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262 lines
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<title>Errors in Measurement</title>
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<div id="content" role="main"><!-- #BeginEditable "Body" -->
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<h1 class="center">Errors in Measurement</h1>
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<table border="0" align="center">
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<tr>
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<td><img src="images/questions.gif" width="107" height="76" alt="questions" /></td>
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<td> </td>
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<td> <b>Error? </b><br />
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No ... you didn't measure it <b>wrong</b> ... this is about <b>accuracy</b>.</td>
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</tr>
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</table>
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<p>Measuring instruments are not exact! </p>
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<h2>Degree of Accuracy</h2>
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<p><a href="../accuracy-precision.html">Accuracy</a> depends on the instrument you are measuring with. But as a general rule:</p>
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<div align="center">
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<div class="center80">The degree of accuracy is <b>half a unit</b> each side of the unit of measure </div>
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</div>
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<h3>Examples:</h3>
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<div class="simple">
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<table border="0" align="center">
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<tr align="center">
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<td>When your instrument measures in "1"s <br />
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then any value between <b>6½</b> and <b>7½</b> is measured as <b>"7"</b></td>
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<td><img src="../images/accuracy-1.svg" alt="accuracy 1: 6.5 to 7.5" /></td>
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</tr>
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<tr align="center">
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<td>When your instrument measures in "2"s <br />
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then any value between <b>7</b> and <b>9</b> is measured as <b>"8"</b></td>
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<td><img src="../images/accuracy-2.svg" alt="accuracy 2: 7 to 8" /></td>
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</tr>
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</table></div>
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<p>Notice that the arrow points to the same spot, but the measured values are different!</p>
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<h2>Plus or Minus</h2>
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<table border="0" align="center">
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<tr>
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<td>We can show the error using the "Plus or Minus" sign:</td>
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<td> </td>
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<td><span class="huge">±</span></td>
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</tr>
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</table>
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<div class="simple">
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<table border="0" align="center">
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<tr align="center">
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<td><p>When the value could be between <b>6½</b> and <b>7½</b>:</p>
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<p class="larger">7 ±0.5</p>
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<p>The error is ±0.5</p>
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</td>
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<td><img src="../images/accuracy-1.svg" alt="accuracy 1: 6.5 to 7.5" /></td>
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</tr>
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<tr align="center">
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<td><p>When the value could be between <b>7</b> and <b>9</b>:</p>
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<p class="larger">8 ±1</p>
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<p>The error is ±1</p>
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</td>
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<td><img src="../images/accuracy-2.svg" alt="accuracy 2: 7 to 8" /></td>
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</tr>
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</table>
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</div>
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<p> </p>
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<div class="example"><img style="float:right; margin: 10px;" src="../algebra/images/fence.jpg" width="150" height="112" alt="fence" />
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<h3>Example: a fence is measured as 12.5 meters long, accurate to 0.1 of a meter</h3>
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<p>Accurate to <b>0.1 m</b> means it could be up to <b>0.05 m</b> either way:</p>
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<p class="center larger">Length = 12.5 ±0.05 m</p>
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<p>So it could really be anywhere between 12.45 m and 12.55 m long. </p>
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</div>
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<p> </p>
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<h2>Absolute, Relative and Percentage Error</h2>
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<div class="dotpoint"> The <b>Absolute Error</b> is the difference between the <b>actual</b> and <b>measured</b> value.
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<p>But ... when measuring we don't know the actual value! So we use the maximum possible error.</p>
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<p>In the example above the Absolute Error is <span class="center larger">0.05 m</span></p>
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<p>What happened to the <span class="center larger">±</span> ... ? Well, we just want the size (the <a href="../numbers/absolute-value.html">absolute value</a>) of the difference.</p>
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</div>
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<div class="dotpoint"> The <b>Relative Error</b> is the Absolute Error divided by the actual measurement.
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<p>We don't know the actual measurement, so the best we can do is use the <b>measured</b> value:</p><p class="center larger"><span class="larger">Relative Error = </span><span class="intbl"><em><span class="larger">Absolute Error</span></em><strong><span class="larger">Measured Value</span></strong></span></p>
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</div>
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<div class="dotpoint"> The <b>Percentage Error</b> is the Relative Error shown as a percentage (see <a href="../numbers/percentage-error.html">Percentage Error</a>). </div>
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<p>Let us see them in an example: </p>
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<div class="example">
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<img style="float:right; margin: 10px;" src="../algebra/images/fence.jpg" width="150" height="112" alt="fence" />
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<h3>Example: fence (continued)</h3>
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<p class="center larger">Length = 12.5 ±0.05 m</p>
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<p>So:</p>
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<p class="center larger"> Absolute Error = 0.05 m</p>
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<p>And: </p><p class="center larger"><span class="larger">Relative Error = </span><span class="intbl"><em><span class="larger">0.05 m</span></em><strong><span class="larger">12.5 m</span></strong></span><span class="larger"> = 0.004</span></p>
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<p>And: </p>
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<p class="center larger">Percentage Error = 0.4% </p>
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</div>
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<p>More examples:</p>
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<div class="example"><img style="float:right; margin: 10px;" src="images/thermometer2.gif" width="100" height="122" alt="thermometer" />
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<h3>Example: The thermometer measures to the nearest 2 degrees. The temperature was measured as 38° C</h3>
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<p>The temperature could be up to 1° either side of 38° (i.e. between 37° and 39°)</p>
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<p> </p>
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<p class="center larger">Temperature = 38 ±1°</p>
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<p>So:</p>
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<p class="center larger"> Absolute Error = 1°</p>
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<p>And: </p>
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<p class="center larger"><span class="larger">Relative Error = </span><span class="intbl"><em><span class="larger">1°</span></em><strong><span class="larger">38°</span></strong></span><span class="larger"> = 0.0263...</span></p>
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<p>And: </p>
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<p class="center larger">Percentage Error = 2.63...% </p>
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</div>
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<div class="example">
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<h3>Example: You measure the plant to be 80 cm high (to the nearest cm)</h3>
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<p>This means you could be up to 0.5 cm wrong (the plant could be between 79.5 and 80.5 cm high)</p>
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<p> </p>
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<p class="center larger">Height = 80 ±0.5 cm</p>
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<p>So:</p>
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<p class="center larger"> Absolute Error = 0.5 cm</p>
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<p>And: </p>
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<p class="center larger"><span class="larger">Relative Error = </span><span class="intbl"><em><span class="larger">0.5 cm</span></em><strong><span class="larger">80 cm</span></strong></span><span class="larger"> = 0.00625</span></p>
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<p>And: </p>
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<p class="center larger">Percentage Error = 0.625% </p>
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</div>
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<p> </p>
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<h2>Area</h2>
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<p>When working out areas you need to think about both the <b>width and length</b> ... they could possibly both be the smallest measure or both the largest.</p>
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<div class="example">
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<h3>Example: Alex measured the field to the nearest meter, and got a width of 6 m and a length of 8 m.</h3>
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<p>Measuring to the nearest meter means the true value could be up to <b>half a meter</b> smaller or larger.</p>
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<p style="float:right; margin: 0 0 5px 10px;"><img src="../algebra/images/error-measure-area.gif" width="257" height="139" alt="area 6x8 measure error 41.25,48,55.25" /></p>
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<p>The width (w) could be from 5.5m to 6.5m:</p>
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<div class="so">5.5 ≤ w < 6.5 </div>
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<p>The length (l) could be from 7.5m to 8.5m:</p>
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<div class="so">7.5 ≤ l < 8.5</div>
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<p>The area is width × length:</p>
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<div class="so">A = w × l</div>
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<p>The smallest possible area is: 5.5m × 7.5m = <b>41.25 m<sup>2</sup></b><br />
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The measured area is: 6m × 8m = <b>48 m<sup>2</sup></b><br />
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And the largest possible area is: 6.5m × 8.5m = <b>55.25 m<sup>2</sup></b><br />
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</p>
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<div class="so"> 41.25 ≤ A < 55.25 </div>
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<p> </p>
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<h3>Absolute, Relative and Percentage Error</h3>
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<p>The only tricky thing here is ... <b>which</b> is the absolute error? </p>
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<ul>
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<li>From 41.25 to 48 = <b>6.75</b></li>
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<li>From 48 to 55.25 = <b>7.25</b></li>
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</ul>
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<p>Answer: pick the biggest one! So:</p>
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<p class="center larger"> Absolute Error = 7.25 m<sup>2</sup></p>
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<p class="center larger"><span class="larger">Relative Error = </span><span class="intbl"><em><span class="center larger">7.25 m<sup>2</sup></span></em><strong><span class="center larger">48 m<sup>2</sup></span></strong></span><span class="larger"> = 0.151...</span></p>
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<p class="center larger">Percentage Error = 15.1% </p>
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<p class="center">(Which is not very accurate, is it?)
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</p>
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</div> <h2>Volume</h2>
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<p>And volume has three measurements: <b>width, length and height</b>! </p>
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<p>Each measurement could possibly be the smallest possible measure, or the largest.</p>
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<div class="example">
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<h3>Example: Sam measured the box to the nearest 2 cm, and got 24 cm × 24 cm × 20 cm</h3>
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<p>Measuring to the nearest 2 cm means the true value could be up to <b>1 cm</b> smaller or larger.</p>
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<p>The three measurements are:</p>
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<ul>
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<li>24 ±1 cm</li>
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<li>24 ±1 cm</li>
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<li>20 ±1 cm</li>
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</ul>
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<p>Volume is width × length × height:</p>
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<div class="so">V = w × l × h</div>
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<p>The smallest possible Volume is: 23cm × 23cm × 19cm = <b>10051 cm<sup>3</sup></b><br />
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The measured Volume is: 24cm × 24cm × 20cm = <b>11520 cm<sup>3</sup></b><br />
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The largest possible Volume is: 25cm × 25cm × 21cm = <b>13125 cm<sup>3</sup></b></p>
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<p style="float:right; margin: 0 0 5px 10px;"><img src="images/volume-error.gif" width="183" height="102" alt="volume measure error 10051, 11520,13125" /></p>
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<p>And so we get:</p>
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<div class="so"> 10051 ≤ V < 13125 </div>
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<p> </p>
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<h3>Absolute, Relative and Percentage Error</h3>
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<p>Absolute error: </p>
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<ul>
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<li>From 10051 to 11520 = <b>1469</b></li>
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<li>From 11520 to 13125 = <b>1605</b></li>
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</ul>
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<p>Pick the biggest one:</p>
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<p class="center larger"> Absolute Error = 1605 cm<sup>3</sup></p>
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<p class="center larger"><span class="larger">Relative Error = </span><span class="intbl"><em><span class="center larger">1605 cm<sup>3</sup></span></em><strong><span class="center larger">11520 cm<sup>3</sup></span></strong></span><span class="larger"> = 0.139...</span></p>
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<p class="center larger">Percentage Error = 13.9% </p></div>
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<p> </p>
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<div class="questions">
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<script type="text/javascript">getQ(7031, 7036, 7157, 7160, 7037, 7041, 7044, 7047, 7052, 7053);</script>
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</div>
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<div class="related">
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<a href="../accuracy-precision.html">Accuracy and Precision</a>
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<a href="index.html">Measure Index</a>
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</div>
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Copyright © 2017 MathsIsFun.com
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