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class="pull-right" style="color: #555" title="Suggested study time: 10 minutes"><i class="fa fa-clock-o"></i> 10'</span> </ol> <article id="main-article"> <p><img alt="" src="../../basics/question-mark-1872665_960_720-1.jpg" style="float: left; width: 250px; height: 138px;">An <i>error</i> is the difference between the value you determine and the true value. Given that we collect data in experiments, and that our measuring instruments are not perfectly sensitive, there will always be an <i>uncertainty </i>in our value.</p> <hr class="hidden-separator"> <div class="panel panel-turquoise panel-has-colored-body"> <div class="panel-heading"> <div> <p>Key Concepts</p> </div> </div> <div class="panel-body"> <div> <div class="panel panel-has-colored-body panel-has-border panel-turquoise"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Errors when measuring</p> </div> </div> <div class="panel-body"> <div> <p>A <em>systematic</em> error causes all measurements to be offset by exactly the same amount. This has the effect of changing the y-axis on a linear graph, for example:</p> <ul> <li>a non-zeroed ruler when measuring length</li> <li>failing to subtract the mass of the measuring cylinder when finding the density of a liquid</li> </ul> <p>A <em>random</em> error is usually caused by the experimenter's interaction with the equipment and could increase or decrease the measurement. Provided that a large number of data points have been collected or that repeats have been done, the overall effect of a random error should increase the size of the error bars, but not the final gradient or intercept. For example:</p> <ul> <li>deciding when a pendulum passes a fiducial marker</li> <li>marking the centre of a light ray from a reflecting surface</li> </ul> </div> </div> <div class="panel-footer"> <div> <p> </p> </div> </div> </div> <div class="panel panel-has-colored-body panel-has-border panel-turquoise panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Uncertainties</p> </div> </div> <div class="panel-body"> <div> <p>An <em>absolute</em> uncertainty (often just referred to as the uncertainty) is the size of the error due to selection of a particular measuring instrument. This is usually equal to the smallest scale division on the measuring instrument. A typical metre ruler has an absolute uncertainty of <span class="math-tex">\(\pm\)</span> 1 mm. An ammeter might have an absolute uncertainty of <span class="math-tex">\(\pm\)</span>0.01 A.</p> <p>However, you may realise that the absolute uncertainty is greater if there is considerable random error. You could quantify this absolute uncertainty by halving the range of repeated results.</p> <p>A <em>fractional</em> uncertainty (or percentage uncertainty, if <span class="math-tex">\(\times 100 \%\)</span>) is the ratio of the absolute uncertainty to the measurement itself: <span class="math-tex">\({\Delta x \over x}\)</span></p> </div> </div> <div class="panel-footer"> <div> <p> </p> </div> </div> </div> <p> </p> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-yellow"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Essentials</p> </div> </div> <div class="panel-body"> <div> <div class="panel panel-has-colored-body panel-has-border panel-yellow"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Propagating uncertainties</p> </div> </div> <div class="panel-body"> <div> <p>Absolute uncertainties can be combined easily for quantities that are added. For example, if Alice and Ben are in charge of timing the first 10 m and last 10 m of a race, you would add together the times <strong>and</strong> the uncertainties:</p> <div class="box"> <p style="text-align: center;"><span class="math-tex">\(4.6 \pm 0.3\)</span> s added to <span class="math-tex">\(4.2 \pm 0.2\)</span> s is <span class="math-tex">\(8.8 \pm 0.5\)</span> s</p> </div> <p>The same applies for subtraction; <strong>add</strong> the uncertainties. As a rule of thumb, the uncertainty should always increase with more data being combined.</p> <p>NB: If quantities are multiplied or divided, absolute uncertainties cannot be combined. Instead, the fractional uncertainties are added:</p> <div class="box"> <p style="text-align: center;">If <span class="math-tex">\(a=bc\)</span> or <span class="math-tex">\(a={ b \over c}\)</span>, <span class="math-tex">\({\Delta a \over a} = {\Delta b \over b} +{\Delta c \over c} \)</span></p> </div> <p>You can then convert back into an absolute uncertainty by multiplying by the value (in this case, <span class="math-tex">\(a\)</span>).</p> <p>A trick with powers is to realise that, for example, <span class="math-tex">\(a^3 = a \times a \times a\)</span>. It is then quite straightforward to spot the following:</p> <p><span class="math-tex">\({\Delta a^3 \over a^3}={\Delta a \over a}+{\Delta a \over a}+{\Delta a \over a} = 3{\Delta a \over a}\)</span></p> <p>In general:</p> <div class="box"> <p style="text-align: center;"><span class="math-tex">\({\Delta a^n \over a^n}=n{\Delta a \over a}\)</span></p> </div> </div> </div> <div class="panel-footer"> <div> <p> </p> </div> </div> </div> <div class="panel panel-has-colored-body panel-has-border panel-yellow panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Gradients and intercepts</p> </div> </div> <div class="panel-body"> <div> <p style="text-align: center;"><strong><img alt="" src="../../basics/max-min.jpg" style="width: 213px; height: 120px;"></strong></p> <p><img class="sibico" src="../../../img/sibico/ee.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="Extended Essay"> In your Internal Assessment or Extended Essay you are likely to want to know the uncertainty in your gradient or intercept (for a linear relationship). Simply draw the maximum and minimum gradients of the line based on your error bars and find the gradient or intercept of each. The largest deviation from the best fit line will provide a good estimate of the uncertainty.</p> <div class="magenta"> <p>The goal is for our uncertainty to encompass the true value. This makes an experiment <em>accurate</em>. For example, finding <span class="math-tex">\(g=8 \pm2 \)</span> ms<sup>-2</sup> is accurate, even though it is much less <em>precise</em> than finding <span class="math-tex">\(g=9.79 \pm 0.01\)</span>ms<sup>-2</sup> (which is inaccurate).</p> </div> </div> </div> <div class="panel-footer"> <div> <p> </p> </div> </div> </div> </div> </div> </div> <div class="panel panel-has-colored-body panel-green"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Test Yourself</p> </div> </div> <div class="panel-body"> <div> <p><i>Use flashcards to practise your recall.</i></p> <div class="tib-flashcard"><a class="show-flashcards btn btn-success btn-xs-block btn-block " data-levels="1" data-mode="" data-topics="757" data-subject-id="6" data-n-flashcards="5" style="text-align:center">Show flashcards</a></div><hr> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="page-container panel-self-assessment" data-id="672"> <div class="panel-heading">MY PROGRESS</div> <div class="panel-body understanding-rate"> <div class="msg"></div> <label class="label-lg">Self-assessment</label><p>How much of <strong>Error and uncertainty</strong> have you understood?</p><div class="slider-container text-center"><div id="self-assessment-slider" class="sib-slider self-assessment " data-value="1" data-percentage=""></div></div> <label class="label-lg">My notes</label> <textarea name="page-notes" class="form-control" rows="3" placeholder="Write your notes here..."></textarea> </div> <div class="panel-footer text-xs-center"> <span id="last-edited" class="mb-xs-3"> </span> <div class="actions mt-xs-3"> <button id="save-my-progress" type="button" class="btn btn-sm btn-primary text-center btn-xs-block"> <i class="fa fa-fw fa-floppy-o"></i> Save </button> </div> </div></div> <div id="modal-feedback" class="modal fade" tabindex="-1" role="dialog"> <div class="modal-dialog" role="document"> <div class="modal-content"> <div class="modal-header"> <h4 class="modal-title">Feedback</h4> <button type="button" class="close hidden-xs hidden-sm" data-dismiss="modal" aria-label="Close"> <span aria-hidden="true">×</span> </button> </div> <div class="modal-body"> <div class="errors"></div> <p><strong>Which of the following best describes your feedback?</strong></p> <form method="post" style="overflow: hidden"> <div class="form-group"> <div class="radio"><label style="color: #121212;"><input type="radio" name="feedback-type" value="Recommendation"> Recommend</label></div><div class="radio"><label style="color: #121212;"><input type="radio" name="feedback-type" value="Problem"> Report a problem</label></div><div class="radio"><label style="color: #121212;"><input type="radio" name="feedback-type" value="Improvement"> Suggest an improvement</label></div><div class="radio"><label style="color: #121212;"><input type="radio" name="feedback-type" value="Other"> Other</label></div> </div> <hr> <div class="row"> <div class="col-md-6"> <div class="form-group"> <label for="feedback-name">Name</label> <input type="text" class="form-control" name="feedback-name" placeholder="Name" value=" "> </div> </div> <div class="col-md-6"> <div class="form-group"> <label for="feedback-email">Email address</label> <input type="email" class="form-control" name="feedback-email" placeholder="Email" value="@airmail.cc"> </div> </div> </div> <div class="form-group"> <label for="feedback-comments">Comments</label> <textarea class="form-control" name="feedback-comments" style="resize: vertical;"></textarea> </div> <input type="hidden" name="feedback-ticket" value="082b9c9c4ae3624d"> <input type="hidden" name="feedback-url" value="https://studyib.net/physics/page/672/error-and-uncertainty"> <input type="hidden" name="feedback-subject" value="6"> <input type="hidden" name="feedback-subject-name" value="Physics"> <div class="pull-left"> </div> </form> </div> <div class="modal-footer"> <button type="button" class="btn btn-primary btn-xs-block feedback-submit mb-xs-3 pull-right"> <i class="fa fa-send"></i> Send </button> <button type="button" class="btn btn-default btn-xs-block m-xs-0 pull-left" data-dismiss="modal"> Close </button> </div> </div> </div></div> <div id="fc-viewer" class="modal fade modal-flashcard" tabindex="-1" role="dialog"><div class="modal-dialog" role="document"><div class="modal-content"><div class="modal-header" style="background-color: #fafafa;"><div class="row" style="width: 100%;"><div class="col-md-12 tags-heading"><div style="display: flex; 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