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    <name>
      <text>36.2.1.3 (not used for Edinburgh) The steam-and-leaf diagram</text>
    </name>
    <questiontext format="html">
      <text><![CDATA[<h4 style="text-align: left;">The steam-and-leaf diagram</h4>
<p>You have already seen a basic stem-and-leaf diagram and you know that it shows the shape of distributions well. Here you will learn how to handle larger amounts of data to form stem-and-leaf diagrams. As you will be able to verify, one set of data can give rise to more than one stem-and-leaf diagram and highlight different aspects of the data. Look at the data set below:</p>
<p>\(11\)&nbsp; \(9\)&nbsp; \(6\)&nbsp; \(27\)&nbsp; \(17\)&nbsp; \(2\)&nbsp; \(19\)&nbsp; \(12\)&nbsp; \(8\)&nbsp; \(17\)&nbsp; \(3\)&nbsp; \(10\)&nbsp; \(23\)&nbsp; \(6\)&nbsp; \(18\)</p>
<p>\(13\)&nbsp; \(11\)&nbsp; \(22\)&nbsp; \(13\)&nbsp; \(19\)&nbsp; \(4\)&nbsp; \(12\)&nbsp; \(23\)&nbsp; \(34\)&nbsp; \(19\)&nbsp; \(15\)&nbsp; \(7\)&nbsp; \(40\)&nbsp; \(16\)&nbsp; \(20\).</p>
<p>Using the numbers to the left of the stem to represent \(10\)s and the numbers to the right to represent units, we obtain the stem-and-leaf diagram shown below:</p><p><br></p><p style="text-align: center;"><img src="@@PLUGINFILE@@/36.2.5%20stem%20and%20leaf%20diagram%202.svg" alt="stem and leaf diagram of the data set (11, 9, 6, 27, 17, 2, 19, 12, 8, 17, 3, 10, 23, 6, 18, 13, 11, 22, 13, 19, 4, 12, 23, 34, 19, 15, 7, 40, 16, 20)" role="presentation" class="atto_image_button_text-bottom" width="300" height="90"></p><p style="text-align: left;"><br></p>

<p>Notice that the skewed nature of the data stands out immediately. What also stands out are the following:</p>
<ul>
<li>the \(10\)s class has the highest number of members;</li>
<li>the modal (most frequently occurring) value is \(19\);</li>
<li>the \(30\)s and \(40\)s tie for the least number of members (one each).</li>
</ul><br>
<p>This is not new information, we could have written these down after properly inspecting the original raw data. The advantage of the stem-and-leaf diagram is that it enables to express these properties in a clear and obvious way. As a further illustrative example, look at the idea in the table below which we will use to draw two stem-and-leaf diagrams.</p><p><br>
</p><table>
<caption></caption>
<thead>
<tr>
<th style="text-align: right;" scope="col"><span style="font-weight: normal;">\(9.5\) &nbsp;&nbsp; &nbsp; &nbsp; </span><br></th>
<th style="text-align: right;" scope="col"><span style="font-weight: normal;">\(11.9\)&nbsp; &nbsp;&nbsp; </span><br></th>
<th style="text-align: right;" scope="col"><span style="font-weight: normal;">\(20.0\)&nbsp;&nbsp;&nbsp; </span><br></th>
<th style="text-align: right;" scope="col"><span style="font-weight: normal;">\(33.4\)&nbsp;&nbsp;&nbsp; </span><br></th>
<th style="text-align: right;" scope="col"><span style="font-weight: normal;">\(40.1\)&nbsp;&nbsp;&nbsp; </span><br></th>
<th style="text-align: right;" scope="col"><span style="font-weight: normal;">\(50.0\)&nbsp;&nbsp;&nbsp; </span><br></th>
<th style="text-align: right;" scope="col"><span style="font-weight: normal;">\(12.7\)&nbsp;&nbsp;&nbsp; </span><br></th>
<th style="text-align: right;" scope="col"><span style="font-weight: normal;">\(21.0\)&nbsp;&nbsp;&nbsp; </span><br></th>
<th style="text-align: right;" scope="col"><span style="font-weight: normal;">\(33.6\)&nbsp;&nbsp;&nbsp; </span><br></th>
<th style="text-align: right;" scope="col"><span style="font-weight: normal;">\(40.6\)&nbsp;&nbsp;&nbsp; </span><br></th>
</tr>
</thead>
<tbody>
<tr>
<td style="text-align: right;">\(50.0\)&nbsp;&nbsp; &nbsp;&nbsp; <br></td>
<td style="text-align: right;">\(15.5\)&nbsp;&nbsp;&nbsp;&nbsp; <br></td>
<td style="text-align: right;">\(26.4\)&nbsp;&nbsp; <br></td>
<td style="text-align: right;">\(35.4\)&nbsp;&nbsp;&nbsp; <br></td>
<td style="text-align: right;">\(41.1\)&nbsp;&nbsp;&nbsp; <br></td>
<td style="text-align: right;">\(50.0\)&nbsp;&nbsp;&nbsp; <br></td>
<td style="text-align: right;">\(17.7\)&nbsp;&nbsp;&nbsp; <br></td>
<td style="text-align: right;">\(37.9\)&nbsp;&nbsp;&nbsp; <br></td>
<td style="text-align: right;">\(41.3\)&nbsp;&nbsp;&nbsp; <br></td>
<td style="text-align: right;">\(50.0\)&nbsp;&nbsp;&nbsp; <br></td>
</tr>
<tr>
<td style="text-align: right;">\(41.9\)&nbsp;&nbsp; &nbsp;&nbsp; <br></td>
<td style="text-align: right;">\(50.4\)&nbsp; &nbsp;&nbsp; <br></td>
<td style="text-align: right;">\(43.0\)&nbsp;&nbsp;&nbsp; <br></td>
<td style="text-align: right;">\(43.3\)&nbsp;&nbsp;&nbsp; <br></td>
<td style="text-align: right;">\(43.6\)&nbsp;&nbsp;&nbsp; <br></td>
<td style="text-align: right;">\(43.7\)&nbsp;&nbsp;&nbsp; <br></td>
<td style="text-align: right;">\(43.8\)&nbsp;&nbsp;&nbsp; <br></td>
<td style="text-align: right;">\(44.7\)&nbsp;&nbsp;&nbsp; <br></td>
<td style="text-align: right;">\(44.9\)&nbsp;&nbsp;&nbsp; <br></td>
<td style="text-align: right;">\(45.0\)&nbsp;&nbsp;&nbsp; <br></td>
</tr>
<tr>
<td style="text-align: right;">\(45.1\)&nbsp; &nbsp; &nbsp; <br></td>
<td style="text-align: right;">\(45.2\)&nbsp;&nbsp;&nbsp;&nbsp; <br></td>
<td style="text-align: right;">\(45.3\)&nbsp;&nbsp;&nbsp; <br></td>
<td style="text-align: right;">\(45.5\)&nbsp;&nbsp;&nbsp; <br></td>
<td style="text-align: right;">\(46.1\)&nbsp;&nbsp;&nbsp; <br></td>
<td style="text-align: right;">\(46.5\)&nbsp;&nbsp;&nbsp; <br></td>
<td style="text-align: right;">\(46.6\)&nbsp;&nbsp;&nbsp; <br></td>
<td style="text-align: right;">\(47.1\)&nbsp;&nbsp;&nbsp; <br></td>
<td style="text-align: right;">\(48.0\)&nbsp;&nbsp;&nbsp; <br></td>
<td style="text-align: right;">\(48.2\)&nbsp;&nbsp;&nbsp; <br></td>
</tr>
<tr>
<td style="text-align: right;">\(48.5\)&nbsp; &nbsp; &nbsp; <br></td>
<td style="text-align: right;">\(48.4\)&nbsp;&nbsp;&nbsp;&nbsp; <br></td>
<td style="text-align: right;">\(48.6\)&nbsp;&nbsp;&nbsp; <br></td>
<td style="text-align: right;">\(48.7\)&nbsp;&nbsp;&nbsp; <br></td>
<td style="text-align: right;">\(48.8\)&nbsp;&nbsp;&nbsp; <br></td>
<td style="text-align: right;">\(48.9\)&nbsp;&nbsp;&nbsp; <br></td>
<td style="text-align: right;">\(49.4\)&nbsp;&nbsp;&nbsp; <br></td>
<td style="text-align: right;">\(49.5\)&nbsp;&nbsp;&nbsp; <br></td>
<td style="text-align: right;">\(49.6\)&nbsp;&nbsp;&nbsp; <br></td>
<td style="text-align: right;">\(49.8\)&nbsp;&nbsp;&nbsp; <br></td>
</tr>
</tbody>
</table>
<br><br><p></p>

<h4>Drawing steam-and-leaf diagram</h4>
<p>We can start by looking at the data as it is displayed by a stem-and-leaf diagram. Here we will use two-digit leaves with the first digit representing units and the second digit representing tenths. The tens are represented by the numbers to the left of the stem.</p><p><br></p><p style="text-align: center;"><img src="data:image/png;base64,[encoded data, 14056 characters]" alt=""><br></p><p style="text-align: left;"><br></p>

<p>Notice that all we have really done is rank the data from the lowest value to the highest value reading from top to bottom. This particular display has over half of its members crushed into one class - the \(4\)-class.</p>
<p>It may be informative to split the classes and look more closely at the data. This can be done by:</p>
<ol>
<li>rounding the raw data to two figures;</li>
<li>splitting each class according to the rule</li></ol><p>&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;second digit \(0 - 4\) ......*&nbsp; &nbsp; &nbsp; &nbsp;second digit \(5 - 9\) .....\(\bullet\)</p><p>The rounded raw now appears as follows</p><p></p><table><thead><tr><th scope="col"><span style="font-weight: normal;"><span>\(10\)&nbsp; &nbsp; &nbsp; </span></span><br></th><th scope="col"><span style="font-weight: normal;">\(12\) &nbsp; &nbsp; &nbsp; </span><br></th><th scope="col"><span style="font-weight: normal;">\(20\) &nbsp; &nbsp;&nbsp; </span><br></th><th scope="col"><span style="font-weight: normal;">\(33\) &nbsp;&nbsp;&nbsp;&nbsp; </span><br></th><th scope="col"><span style="font-weight: normal;">\(40\) &nbsp; &nbsp;&nbsp; </span><br></th><th scope="col"><span style="font-weight: normal;">\(50\) &nbsp; &nbsp;&nbsp; </span><br></th><th scope="col"><span style="font-weight: normal;">\(13\) &nbsp;&nbsp;&nbsp;&nbsp; </span><br></th><th scope="col"><span style="font-weight: normal;">\(21\) &nbsp;&nbsp;&nbsp; </span><br></th><th scope="col"><span style="font-weight: normal;">\(34\) &nbsp;&nbsp;&nbsp; </span><br></th><th scope="col"><span style="font-weight: normal;">\(41\) &nbsp;&nbsp; </span><br></th></tr><tr><td style="text-align: center;">\(50\)&nbsp; &nbsp; &nbsp; <br></td><td style="text-align: center;">\(16\)&nbsp; &nbsp;&nbsp;&nbsp;&nbsp; <br></td><td style="text-align: center;">\(26\) &nbsp;&nbsp; &nbsp; <br></td><td style="text-align: center;">\(35\) &nbsp;&nbsp;&nbsp;&nbsp; <br></td><td style="text-align: center;">\(41\) &nbsp; &nbsp;&nbsp; <br></td><td style="text-align: center;">\(50\) &nbsp;&nbsp; &nbsp; <br></td><td style="text-align: center;">\(18\) &nbsp;&nbsp;&nbsp;&nbsp; <br></td><td style="text-align: center;">\(38\) &nbsp;&nbsp;&nbsp; <br></td><td style="text-align: center;">\(41\) &nbsp;&nbsp;&nbsp; <br></td><td style="text-align: center;">\(50\) &nbsp;&nbsp; <br></td></tr><tr><td style="text-align: center;">\(42\)&nbsp; &nbsp; &nbsp; <br></td><td style="text-align: center;">\(50\) &nbsp; &nbsp;&nbsp;&nbsp; <br></td><td style="text-align: center;">\(43\) &nbsp;&nbsp;&nbsp;&nbsp; <br></td><td style="text-align: center;">\(43\) &nbsp;&nbsp;&nbsp;&nbsp; <br></td><td style="text-align: center;">\(44\) &nbsp; &nbsp;&nbsp; <br></td><td style="text-align: center;">\(44\) &nbsp;&nbsp;&nbsp;&nbsp; <br></td><td style="text-align: center;">\(44\) &nbsp;&nbsp;&nbsp;&nbsp; <br></td><td style="text-align: center;">\(45\) &nbsp;&nbsp;&nbsp; <br></td><td style="text-align: center;">\(45\) &nbsp;&nbsp;&nbsp; <br></td><td style="text-align: center;">\(45\) &nbsp;&nbsp; <br></td></tr><tr><td style="text-align: center;">\(45\) &nbsp;&nbsp;&nbsp;&nbsp; <br></td><td style="text-align: center;">\(45\)&nbsp; &nbsp; &nbsp; <br></td><td style="text-align: center;">\(45\) &nbsp;&nbsp;&nbsp;&nbsp; <br></td><td style="text-align: center;">\(46\) &nbsp;&nbsp;&nbsp;&nbsp; <br></td><td style="text-align: center;">\(46\) &nbsp; &nbsp;&nbsp; <br></td><td style="text-align: center;">\(47\) &nbsp;&nbsp;&nbsp;&nbsp; <br></td><td style="text-align: center;">\(47\) &nbsp;&nbsp;&nbsp;&nbsp; <br></td><td style="text-align: center;">\(47\) &nbsp;&nbsp;&nbsp; <br></td><td style="text-align: center;">\(48\) &nbsp;&nbsp;&nbsp; <br></td><td style="text-align: center;">\(48\) &nbsp;&nbsp; <br></td></tr><tr><td style="text-align: center;">\(49\)&nbsp; &nbsp; &nbsp; <br></td><td style="text-align: center;">\(48\) &nbsp; &nbsp; &nbsp; <br></td><td style="text-align: center;">\(49\) &nbsp;&nbsp;&nbsp;&nbsp; <br></td><td style="text-align: center;">\(49\) &nbsp;&nbsp;&nbsp;&nbsp; <br></td><td style="text-align: center;">\(49\)&nbsp; &nbsp;&nbsp; <br></td><td style="text-align: center;">\(49\) &nbsp;&nbsp;&nbsp;&nbsp; <br></td><td style="text-align: center;">\(49\) &nbsp;&nbsp;&nbsp;&nbsp; <br></td><td style="text-align: center;">\(50\) &nbsp;&nbsp;&nbsp; <br></td><td style="text-align: center;">\(50\) &nbsp;&nbsp;&nbsp; <br></td><td style="text-align: center;">\(50\) &nbsp;&nbsp;&nbsp; <br></td></tr></thead></table><p><br></p><p>The stem and leaf diagram now becomes <br></p><p style="text-align: center;"><img src="@@PLUGINFILE@@/Block221_Fig19-1.svg" alt="Stem and leaf diagram of the data in the table before the picture" class="img-responsive atto_image_button_text-bottom" width="393" height="200"></p><p style="text-align: left;">Essentially, the classes have been split according to the usual rule for rounding decimals. This process can make certain information contained in the data a little more obvious than the previous stem and leaf diagram. For example:</p><ul><li>the values in the \(3\)-class are evenly distributed between both halves of the class in the sense that each half has two members;</li><li>the \(4\)-class is split in the ratio \(2:1\) in favour of the upper half of the class;</li><li>the values in the \(5\)-class are all in the lower half of the class.</li></ul><p>You should have realised that:</p><ul><li>this is not <i>new </i>information - the new display has merely highlighted certain aspects of the raw data;</li><li>some of the conclusions may have been affected by the rounding process.</li></ul><p><br></p><p>Looking at the original stem and leaf diagram of the Inter-party data it is easy to produce a five-number summary of the data. The summary is:</p><ol><li>The lowest value, this is \(9.50\);</li><li>The lower hinge, this is \(39\) (to find the lower hinge average the \(12\)th and \(13\)th values);</li><li>The median, this is \(45.05\) (the average of the \(25\)th and \(26\)th values);</li><li>The upper hinge, this is \(48.55\) (to find the upper hinge average the \(37\)th and \(38\)th values);</li><li>The highest values, this is \(50.04\).</li></ol><p>The corresponding spreads are:</p><ol><li>The low-spread, this is \(45.05-9.50=35.55\);</li><li>The mid-spread, this is \(48.55-39.00=9.55\);</li><li>The high-spread, this is \(50.40-45.05=5.35\).</li></ol><p>Notice that the spreads indicate a considerable deviation from normality. For an ideal normal distribution, we would expect:</p><ul><li>The distances between the median and hinges to be equal</li><li>The high-spread and low-spread to be equal</li><li>The distance between the hinges and the extremes to be equal</li></ul><p>as shown in the following diagram</p><p style="text-align: center;"><img src="@@PLUGINFILE@@/36.2.2.3%20normal%20distribution.svg" alt="normal distribution with the five essential statistics and the spreads" class="img-responsive atto_image_button_text-bottom" width="401" height="188"><br></p><p></p>]]></text>
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