it is replaced by a new, but otherwise identical, one and the waiting time for its failure begins. In these examples, an observer is more likely to enter during a longer lifetime or waiting time. In these settings, the inspection paradox is sometimes called the <emph>renewal-time paradox</emph>. With the assumption that the lengths are the outcome of a renewal process in which events occur as a Poisson process, which is often true or true for practical purposes, the inspection paradox is derived in (Feller, [<reflink idref="bib8" id="ref50">8</reflink>], pp. 11–15; Karlin & Taylor, [<reflink idref="bib16" id="ref51">16</reflink>], pp. 173–175; Marengo et al., [<reflink idref="bib25" id="ref52">25</reflink>]; Ross, [<reflink idref="bib42" id="ref53">42</reflink>], pp. 447–449; Taylor & Karlin, [<reflink idref="bib50" id="ref54">50</reflink>], pp. 432–434). The usual proofs of the paradox, which is stated as an inequality of probabilities, without assuming a particular probability distribution for the waiting times, employs conditional probabilities (Karlin & Taylor, [<reflink idref="bib16" id="ref55">16</reflink>], pp. 192–195; Ross, [<reflink idref="bib41" id="ref56">41</reflink>], [<reflink idref="bib42" id="ref57">42</reflink>], pp. 447–448), and appears in Example 4.</p> <hd id="AN0185445721-8">Example 4 For a renewal process</hd> <p>Assume that the lifetime or waiting time between the events has a probability distribution, which might be known. Designate the cumulative distribution function of the random variable <emph>T</emph> of the lifetime between events by <emph>F</emph>(<emph>t</emph>). The total lifetime of an object, e.g. a lightbulb, or the time between events, e.g. the interval between buses, is the sum of the current elapsed time or age <emph>a</emph> when we intervene or arrive and the remaining time until the end of the lifetime. For the bus example, we might not know the age <emph>a</emph>, i.e. the elapsed time since the previous bus left our bus stop. The numerical value of the current age <emph>a</emph> is not required to illustrate the inspection paradox. To display the inspection paradox, use the conditional probability that the lifetime of the item or time between events is greater than <emph>a </emph>+<emph> r</emph>, given that the lifetime is greater than the elapsed time <emph>a</emph>, which is perhaps unknown. Symbolically, this is</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Pr</mi><mo>(</mo><mi>T</mi><mo>></mo><mi>a</mi><mo>+</mo><mi>r</mi><mrow><mo>|</mo></mrow><mi>T</mi><mo>></mo><mi>a</mi><mo>)</mo><mo>,</mo></math> </ephtml> </p> <p>where <emph>T</emph> is the random variable which is the lifetime of the item or length of the time between events, <emph>a</emph> is the time between the prior event and our arrival, and <emph>r</emph> is the remaining lifetime. Recalling the definition of conditional probability, which is Pr(<emph>A</emph>| <emph>B</emph>) = Pr(<emph>A</emph> ∩ <emph>B</emph>)/Pr(<emph>B</emph>) for events <emph>A</emph> and <emph>B</emph>,</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"><mtr><mtd><mo movablelimits="true" form="prefix">Pr</mo><mo>(</mo><mi>T</mi><mo>></mo><mi>a</mi><mo>+</mo><mi>r</mi><mrow><mo>|</mo></mrow><mi>T</mi><mo>></mo><mi>a</mi><mo>)</mo></mtd><mtd><mi /><mo>=</mo><mo movablelimits="true" form="prefix">Pr</mo><mo>(</mo><mo>(</mo><mi>T</mi><mo>></mo><mi>a</mi><mo>+</mo><mi>r</mi><mo>)</mo><mo>∩</mo><mo>(</mo><mi>T</mi><mo>></mo><mi>a</mi><mo>)</mo><mo>)</mo><mrow><mo>/</mo></mrow><mo movablelimits="true" form="prefix">Pr</mo><mo>(</mo><mi>T</mi><mo>></mo><mi>a</mi><mo>)</mo></mtd></mtr><mtr><mtd /><mtd><mi /><mo>=</mo><mrow><mtext>Pr</mtext></mrow><mo>(</mo><mrow><mi>T</mi><mo>></mo><mi>a</mi><mo>+</mo><mi>r</mi></mrow><mo>)</mo><mrow><mo>/</mo></mrow><mrow><mtext>Pr</mtext></mrow><mo>(</mo><mrow><mi>T</mi><mo>></mo><mi>a</mi></mrow><mo>)</mo></mtd></mtr><mtr><mtd /><mtd><mi /><mo>=</mo><mo>(</mo><mn>1</mn><mo>−</mo><mi>F</mi><mo>(</mo><mrow><mi>a</mi><mo>+</mo><mi>r</mi><mo>)</mo></mrow><mo>)</mo><mrow><mo>/</mo></mrow><mo>(</mo><mrow><mn>1</mn><mo>−</mo><mi>F</mi><mo>(</mo><mi>a</mi><mo>)</mo></mrow><mo>)</mo><mo>.</mo></mtd></mtr></mtable></math> </ephtml> </p> <p>This supplies a formula that helps to quantify the paradox when <emph>F</emph> is known. Because</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"><mtr><mtd /><mtd><mn>0</mn><mo><</mo><mrow><mtext>Pr</mtext></mrow><mo>(</mo><mrow><mi>T</mi><mo>></mo><mi>a</mi></mrow><mo>)</mo><mo>=</mo><mn>1</mn><mo>−</mo><mi>F</mi><mo>(</mo><mi>a</mi><mo>)</mo><mo><</mo><mn>1</mn><mo>,</mo></mtd></mtr></mtable></math> </ephtml> </p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"><mtr><mtd /><mtd><mrow><mtext>Pr</mtext></mrow><mo>(</mo><mi>T</mi><mo>></mo><mi>a</mi><mo>+</mo><mi>r</mi><mrow><mo>|</mo></mrow><mi>T</mi><mo>></mo><mi>a</mi><mo>)</mo><mo>></mo><mn>1</mn><mo>−</mo><mi>F</mi><mo>(</mo><mrow><mi>a</mi><mo>+</mo><mi>r</mi></mrow><mo>)</mo><mo>=</mo><mrow><mtext>Pr</mtext></mrow><mo>(</mo><mrow><mi>T</mi><mo>></mo><mi>a</mi><mo>+</mo><mi>r</mi></mrow><mo>)</mo><mo>.</mo></mtd></mtr></mtable></math> </ephtml> </p> <p>The lifetime or length of the interval in which we have intervened or arrived is greater in probability than a typical interval, which is a form of the inspection paradox. ▪</p> <p>The lengths-of-logs example is a renewal process. The log that is passing the given point at the randomly chosen time is selected. Picking a point on a log is analogous to arriving at a bus stop at a selected time. The log's length is the sum of the lengths from each end to the point, one length corresponds to <emph>a</emph>, but for the logs, the whole length is imagined to be determined in one measurement.</p> <hd id="AN0185445721-9">4. Two points of view</hd> <p>The difference ±</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo>(</mo><mi>ϵ</mi></mrow><mtext>--</mtext><mi>E</mi><mo>)</mo></math> </ephtml> between the two estimates based on the two sampling procedures might be called the <emph>bias</emph> if we want an estimate either as the experience seen by someone with an overview or by a participant. These are merely two sampling methods or points of view that produce different, but correct, answers to different questions. Thus, this is a veridical paradox in Quine's typology. By employing one sampling method but seeking the answer that would be supplied by the other sampling method, we could decide that there is a paradox. Often, the emphasis is on situations in which length-based sampling can lead to incorrect conclusions. This can be very serious and damaging, as the discussions in (Zelen & Feinleib, [<reflink idref="bib54" id="ref58">54</reflink>]) and (Gates, [<reflink idref="bib10" id="ref59">10</reflink>]) show, where slowly progressing types of cancer are more likely to be detected, studied, or treated than fast moving cancers that have a smaller time to be detected.</p> <p>There are examples in which length-based sampling is 'correct', as we have seen in the examples in Section 2. The number of fellow students in classes and the actual waiting times for buses that are obtained with length-based sampling are more important to the individuals than the alternative method.</p> <p>As with the class sizes and waiting times for buses, researchers who want to determine the experience of individuals favour length-based sampling. In many circumstances, it is the most effective sampling method. Keillor et al. ([<reflink idref="bib17" id="ref60">17</reflink>]) use it to determine consumer behaviour. Pal et al. ([<reflink idref="bib32" id="ref61">32</reflink>]) and Reuveni ([<reflink idref="bib40" id="ref62">40</reflink>]) use the word 'restart' to mean to intervene and, hence select longer-lived physical, chemical, and biological systems via the inspection paradox. Downey ([<reflink idref="bib6" id="ref63">6</reflink>]) makes the case for length-based sampling to determine individual's experience with COVID-19. Lewis ([<reflink idref="bib21" id="ref64">21</reflink>]) argues that backward COVID tracing, which by its nature uses length-based sampling because a few people spread the disease to a disproportionately large number of individuals, is very effective at finding superspreaders of COVID. Morozov and Potakhina ([<reflink idref="bib28" id="ref65">28</reflink>]) show that length-based sampling was occurring in a system of optical buffers, and thereby the authors create a more realistic model of the system.</p> <hd id="AN0185445721-10">5. Observations on teaching the paradox</hd> <p>We have found students' reactions to our presentations in open-ended questions on snap semi-monthly surveys of classes, end-of-semester course evaluations, and informal chats with students. We have discovered that, unlike some paradoxes, students have no difficulty understanding or appreciating the inspection paradox, when it is presented to them in a manner reflecting Sections 1–4 at a point in the course where they have sufficient background. The topic tends to grab students' interest, perhaps because it might seem enigmatic, and the word <emph>paradox</emph> can arouse curiosity (Klymchuk & Kachapova, [<reflink idref="bib19" id="ref66">19</reflink>]; Movshovitz-Hadar & Hadass, [<reflink idref="bib29" id="ref67">29</reflink>]). Students appear to like the mysterious or almost occult nature of paradoxes, especially if they believe that they are not traps to ensnare them personally or hurt their grade. They like feeling as though they are insiders to some fact about which even some practitioners are unaware, as shown by some articles such as Bytheway ([<reflink idref="bib2" id="ref68">2</reflink>]) and Jenkins and Tuten ([<reflink idref="bib15" id="ref69">15</reflink>]).</p> <p>Our experience from talking to students months after the courses were held has revealed that students remember unusual or different portions of the subject. The Monty-Hall problem, the birthday problem, and this paradox are among them. Additionally, they might remember them well because those problems and applications can serve as touchstones in class discussions and be part of the course's culture.</p> <p>As Section 4 emphasises, the paradox can be considered to be not paradoxical at all, but rather simply choices among goals of a study and sampling methodologies. That realisation can make the topic nearly mundane for both the teachers and the learners. It unpacks the paradox as simply selections that may all be correct. We consider it a success when a student realises that before we say so. Students have asked why it is called a <emph>paradox</emph> and what person decided that it is one.</p> <p>As some students have told us, the inspection paradox just tells scientists or experimenters to take samples from the population that they wish to investigate and to measure what they want to estimate. To these students, it is just the correct thing to do, and, as one student said, it is not that different [to her] from being cautious not to divide by zero.</p> <p>There are side benefits to using and emphasising an occasional paradox or more popularly-based examples, such as the Monty Hall problem and this paradox. They bring the course to a more everyday level and supply students with something to tell other students or even people back home. One student told one of us that the inclusion of the inspection paradox and the Monty Hall problem may be responsible for his obtaining his current employment, because during the job interview, when he felt that he had nothing to say, he had an opening to talk about both and appeared to have done a sufficiently good job for the interviewer.</p> <hd id="AN0185445721-11">6. Concluding comments</hd> <p>Although our theme is that this paradox can be made non-paradoxical rather quickly, it continues to be called a paradox and, as we have shown, can be an important topic that interests students. It is ripe with possibilities for further study, such as in service courses for particular disciplines. Gelman and Glickman ([<reflink idref="bib11" id="ref70">11</reflink>]) present the use of the family size example in the classroom. Masuda and Porter ([<reflink idref="bib26" id="ref71">26</reflink>]) suggest ways to introduce it in precollege classes. It can be viewed solely from the perspective of the student's area of study, such as in medicine (Gates, [<reflink idref="bib10" id="ref72">10</reflink>]), in control charts (Herff et al., [<reflink idref="bib14" id="ref73">14</reflink>]), and in the study of families (Smith, [<reflink idref="bib48" id="ref74">48</reflink>]).</p> <p>The topic is ripe for use in independent study and undergraduate research at almost any academic level. Here are some suggestions.</p> <p>Sometimes, weights <emph>w</emph>(<emph>x</emph>) other than lengths, i.e. <emph>w</emph>(x) = <emph>x</emph>, are useful (Navarro et al., [<reflink idref="bib30" id="ref75">30</reflink>]; Patil & Rao, [<reflink idref="bib34" id="ref76">34</reflink>]; Rao, [<reflink idref="bib38" id="ref77">38</reflink>]). Then, the probability distribution for the measurements is</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mi>w</mi><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mrow><mi>E</mi><mo>(</mo><mrow><mi>w</mi><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>)</mo></mrow></mfrac><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></math> </ephtml> , assuming that the expectation in the denominator exists. The weights <emph>w</emph>(<emph>x</emph>) = <emph>x</emph><sups>2</sups> are used in spatial or area statistics where area contributes to the selection in the sampling (Akman et al., [<reflink idref="bib1" id="ref78">1</reflink>]; Shanker, [<reflink idref="bib44" id="ref79">44</reflink>]). The inspection paradox can be reversed if the weights <emph>w</emph>(<emph>x</emph>) create a bias toward smaller objects, thus the usual inspection paradox is not inevitable. If <emph>w</emph>(<emph>x</emph>) decreases with <emph>x</emph>, smaller objects or intervals are more likely to be selected (Patil & Rao, [<reflink idref="bib34" id="ref80">34</reflink>]). A student could examine these alternative weights mathematically and in applications.</p> <p>Students might explore remedial measures. From their study of correlation and regression, they are accustomed to diminishing the effect of larger measurements by taking the logarithm or square root of the observations. This could be applied here. Students could be asked to imagine and evaluate practical solutions to the paradox or ways to avoid it. For example, a suggestion for renewal processes that appear in Example 4 is not to take the item selected, but, the next one, as that one might not suffer from the paradox (Marengo et al., [<reflink idref="bib25" id="ref81">25</reflink>]). In Example 3 for exponentially distributed lifetimes, a suggestion might be to divide the observed length of the selected object or waiting time by two. This might be an excellent approximation if the events follow a Poisson process, which has exponentially distributed waiting times (Ross, [<reflink idref="bib42" id="ref82">42</reflink>], pp. 301–302; Shiavi & Negin, [<reflink idref="bib45" id="ref83">45</reflink>]).</p> <p>Various averages or means arose in Examples 1 and 2. Those might be productively analyzed in a student-based study (Lann & Falk, [<reflink idref="bib20" id="ref84">20</reflink>]; Pathak & Singh, [<reflink idref="bib33" id="ref85">33</reflink>]; Sen, [<reflink idref="bib43" id="ref86">43</reflink>]). Students might adapt hypothesis tests, which are familiar to them, to detecting that length-based sampling had been used. Guidance and examples can be found in (Akman et al., [<reflink idref="bib1" id="ref87">1</reflink>]; Navarro et al., [<reflink idref="bib30" id="ref88">30</reflink>]).</p> <p>Thus, although this is a relatively easily explained veridical paradox, it continues to seem strange and mysterious when first seen. It is an excellent entry point for in-the-classroom and out-of-the-classroom learning and discovery.</p> <hd id="AN0185445721-12">Disclosure statement</hd> <p>No potential conflict of interest was reported by the authors.</p> <ref id="AN0185445721-13"> <title> References </title> <blist> <bibl id="bib1" idref="ref2" type="bt">1</bibl> <bibtext> Akman, O., Gamage, J., Jannot, J., Juliano, S., Thurman, A., & Whitman, D. (2007). A simple test for detection of length-based sampling. JP Journal of Biostatistics, 1 (2), 189 – 195.</bibtext> </blist> <blist> <bibl id="bib2" idref="ref21" type="bt">2</bibl> <bibtext> Bytheway, B. (1974). A statistical trap associated with family size. 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  Group: Ti
  Data: The Useful and Not so Paradoxical Inspection Paradox
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  Data: English
– Name: Author
  Label: Authors
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  Data: <searchLink fieldCode="AR" term="%22James+E%2E+Marengo%22">James E. Marengo</searchLink> (ORCID <externalLink term="https://orcid.org/0000-0003-0498-0424">0000-0003-0498-0424</externalLink>)<br /><searchLink fieldCode="AR" term="%22David+L%2E+Farnsworth%22">David L. Farnsworth</searchLink> (ORCID <externalLink term="https://orcid.org/0000-0003-0251-8770">0000-0003-0251-8770</externalLink>)
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  Data: <searchLink fieldCode="SO" term="%22International+Journal+of+Mathematical+Education+in+Science+and+Technology%22"><i>International Journal of Mathematical Education in Science and Technology</i></searchLink>. 2025 56(6):1196-1208.
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  Data: Taylor & Francis. Available from: Taylor & Francis, Ltd. 530 Walnut Street Suite 850, Philadelphia, PA 19106. Tel: 800-354-1420; Tel: 215-625-8900; Fax: 215-207-0050; Web site: http://www.tandf.co.uk/journals
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  Group: SrcInfo
  Data: Y
– Name: Pages
  Label: Page Count
  Group: Src
  Data: 13
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  Label: Publication Date
  Group: Date
  Data: 2025
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  Data: Journal Articles<br />Reports - Descriptive
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  Label: Education Level
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  Data: <searchLink fieldCode="EL" term="%22Higher+Education%22">Higher Education</searchLink><br /><searchLink fieldCode="EL" term="%22Postsecondary+Education%22">Postsecondary Education</searchLink>
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  Data: <searchLink fieldCode="DE" term="%22Statistics%22">Statistics</searchLink><br /><searchLink fieldCode="DE" term="%22Mathematics+Instruction%22">Mathematics Instruction</searchLink><br /><searchLink fieldCode="DE" term="%22Student+Projects%22">Student Projects</searchLink><br /><searchLink fieldCode="DE" term="%22Undergraduate+Students%22">Undergraduate Students</searchLink><br /><searchLink fieldCode="DE" term="%22Student+Research%22">Student Research</searchLink><br /><searchLink fieldCode="DE" term="%22Mathematical+Concepts%22">Mathematical Concepts</searchLink>
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  Data: 10.1080/0020739X.2024.2327551
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  Data: 0020-739X<br />1464-5211
– Name: Abstract
  Label: Abstract
  Group: Ab
  Data: We offer ways that the inspection paradox can be usefully presented, even in the most elementary statistics courses, and provide guidance about how it can be revisited subsequently in the same course and in succeeding courses. Numerous situations in which the inspection paradox might occur are mentioned, and mathematically simple demonstrations are furnished in order to help convince students of the paradox's veracity. Although remedial actions for the paradox are given, the main point of view is that the two sampling procedures leading to this apparent paradox are simply alternative methods that yield differing, but bona fide, measures and interpretations for the populations from which the data were obtained. Suggestions for student projects and undergraduate research are given.
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      Pagination:
        PageCount: 13
        StartPage: 1196
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        Type: general
      – SubjectFull: Mathematics Instruction
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      – SubjectFull: Student Projects
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      – TitleFull: The Useful and Not so Paradoxical Inspection Paradox
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