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742 lines
23 KiB
HTML
742 lines
23 KiB
HTML
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<meta http-equiv="content-type" content="text/html; charset=utf-8">
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<!-- #BeginEditable "doctitle" -->
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<title>Activity: Lengths of Leaves</title>
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<script>reSpell=[["millimeter","millimetre"],["millimeters","millimetres"],["neighbor","neighbour"]];</script>
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<article id="content" role="main">
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<!-- #BeginEditable "Body" -->
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<h1 class="center">Activity: Lengths of Leaves</h1>
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<p>In
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this Activity we will
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investigate the lengths of leaves and work out the <a href="../mean.html">mean</a> (average)
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length.</p>
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<p>You are also going to learn a method to find an estimate of the mean
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length.</p>
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<p><strong>The first thing you need to do is to find some leaves ...</strong></p>
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<h2>What kind of leaves?</h2>
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<p>You
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can use leaves from any tree or plant; but if it's a tree, then choose
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one that is easy to reach (we don't want any accidents), and don't take
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your neighbor's leaves without asking! Also it's best to use simple
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leaves. Don't try to use leaves with strange shapes. The kind of leaves
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you see in the photograph below are ideal.</p>
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<div class="center80">
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<h4>Rules</h4>
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<p>Some of the rules you might
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want to consider are:</p>
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<ul>
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<li>I
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must choose all my leaves from the
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same tree or plant.</li>
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<li>I must make sure the
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leaves
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are whole - broken leaves just won't do.</li>
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<li>I must try to get a <a href="../definitions/random-sample.html">random</a> sample of leaves - not all big ones and not all small ones.</li>
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</ul>
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</div>
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<p> </p>
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<p><strong>The
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following photograph shows four
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leaves of different sizes that I collected from a tree in my garden:</strong></p>
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<p class="center"><img alt="leaves" src="images/leaves1.jpg" style="max-width:100%"></p>
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<p>You will see that some leaves are small and some are larger.<br>
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The next thing you need to decide is how many leaves to pick and how to
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get a random sample.</p>
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<h2>How
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many?</h2>
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<p><br>
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You
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could pick as many as you like. Ideally you
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would use all the leaves on the tree, but that would be rather a large
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number to measure, and your parents might not be very
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happy if you stripped the tree bare!</p>
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<p><strong>So let's try 100 leaves.</strong></p>
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<h2>How
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to get a random sample?<br>
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</h2>
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<div class="example">
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<p><strong>Here is one way you could get a <a href="../definitions/random-sample.html">random</a> sample:</strong></p>
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<ul>
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<li>Try
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taking all the leaves on one branch of the
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tree. After you've discarded any that are broken, then see how many
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there are. If there are less than 100, then get the leaves from another
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branch until you have more than 100. Then write a different number
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(starting with 1) on each leaf
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with a marking pen.</li>
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<li>Write
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the same numbers on small pieces of paper.</li>
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<li>Fold the pieces of paper.</li>
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<li>Put them into a hat and mix them up.</li>
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<li>Draw
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100 pieces of paper (without looking) from the hat. The first 100 pieces
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of paper you draw would
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give you the numbers of the leaves to use in the experiment.</li>
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</ul>
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</div>
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The
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next thing you need to decide is how to measure the leaves, and<br>
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how
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accurate your measurements should be:
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<h2>How
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to measure?</h2>
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<p>Keeping your leaves flat, use
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a ruler to measure the length of each leaf from the pointy part at one
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end of the leaf to the point where the leaf joins the stalk at the
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other end. Maybe your leaves bend a bit, but don't follow the main rib
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of the leaf as this would make measurement too difficult.</p>
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<p><strong>Just measure in a straight
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line as shown in the
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following diagram:</strong></p>
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<p class="center"><img alt="leaf" src="images/leaves2.jpg"></p>
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<h2>How accurate?</h2>
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<h3>You should measure the length
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of each leaf to the nearest millimeter.
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<strong><br>
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</strong></h3>
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<p><strong>Now you're ready to begin.</strong>
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<br>
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Measure the length of each leaf to the nearest millimeter and record
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you results in a table, as follows:</p>
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<p>(5 columns of 20 each equals 100 measurements)</p>
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<table align="center" cellspacing="2" cellpadding="2" border="1">
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<tbody>
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<tr>
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<td align="center"> </td>
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<td align="center" width="80"><strong>Length<br>
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(mm)</strong></td>
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<td align="center" width="80"><strong>Length<br>
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(mm)</strong></td>
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<td align="center" width="80"><strong>Length<br>
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(mm)</strong></td>
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<td align="center" width="80"><strong>Length<br>
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(mm)</strong></td>
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<td align="center" width="80"><strong>Length<br>
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(mm)</strong></td>
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</tr>
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<tr>
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<td height="28">1</td>
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<td><br></td>
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<td><br></td>
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<td><br></td>
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<td><br></td>
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<td><br></td>
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</tr>
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<tr>
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<td height="28">2</td>
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<td><br></td>
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<td><br></td>
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<td><br></td>
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<td><br></td>
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<td><br></td>
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</tr>
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<tr>
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<td height="28">3</td>
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<td><br></td>
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<td><br></td>
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<td><br></td>
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<td><br></td>
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<td><br></td>
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</tr>
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<tr>
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<td height="28">4</td>
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<td><br></td>
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<td><br></td>
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<td><br></td>
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<td><br></td>
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<td><br></td>
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</tr>
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<tr>
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<td height="28">5</td>
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<td><br></td>
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<td><br></td>
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<td><br></td>
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<td><br></td>
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<td><br></td>
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</tr>
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<tr>
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<td height="28">6</td>
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<td><br></td>
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<td><br></td>
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<td><br></td>
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<td><br></td>
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<td><br></td>
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</tr>
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<tr>
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<td height="28">7</td>
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<td><br></td>
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<td><br></td>
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<td><br></td>
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<td><br></td>
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<td><br></td>
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</tr>
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<tr>
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<td height="28">8</td>
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<td><br></td>
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<td><br></td>
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<td><br></td>
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<td><br></td>
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<td><br></td>
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</tr>
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<tr>
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<td height="28">9</td>
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<td><br></td>
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<td><br></td>
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<td><br></td>
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<td><br></td>
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<td><br></td>
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</tr>
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<tr>
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<td height="28">10</td>
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<td><br></td>
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<td><br></td>
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<td><br></td>
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<td><br></td>
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<td><br></td>
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</tr>
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<tr>
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<td height="28">11</td>
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<td><br></td>
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<td><br></td>
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<td><br></td>
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<td><br></td>
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<td><br></td>
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</tr>
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<tr>
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<td height="28">12</td>
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<td><br></td>
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<td><br></td>
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<td><br></td>
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<td><br></td>
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<td><br></td>
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</tr>
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<tr>
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<td height="28">13</td>
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<td><br></td>
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<td><br></td>
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<td><br></td>
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<td><br></td>
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<td><br></td>
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</tr>
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<tr>
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<td height="28">14</td>
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<td><br></td>
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<td><br></td>
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<td><br></td>
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<td><br></td>
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<td><br></td>
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</tr>
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<tr>
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<td height="28">15</td>
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<td><br></td>
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<td><br></td>
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<td><br></td>
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<td><br></td>
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<td><br></td>
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</tr>
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<tr>
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<td height="28">16</td>
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<td><br></td>
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<td><br></td>
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<td><br></td>
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<td><br></td>
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<td><br></td>
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</tr>
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<tr>
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<td height="28">17</td>
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<td><br></td>
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<td><br></td>
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<td><br></td>
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<td><br></td>
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<td><br></td>
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</tr>
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<tr>
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<td height="28">18</td>
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<td><br></td>
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<td><br></td>
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<td><br></td>
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<td><br></td>
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<td><br></td>
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</tr>
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<tr>
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<td height="28">19</td>
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<td><br></td>
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<td><br></td>
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<td><br></td>
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<td><br></td>
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<td><br></td>
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</tr>
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<tr>
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<td height="28">20</td>
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<td><br></td>
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<td><br></td>
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<td><br></td>
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<td><br></td>
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<td><br></td>
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</tr>
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<tr>
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<td height="28">Sums:</td>
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<td> </td>
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<td> </td>
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<td> </td>
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<td> </td>
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<td> </td>
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</tr>
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</tbody>
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</table>
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<p>Add up each column, then add those sums together for the grand total:</p>
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<p class="large center">_______ + _______ + _______ + _______ + _______ = _________</p>
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<p> </p>
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<table align="center" cellspacing="2" cellpadding="2" border="1">
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<tbody>
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<tr>
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<td height="30">Grand Total:</td>
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<td width="100"> </td>
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</tr>
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</tbody>
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</table>
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<h2>Finding the Mean</h2>
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<p>Now
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you should be able to calculate the <a href="../mean.html">mean</a> length of your leaves. Simply divide the sum of lengths by
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<strong>100</strong>
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.</p>
|
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<table align="center" cellspacing="2" cellpadding="2" border="1">
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<tbody>
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<tr>
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<td height="30">Mean (= Grand Total / 100):</td>
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<td width="100"> </td>
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</tr>
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</tbody>
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</table>
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<p> </p>
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<h2>Is
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There Another Way?</h2>
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<p>There is a
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way to <b>estimate</b> the
|
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mean length by grouping
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your results.</p>
|
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<p>This makes the calculation quicker, but is not as accurate.</p>
|
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<h3>Example</h3>
|
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<p>I measured the lengths of
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<strong>100</strong>
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leaves each to the nearest millimeter and grouped the results into
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groups of size
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<strong>10</strong>
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millimeters using a <a href="../data/tally-marks.html">tally</a> and constructed a <a href="../data/frequency-distribution.html">frequency</a> table,
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as follows:</p>
|
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<table align="center" cellspacing="2" cellpadding="2" border="1">
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<tbody>
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<tr>
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<td><strong>Length<br>
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(mm)</strong></td>
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<td><strong>Tally</strong></td>
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<td align="right"><strong>Number <br>
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of
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leaves<br>
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</strong>
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<strong>(frequency)</strong></td>
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</tr>
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<tr>
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<td>20
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- 29</td>
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<td><img src="../data/images/tally3.gif" alt="tally 3" height="20" width="22"></td>
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<td align="right">3</td>
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</tr>
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<tr>
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<td>30
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- 39</td>
|
|
<td><img src="../data/images/tally5.gif" alt="tally 5" height="20" width="30"><img src="../data/images/tally3.gif" alt="tally 3" height="20" width="22"></td>
|
|
<td align="right">8</td>
|
|
</tr>
|
|
<tr>
|
|
<td>40
|
|
- 49</td>
|
|
<td><img src="../data/images/tally5.gif" alt="tally 5" height="20" width="30"><img src="../data/images/tally5.gif" alt="tally 5" height="20" width="30"><img src="../data/images/tally5.gif" alt="tally 5" height="20" width="30"></td>
|
|
<td align="right">15</td>
|
|
</tr>
|
|
<tr>
|
|
<td>50
|
|
- 59</td>
|
|
<td><img src="../data/images/tally5.gif" alt="tally 5" height="20" width="30"><img src="../data/images/tally5.gif" alt="tally 5" height="20" width="30"><img src="../data/images/tally5.gif" alt="tally 5" height="20" width="30"><img src="../data/images/tally5.gif" alt="tally 5" height="20" width="30"><img src="../data/images/tally5.gif" alt="tally 5" height="20" width="30"><img src="../data/images/tally1.gif" alt="tally 1" height="20" width="9"></td>
|
|
<td align="right">26</td>
|
|
</tr>
|
|
<tr>
|
|
<td>60
|
|
- 69</td>
|
|
<td><img src="../data/images/tally5.gif" alt="tally 5" height="20" width="30"><img src="../data/images/tally5.gif" alt="tally 5" height="20" width="30"><img src="../data/images/tally5.gif" alt="tally 5" height="20" width="30"><img src="../data/images/tally5.gif" alt="tally 5" height="20" width="30"><img src="../data/images/tally3.gif" alt="tally 3" height="20" width="22"></td>
|
|
<td align="right">23</td>
|
|
</tr>
|
|
<tr>
|
|
<td>70
|
|
- 79</td>
|
|
<td><img src="../data/images/tally5.gif" alt="tally 5" height="20" width="30"><img src="../data/images/tally5.gif" alt="tally 5" height="20" width="30"><img src="../data/images/tally5.gif" alt="tally 5" height="20" width="30"><img src="../data/images/tally1.gif" alt="tally 1" height="20" width="9"></td>
|
|
<td align="right">16</td>
|
|
</tr>
|
|
<tr>
|
|
<td>80
|
|
- 89</td>
|
|
<td><img src="../data/images/tally5.gif" alt="tally 5" height="20" width="30"><img src="../data/images/tally2.gif" alt="tally 2" height="20" width="14"></td>
|
|
<td align="right">7</td>
|
|
</tr>
|
|
<tr>
|
|
<td>90
|
|
- 99</td>
|
|
<td><img src="../data/images/tally2.gif" alt="tally 2" height="20" width="14"></td>
|
|
<td align="right">2</td>
|
|
</tr>
|
|
<tr>
|
|
<td>Total</td>
|
|
<td><br></td>
|
|
<td align="right">100</td>
|
|
</tr>
|
|
</tbody>
|
|
</table>
|
|
<div class="center80">
|
|
<p><strong>If
|
|
this is not clear, then let me explain:</strong></p>
|
|
<p>The three shortest leaves in my sample have lengths
|
|
<strong>22 mm</strong>,
|
|
<strong>25
|
|
mm</strong>
|
|
and
|
|
<strong>27 mm</strong>. These are all between
|
|
<strong>20 mm</strong>
|
|
and
|
|
<strong>29 mm</strong>,
|
|
so each of them adds a tally mark for the group
|
|
<strong>20 -
|
|
29</strong>.</p>
|
|
<p>So there are three tally marks altogether (<img src="../data/images/tally3.gif" alt="tally 3" height="20" width="22">) in the group
|
|
<strong>20 - 29</strong>, and the <a href="../data/frequency-distribution.html">frequency</a> for this group is
|
|
<strong>3</strong>
|
|
(i.e. there are 3 leaves in this group).</p>
|
|
<p>Similarly, there are eight leaves whose lengths are between
|
|
<strong>30 mm</strong>
|
|
and
|
|
<strong>39
|
|
mm</strong>, so there are
|
|
<strong>8</strong>
|
|
tally marks for this group
|
|
and the frequency is
|
|
<strong>8</strong>.</p>
|
|
<p>And
|
|
so on.</p>
|
|
</div>
|
|
<h2>Once you've grouped your lengths, how can you estimate the mean?</h2>
|
|
<p>What we do is we assume all leaves in a group have the same length,
|
|
which is the average for that group. This value is called the midpoint
|
|
of the group and is simply found by taking the average (<a href="../mean.html">mean</a>) of the
|
|
smallest and greatest lengths in that group.</p>
|
|
<div class="center80">
|
|
<p><strong>For example:</strong></p>
|
|
<p>The midpoint for the group
|
|
<strong>20 - 29</strong>
|
|
is
|
|
<strong>(20 +29)/2 = 49/2 = 24.5</strong></p>
|
|
<p>The midpoint for the
|
|
group
|
|
<strong>30 - 39</strong>
|
|
is
|
|
<strong>(30 +39)/2 = 69/2 = 34.5</strong></p>
|
|
<p>etc</p>
|
|
</div>
|
|
<p>So, returning to my example, I am going to assume that there are 3
|
|
leaves with a mean length of
|
|
<strong>24.5 mm</strong>,
|
|
<strong>8</strong>
|
|
leaves with a mean length of
|
|
<strong>34.5 mm</strong>,
|
|
<strong>15</strong>
|
|
leaves with a mean length of
|
|
<strong>44.5 mm</strong>
|
|
, etc. I can then work
|
|
out my estimate of the mean length from a <a href="../data/mean-frequency-table.html">frequency table</a>, as follows:</p>
|
|
|
|
<table align="center" cellspacing="2" cellpadding="2" border="1">
|
|
<tbody>
|
|
<tr>
|
|
<td align="center" width="70"><strong>Length<br>
|
|
(mm)</strong></td>
|
|
<td align="center" width="70"><strong>Midpoint<br>
|
|
x</strong></td>
|
|
<td align="center" width="70"><strong>Frequency<br>
|
|
f</strong></td>
|
|
<td align="center"><strong> f
|
|
× x</strong></td>
|
|
</tr>
|
|
<tr>
|
|
<td align="center">20
|
|
- 29</td>
|
|
<td align="center">24.5</td>
|
|
<td align="center">3</td>
|
|
<td align="right">73.5</td>
|
|
</tr>
|
|
<tr>
|
|
<td align="center">30
|
|
- 39</td>
|
|
<td align="center">34.5</td>
|
|
<td align="center">8</td>
|
|
<td align="right">276.0</td>
|
|
</tr>
|
|
<tr>
|
|
<td align="center">40
|
|
- 49</td>
|
|
<td align="center">44.5</td>
|
|
<td align="center">15</td>
|
|
<td align="right">667.5</td>
|
|
</tr>
|
|
<tr>
|
|
<td align="center">50
|
|
- 59</td>
|
|
<td align="center">54.5</td>
|
|
<td align="center">26</td>
|
|
<td align="right">1,417.0</td>
|
|
</tr>
|
|
<tr>
|
|
<td align="center">60
|
|
- 69</td>
|
|
<td align="center">64.5</td>
|
|
<td align="center">23</td>
|
|
<td align="right">1,483.5</td>
|
|
</tr>
|
|
<tr>
|
|
<td align="center">70
|
|
- 79</td>
|
|
<td align="center">74.5</td>
|
|
<td align="center">16</td>
|
|
<td align="right">1,192.0</td>
|
|
</tr>
|
|
<tr>
|
|
<td align="center">80
|
|
- 89</td>
|
|
<td align="center">84.5</td>
|
|
<td align="center">7</td>
|
|
<td align="right">591.5</td>
|
|
</tr>
|
|
<tr>
|
|
<td align="center">90
|
|
- 99</td>
|
|
<td align="center">94.5</td>
|
|
<td align="center">2</td>
|
|
<td align="right">189.0</td>
|
|
</tr>
|
|
<tr>
|
|
<td><br></td>
|
|
<td align="center"><br></td>
|
|
<td align="center">Σf
|
|
= 100</td>
|
|
<td align="right">Σ(f
|
|
× x) = 5,890</td>
|
|
</tr>
|
|
</tbody>
|
|
</table>
|
|
<p>So the estimate of the mean length =
|
|
<strong>Σ(f ×
|
|
x)/Σf = 5,890/100 = 58.9 mm</strong></p>
|
|
<p>This value differed a little from the exact value of the mean that I
|
|
obtained earlier. It's important to understand that this is just an
|
|
estimate, but it can be much
|
|
quicker to calculate.</p>
|
|
<h2>Your Turn</h2>
|
|
<h3>Try using the method for your sample and see how good an estimate you
|
|
get.</h3>
|
|
<p>You can use this table. First decide which <b>length groups</b> to use. You may need fewer groups
|
|
or more
|
|
groups than we did. For example, if you have a leaf of length 105 mm, then you will
|
|
need an extra group 100 - 109. Or you might not have
|
|
any leaves as short as mine, so you may not need the group 20 -
|
|
29. Choose the right groups for your results.</p>
|
|
<table align="center" cellspacing="2" cellpadding="2" border="1">
|
|
<tbody>
|
|
<tr>
|
|
<td align="center" width="80"><strong>Length<br>
|
|
(mm)</strong></td>
|
|
<td align="center" width="80"><strong>Midpoint<br>
|
|
x</strong></td>
|
|
<td width="180"><strong>Tally</strong></td>
|
|
<td align="center" width="80"><strong>Number <br>
|
|
of
|
|
leaves<br>
|
|
</strong>
|
|
<strong>f</strong></td>
|
|
<td align="center" width="80"><strong> f
|
|
× x</strong></td>
|
|
</tr>
|
|
<tr>
|
|
<td height="30"> </td>
|
|
<td align="right"> </td>
|
|
<td> </td>
|
|
<td align="right"> </td>
|
|
<td align="right"> </td>
|
|
</tr>
|
|
<tr>
|
|
<td height="30"> </td>
|
|
<td align="right"> </td>
|
|
<td> </td>
|
|
<td align="right"> </td>
|
|
<td align="right"> </td>
|
|
</tr>
|
|
<tr>
|
|
<td height="30"> </td>
|
|
<td align="right"> </td>
|
|
<td> </td>
|
|
<td align="right"> </td>
|
|
<td align="right"> </td>
|
|
</tr>
|
|
<tr>
|
|
<td height="30"> </td>
|
|
<td align="right"> </td>
|
|
<td> </td>
|
|
<td align="right"> </td>
|
|
<td align="right"> </td>
|
|
</tr>
|
|
<tr>
|
|
<td height="30"> </td>
|
|
<td align="right"> </td>
|
|
<td> </td>
|
|
<td align="right"> </td>
|
|
<td align="right"> </td>
|
|
</tr>
|
|
<tr>
|
|
<td height="30"> </td>
|
|
<td align="right"> </td>
|
|
<td> </td>
|
|
<td align="right"> </td>
|
|
<td align="right"> </td>
|
|
</tr>
|
|
<tr>
|
|
<td height="30"> </td>
|
|
<td align="right"> </td>
|
|
<td> </td>
|
|
<td align="right"> </td>
|
|
<td align="right"> </td>
|
|
</tr>
|
|
<tr>
|
|
<td height="30"> </td>
|
|
<td align="right"> </td>
|
|
<td> </td>
|
|
<td align="right"> </td>
|
|
<td align="right"> </td>
|
|
</tr>
|
|
<tr>
|
|
<td height="30"> </td>
|
|
<td align="right"> </td>
|
|
<td> </td>
|
|
<td align="right"> </td>
|
|
<td align="right"> </td>
|
|
</tr>
|
|
<tr>
|
|
<td height="30"> </td>
|
|
<td align="right"> </td>
|
|
<td> </td>
|
|
<td align="right"> </td>
|
|
<td align="right"> </td>
|
|
</tr>
|
|
<tr>
|
|
<td height="30"> </td>
|
|
<td align="right"> </td>
|
|
<td> </td>
|
|
<td align="right"> </td>
|
|
<td align="right"> </td>
|
|
</tr>
|
|
<tr>
|
|
<td height="50">Total</td>
|
|
<td align="right"> </td>
|
|
<td><br></td>
|
|
<td align="center" valign="top">Σf
|
|
= <br>
|
|
<b>100</b></td>
|
|
<td align="left" valign="top">Σ(f
|
|
× x) = <br>
|
|
</td>
|
|
</tr>
|
|
</tbody>
|
|
</table>
|
|
<p>Divide Σ(f
|
|
× x) by Σf to get the mean length:</p>
|
|
<p class="large center">Σ(f
|
|
× x) / Σ(f) = ________ / 100 = _________</p>
|
|
<h2>Why use an Estimate?</h2>
|
|
<p>Faster! Easier! (But less accurate.)</p>
|
|
<p>It might not have made much difference in our example, with only 100
|
|
leaves, but it could be a useful method to use when the number in a
|
|
sample is much much larger.</p>
|
|
<p> </p>
|
|
<div class="related">
|
|
<a href="../mean.html">Mean</a>
|
|
<a href="index.html">Activity Index</a>
|
|
</div>
|
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