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class="page_title"> Uncertainties, errors and graphs <a href="#" class="mark-page-favorite pull-right" data-pid="908" title="Mark as favorite" onclick="return false;"><i class="fa fa-star-o"></i></a> </h1> <ol class="breadcrumb"> <li><a href="../../../chemistry.html"><i class="fa fa-home"></i></a><i class="fa fa-fw fa-chevron-right divider"></i></li><li><a href="../363/measurement.html">Measurement</a><i class="fa fa-fw fa-chevron-right divider"></i></li><li><span class="gray">Uncertainties, errors and graphs</span></li> <span class="pull-right" style="color: #555" title="Suggested study time: 60 minutes"><i class="fa fa-clock-o"></i> 60'</span> </ol> <article id="main-article"> <p><img alt="" src="../../images/measurement-and-data/precision-1.png" style="width: 160px; height: 175px; float: left;">The content that is here is mostly useful for internal assessment, and not assessed a great deal in examinations. <strong>So you do not have to learn all this for the exams!</strong> There is an extraordinary amount of vocabulary in this section of the course, but once you have got to grips with it all, the application of it is simpler than it at first appears. Use the flashcards to learn the terms and use the quiz to give you an idea of how the content is likely to be applied to questions.</p> <hr class="hidden-separator"> <div class="panel panel-has-colored-body panel-turquoise"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Key concepts</p> </div> </div> <div class="panel-body"> <div> <div class="panel-body"> <div> <p>Ensure you are confident using the terms below and learn the asterisked* definitions</p> <p>Accuracy, Precision, Error, Random error, Systematic error, Uncertainty, Absolute uncertainty, Percentage uncertainty, Dependent variable, Independent variable, Controlled variable, Extrapolation</p> <div class="tib-flashcard"><a class="show-flashcards btn btn-success btn-xs-block btn-block " data-levels="1" data-mode="Normal" data-topics="561,562" data-subject-id="7" data-n-flashcards="12" style="text-align:center">Show flashcards</a></div><hr> </div> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-yellow panel-has-colored-body"> <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> <p> The revision cards contain all of the essential content:</p> <div id="carousel-207" class="dynamic-gallery carousel slide" data-id="207"><div class="carousel-inner" role="listbox"><div class="item active"><a class="fancy" href="../../../std-galleries/7-207/screenshot-2020-08-10-at-170346.png" data-fancybox="gallery-207" title="" data-caption=""><img alt="" src="../../../std-galleries/7-207/screenshot-2020-08-10-at-170346.png"></a></div><div class="item "><a class="fancy" href="../../../std-galleries/7-207/screenshot-2020-08-10-at-170401.png" data-fancybox="gallery-207" title="" data-caption=""><img alt="" src="../../../std-galleries/7-207/screenshot-2020-08-10-at-170401.png"></a></div><div class="item "><a class="fancy" 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class="q-question"><p>Sets of darts in a dartboard can be used to illustrate accuracy and precision, where the bullseye is the true value:</p><table border="0" cellpadding="0" cellspacing="0" style="width: 468px;"><tbody><tr><td colspan="3" style="text-align: center;"><img alt="" src="../../images/00-qbank-images/darts-01.jpg" style="width: 468px; height: 154px;"></td></tr><tr><td style="text-align: center;"><strong>A</strong></td><td style="text-align: center;"><strong>B</strong></td><td style="text-align: center;"><strong>C</strong></td></tr></tbody></table><strong><span style="font-size:12.0pt;font-family:Calibri;
mso-ascii-theme-font:major-latin;mso-fareast-font-family:">Which set shows neither accuracy nor precision?</span></strong></div><div class="q-answer"><p><label class="radio"> <input type="radio"> <span>None</span></label> </p><p><label class="radio"> <input type="radio"> <span>B</span></label> </p><p><label class="radio"> <input class="c" type="radio"> <span>C</span></label> </p><p><label class="radio"> <input type="radio"> <span>A</span></label> </p></div><div class="q-explanation"><p>C shows data points that are not close together (precise) nor close to the true value (accurate).</p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">2</div><div class="exercise shadow-bottom"><div class="q-question"><p>Which uncertainty is the most appropriate for taking a 3.000g mass reading on a three decimal place balance?</p></div><div class="q-answer"><p><label class="radio"> <input type="radio"> <span>±0.01g</span></label> </p><p><label class="radio"> <input type="radio"> <span>±0.0001g</span></label> </p><p><label class="radio"> <input type="radio"> <span>±0.010g</span></label> </p><p><label class="radio"> <input class="c" type="radio"> <span>±0.001g</span></label> </p></div><div class="q-explanation"><p>It is conventional in chemistry to quote raw uncertainties to one significant figure, and to the same precision (decimal places) as the reading - thus ±0.001g is the correct answer. Although It is acceptable to quote uncertainties to half the smallest increment (±0.0005g) but that is not given as an option here.</p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">3</div><div class="exercise shadow-bottom"><div class="q-question"><p>The table displays a set of volume measurements for a titration where the burette tap is consistently closed one second too late:</p><table border="1" cellpadding="0" cellspacing="0"><tbody><tr><td style="width:174px;height:19px;"><p style="text-align: center;">Volume (±0.05 cm<sup>3</sup>)</p></td></tr><tr><td style="width:174px;height:22px;"><p style="text-align: center;">20.15</p></td></tr><tr><td style="width:174px;height:18px;"><p style="text-align: center;">20.10</p></td></tr><tr><td style="width:174px;height:18px;"><p style="text-align: center;">20.10</p></td></tr></tbody></table><p>The set of data is:</p></div><div class="q-answer"><p><label class="radio"> <input type="radio"> <span>Neither precise nor accurate</span></label> </p><p><label class="radio"> <input type="radio"> <span>Both precise and accurate</span></label> </p><p><label class="radio"> <input class="c" type="radio"> <span>Precise but inaccurate</span></label> </p><p><label class="radio"> <input type="radio"> <span>Imprecise but accurate</span></label> </p></div><div class="q-explanation"><p>Precision can be defined as data points being close together, which this data shows. However, the systematic error due to the tap consistently being closed too late (as mentioned in the question) renders the data inaccurate. (Accuracy is closeness to the true value.)</p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">4</div><div class="exercise shadow-bottom"><div class="q-question"><p>How many significant figures in the value 5.100?</p></div><div class="q-answer"><p><label class="radio"> <input class="c" type="radio"> <span>4</span></label> </p><p><label class="radio"> <input type="radio"> <span>2</span></label> </p><p><label class="radio"> <input type="radio"> <span>1</span></label> </p><p><label class="radio"> <input type="radio"> <span>3</span></label> </p></div><div class="q-explanation"><p>There are <strong>4 </strong>significant figures in this value. See the highlighted rule below.</p><p><strong>There are some simple rules to follow when considering significant digits/figures:</strong></p><p>Numeric (non-zero) digits are always significant.</p><p>Zero digits are always significant when between two numeric (non-zero) digits.</p><p>Zero digits in front of a number are never significant (they only indicate powers of 10).</p><p><strong>Zero digits that follow a number after a decimal place are always significant (by convention as they have been measured and found to be zero, which is why they are included), </strong>but</p><p>Zero digits that follow a number without a decimal place may be significant or may not be – depending on whether they have been measured and found to be zero, or are just there to indicate powers of 10.</p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">5</div><div class="exercise shadow-bottom"><div class="q-question"><p>How many significant figures in the value 1500?</p></div><div class="q-answer"><p><label class="radio"> <input type="radio"> 3 </label></p><p><label class="radio"> <input type="radio"> <span>2</span></label> </p><p><label class="radio"> <input type="radio"> <span>4</span></label> </p><p><label class="radio"> <input class="c" type="radio"> <span>We cannot know for sure; could be 2,3 or 4</span></label> </p></div><div class="q-explanation"><p><strong>We cannot know for sure; could be 2,3 or 4</strong> is the correct answer. The problem is that we do not know if the zero digits have been measured and found to be zero, or are 'place holders' for powers of 10. This is why we use standard form: 1.5×10<sup>3</sup> tells us that there are <strong>two </strong>sig figs; 1.500×10<sup>3</sup> tells us that there are <strong>four </strong>sig figs.</p><p><strong>There are some simple rules to follow when considering significant digits/figures:</strong></p><p>Numeric (non-zero) digits are always significant.</p><p>Zero digits are always significant when between two numeric (non-zero) digits.</p><p>Zero digits in front of a number are never significant (they only indicate powers of 10).</p><p>Zero digits that follow a number after a decimal place are always significant (by convention as they have been measured and found to be zero, which is why they are included)<strong>, </strong>but</p><p><strong>Zero digits that follow a number without a decimal place may be significant or may not be – depending on whether they have been measured and found to be zero, or are just there to indicate powers of 10.</strong></p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">6</div><div class="exercise shadow-bottom"><div class="q-question"><p>A student carries out an experiment to find heat energy, q, using the following data:</p><p>mass (m) is 5.442g; specific heat capacity (c) is 4.184 Jg<sup>–1</sup>K<sup>–1</sup>; initial temperature is 295K and final temperature is 307K</p><p>The calculation is as follows: q = mcΔT = 5.442 × 4.184 × 12 = 273.23194 J<span calibri="" style="font-size:12.0pt;font-family:"></span></p><p>Which final answer shows the appropriate number of significant figures?</p></div><div class="q-answer"><p><label class="radio"> <input class="c" type="radio"> <span>273 J</span></label> </p><p><label class="radio"> <input type="radio"> <span>273.23194 J</span></label> </p><p><label class="radio"> <input type="radio"> <span>273.2 J</span></label> </p><p><label class="radio"> <input type="radio"> <span>270 J</span></label> </p></div><div class="q-explanation"><p>The calculator value would be used if further calculations were going to be made.</p><p>However, as a final answer, the value should be given to 3 significant figures as this is the lowest number of significant figures in raw data measurements (of temperature).</p><p>Thus the correct answer is<strong> 273 J</strong>.</p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">7</div><div class="exercise shadow-bottom"><div class="q-question"><p>What is the percentage uncertainty of a temperature reading of 16.0±0.4°C?</p></div><div class="q-answer"><p><label class="radio"> <input class="c" type="radio"> <span>2.5 %</span></label> </p><p><label class="radio"> <input type="radio"> <span>25 %</span></label> </p><p><label class="radio"> <input type="radio"> <span>0.025 %</span></label> </p><p><label class="radio"> <input type="radio"> 2.5×10<sup>–2</sup> %<span new="" roman="" style="font-size:12.0pt;font-family:" times=""></span></label> </p></div><div class="q-explanation"><p>Percentage uncertainty can be calculated:</p><p style="text-align: center;">Percentage uncertainty = (absolute uncertainty / measurement value) ×100%</p><p>In this example % uncertainty = 0.4 / 16.0 ×100% = <strong>2.5 %</strong></p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">8</div><div class="exercise shadow-bottom"><div class="q-question"><p>A student carries out a titration experiment. If the initial titration reading is equal to 1.60±0.05cm<sup>3</sup> and the final titration reading is equal to 16.25±0.05cm<sup>3</sup>, what is the total titre with correct uncertainty?</p></div><div class="q-answer"><p><label class="radio"> <input type="radio"> <span>17.85±0.10cm<sup>3</sup></span></label> </p><p><label class="radio"> <input class="c" type="radio"> <span>14.65±0.10cm<sup>3</sup></span></label> </p><p><label class="radio"> <input type="radio"> <span>14.65±0.05cm<sup>3</sup></span></label> </p><p><label class="radio"> <input type="radio"> <span>17.85±0.05cm<sup>3</sup></span></label> </p></div><div class="q-explanation"><p>To calculate the total titre, the initial reading should be subtracted from the final reading: 16.25−1.60=14.65</p><p>The two uncertainties should be<strong> added together</strong> to find the total uncertainty: 0.05+0.05=0.10</p><p>Thus the correct answer is <strong>14.65±0.10cm<sup>3</sup></strong></p><p><strong>Processing uncertainties</strong></p><p><u>Averaging values:</u> You may record the uncertainty of the mean (average) as the <strong>absolute uncertainty </strong>of a single reading. (Uncertainties for mean values can be processed to give a decreasing uncertainty value with an increasing number of measurements, but there is no requirement to do this at your level of study.)</p><p><u>Adding or subtracting values:</u> You should <strong>add </strong>the <strong>absolute uncertainties </strong>together. E.g. a total titration reading consists of an initial and a final titration reading. If the uncertainty in the individual readings is ±0.05cm<sup>3</sup>, then the uncertainty in the total titre will be 0.05+0.05=0.10. Notice that this <em>processed uncertainty</em> has more than one significant figure. When processing, do not round your uncertainties until you reach the end of your calculation – your final answer.</p><p><u>Multiplying or dividing values</u>: You should <strong>add </strong>the <strong>percentage uncertainties</strong>.</p><p><u>Raising a value to a particular power</u>: You should <strong>multiply </strong>the <strong>percentage uncertainty</strong> by that particular power.</p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">9</div><div class="exercise shadow-bottom"><div class="q-question"><p>A student records their final answer for a rate experiment calculation as follows:</p><p>Rate of reaction = 0.56±0.009 mol dm<sup>−3</sup> min<sup>−1</sup></p><p>Assuming the values are correct, but that student has not rounded appropriately; how should the student have recorded their final answer?</p></div><div class="q-answer"><p><label class="radio"> <input type="radio"> <span>0.6±0.1 mol dm<sup>−3</sup> min<sup>−1</sup></span></label> </p><p><label class="radio"> <input type="radio"> <span>0.56±0.009 mol dm<sup>−3</sup> min<sup>−1</sup> (no change is needed)</span></label> </p><p><label class="radio"> <input type="radio"> <span>0.560±0.009 mol dm<sup>−3</sup> min<sup>−1</sup></span></label> </p><p><label class="radio"> <input class="c" type="radio"> <span>0.56±0.01 mol dm<sup>−3</sup> min<sup>−1</sup></span></label> </p></div><div class="q-explanation"><p>The student has recorded their final answer with a different number of decimal places in the answer and the uncertainty. The<em> </em><strong>final uncertainty </strong>in any calculation or experiment should never contain more or fewer decimal places than the decimal places in the final answer: The decimal places in both should be the same.</p><p>Here the uncertainty contains more decimal places, so should be rounded down to two decimal places (the same as the value). The correct answer is <strong>0.56±0.01 mol dm<sup>−3</sup></strong> <strong>min<sup>−1</sup></strong></p><p>Note: It is not correct to add zero to the value (0.560±0.009) as this suggests that the zero has been measured and found to be zero (which it has not - it has just been added).</p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">10</div><div class="exercise shadow-bottom"><div class="q-question"><p>Has the following data set been affected by random error and/or systematic error?<span calibri="" style="font-family:
"></span></p><p>The repeated calculation from titration data of the concentration of a limewater solution at 20°C giving values of 0.042; 0.052; 0.033; 0.041; 0.039 ±0.002mol dm<sup>–3</sup>, when the true value is 0.023 mol dm<sup>–3</sup></p></div><div class="q-answer"><p><label class="radio"> <input type="radio"> <span>Neither random nor systematic error</span></label> </p><p><label class="radio"> <input type="radio"> <span>Random error</span></label> </p><p><label class="radio"> <input class="c" type="radio"> <span>Random and systematic error</span></label> </p><p><label class="radio"> <input type="radio"> <span>Systematic error</span></label> </p></div><div class="q-explanation"><p>The data set appears to be distributed (very) randomly about a value that is higher than the true value, so is likely to have been affected by random error and also by systematic error (perhaps an inappropriately chosen indicator).</p><p>All measurements are affected by random error and may also be affected by systematic error.</p><p><u>Random errors</u> vary by chance and do not apply equally to any given set of the same repeated measurements. Random errors are responsible for the uncertainty in instrument readings.</p><p><u>Systematic errors</u> do not vary by chance and often apply equally (or proportionally) to any given set of the same repeated measurements.</p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="totals"><span class="score"></span><button class="btn btn-success btn-block text-center check-total"><i class="fa fa-check-square-o"></i> Check</button></div></div><hr> </div> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-default"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Exam-style questions</p> </div> </div> <div class="panel-body"> <div> <h4>Paper 1</h4> <h5>Core (SL&HL): <a href="../2775/measurement-core-sl-and-hl-paper-1-questions.html" title="Measurement core (SL and HL) paper 1 questions">Measurement core (SL and HL) paper 1 questions</a></h5> <h5>AHL (HL only): <a href="../2776/measurement-ahl-hl-only-paper-1-questions.html" title="Measurement AHL (HL only) paper 1 questions">Measurement AHL (HL only) paper 1 questions</a></h5> <h4>Paper 2</h4> <h5>Core (SL&HL): <a href="../2777/measurement-core-sl-hl-paper-2-questions.html" title="Measurement core (SL & HL) paper 2 questions">Measurement core (SL & HL) paper 2 questions</a></h5> <h5>AHL (HL only): <a href="../2778/measurement-ahl-hl-only-paper-2-questions.html" title="Measurement AHL (HL only) paper 2 questions">Measurement AHL (HL only) paper 2 questions</a></h5> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="page-container panel-self-assessment" data-id="908"> <div class="panel-heading">MY PROGRESS</div> <div class="panel-body understanding-rate"> <div 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