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      <text>36.2.3.1 Skewness, gaps and multiple peaks</text>
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      <text><![CDATA[<h4>Skewness, gaps and multiple peaks</h4>
<br>
<p>When exploring a data set, four properties worth looking for are outliers, skewness, gaps and multiple
peaks. Outliers have been dealt with in some detail above so the comments below briefly address skewness, gaps and multiple peaks.</p>
<br>
<p><b>Skewness</b></p>
<p>If a skewed distribution is represented purely by two numbers, say the mean and standard deviation,
then the representation will be inadequate. Remember that the term ‘skewness’ refers to the location
of the ‘tail’ of a distribution.</p><p style="text-align: center;"><img src="@@PLUGINFILE@@/36.2.3%20Skewness.svg" alt="In the picture two distributions characterised by right skew and left skew respectively" class="img-responsive atto_image_button_text-bottom" width="345" height="69"></p><p style="text-align: left;">

As an example, the data set below gives the current required to burn out a component under test.<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; </span><br></th>
<th style="text-align: right;" scope="col"><span style="font-weight: normal;">\(33.4\) &nbsp;&nbsp; </span><br></th>
<th style="text-align: right;" scope="col"><span style="font-weight: normal;">\(40.1\) &nbsp;&nbsp; </span><br></th>
<th style="text-align: right;" scope="col"><span style="font-weight: normal;">\(50.0\) &nbsp;&nbsp; </span><br></th>
<th style="text-align: right;" scope="col"><span style="font-weight: normal;">\(12.7\) &nbsp;&nbsp; </span><br></th>
<th style="text-align: right;" scope="col"><span style="font-weight: normal;">\(21.0\) &nbsp;&nbsp; </span><br></th>
<th style="text-align: right;" scope="col"><span style="font-weight: normal;">\(33.6\) &nbsp;&nbsp; </span><br></th>
<th style="text-align: right;" scope="col"><span style="font-weight: normal;">\(40.6\) &nbsp;&nbsp; </span><br></th>
</tr>
</thead>
<tbody>
<tr>
<td style="text-align: right;">\(50.0\) &nbsp; &nbsp;&nbsp; <br></td>
<td style="text-align: right;">\(15.5\) &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; <br></td>
<td style="text-align: right;">\(41.1\) &nbsp;&nbsp; <br></td>
<td style="text-align: right;">\(50.0\) &nbsp;&nbsp; <br></td>
<td style="text-align: right;">\(17.7\) &nbsp;&nbsp; <br></td>
<td style="text-align: right;">\(37.9\) &nbsp;&nbsp; <br></td>
<td style="text-align: right;">\(41.3\) &nbsp;&nbsp; <br></td>
<td style="text-align: right;">\(50.0\) &nbsp;&nbsp; <br></td>
</tr>
<tr>
<td style="text-align: right;">\(41.9\) &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; <br></td>
<td style="text-align: right;">\(43.3\) &nbsp;&nbsp; <br></td>
<td style="text-align: right;">\(43.6\) &nbsp;&nbsp; <br></td>
<td style="text-align: right;">\(43.7\) &nbsp;&nbsp; <br></td>
<td style="text-align: right;">\(43.8\) &nbsp;&nbsp; <br></td>
<td style="text-align: right;">\(44.7\) &nbsp;&nbsp; <br></td>
<td style="text-align: right;">\(44.9\) &nbsp;&nbsp; <br></td>
<td style="text-align: right;">\(45.0\) &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; <br></td>
<td style="text-align: right;">\(45.3\) &nbsp;&nbsp; <br></td>
<td style="text-align: right;">\(45.5\) &nbsp;&nbsp; <br></td>
<td style="text-align: right;">\(46.1\) &nbsp;&nbsp; <br></td>
<td style="text-align: right;">\(46.5\) &nbsp;&nbsp; <br></td>
<td style="text-align: right;">\(46.6\) &nbsp;&nbsp; <br></td>
<td style="text-align: right;">\(47.1\) &nbsp;&nbsp; <br></td>
<td style="text-align: right;">\(48.0\) &nbsp;&nbsp; <br></td>
<td style="text-align: right;">\(48.2\) &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; <br></td>
<td style="text-align: right;">\(48.6\) &nbsp;&nbsp; <br></td>
<td style="text-align: right;">\(48.7\) &nbsp;&nbsp; <br></td>
<td style="text-align: right;">\(48.8\) &nbsp;&nbsp; <br></td>
<td style="text-align: right;">\(48.9\) &nbsp;&nbsp; <br></td>
<td style="text-align: right;">\(49.4\) &nbsp;&nbsp; <br></td>
<td style="text-align: right;">\(49.5\) &nbsp;&nbsp; <br></td>
<td style="text-align: right;">\(49.6\) &nbsp;&nbsp; <br></td>
<td style="text-align: right;">\(49.8\) &nbsp;&nbsp; <br></td>
</tr>
</tbody>
</table>
<br>
<p>The data were obtained by measuring the current in mA applied to an electronic component under
conditions of destructive testing give the following values for the mean, standard deviation, median
and IQR:</p>
<p>\(\bar{x}\)=\(40.72\) &nbsp; \(s_{n-1}\)=\(11.49\) &nbsp; median=\(45.05\) &nbsp; and &nbsp;&nbsp; IQR=\(8.5\)&nbsp; .</p>

<p>The values of \(\bar{x}\) and \(s_{n-1}\) indicate that a lower average current with a greater spread will result in the destruction of the component than that indicated by the median and IQR. Clearly, further
investigations are necessary to resolve this situation.</p>
<br>
<p><b>Gaps and multiple peaks</b></p>
<p>Distributions with gaps and multiple peaks can be very difficult to summarise easily. The stem-and-
leaf and box-and-whisker plots shown below summarise some 1972 data concerning adult literacy.
The leaves are single digit and the range of achievement reached in the field of literacy ranges from
2% to 100%.</p><p style="text-align: center;"><img src="@@PLUGINFILE@@/36.2.3%201%20stem%20and%20leaf.svg" alt="On the left in vertical the box-and-whisker diagram and on the right the stem-and-leaf diagram of some 1972 data concerning adult literacy" role="presentation" class="img-responsive atto_image_button_text-bottom" width="431" height="187"></p><p style="text-align: left;"><br></p>

<p>The virtual lack of data between \(50\) and \(60\) indicates a gap and suggests that we are in fact dealing
with two separate distributions which have the following properties:</p>
<ol>
<li>\(2%-50%\) literacy having <b>right</b> skew,</li>
<li>\(60%-100%\) literacy having <b>left</b> skew,</li>
</ol>
<p>Notice that the term skewness refers to the <b>tail</b> of a distribution.</p>
<p>The usual summary statistics that you might be tempted to calculate are:</p>
<p>&nbsp;&nbsp; \(\bar{x}\)=\(54\) &nbsp;&nbsp; and &nbsp;&nbsp; \(s_{n-1}\)=\(34\) &nbsp;&nbsp; or &nbsp;&nbsp; median=\(60\) &nbsp;&nbsp; mid-spread=\(65\)</p>

<p>In this case, neither set of statistics is of much use since neither set indicates the gap or the skewness.
Without visual representation, a single-peaked distribution tends to be assumed, this is, of course,
opposite to the truth in this case.</p><br>
<p>The stem-and-leaf plot is more informative than the box-and-whisker plot since it shows the gap.
In practice, we would work with the two constituent distributions and attempt to relate the results in
a practical way.</p>
<br>
<p><b>Final comments on data representations</b></p>
<ol>
<li>You should not rely on summary statistics such as the mean and standard deviation or median
and mid-spread alone to represent a data set. Remember that if a distribution has outliers,
gaps, skewness or multiple peaks, then the shape is probably more important than location and
spread.</li>
<li><p>The shape of a distribution is better shown visually than numerically. Remember that a stem-
and-leaf diagram retains the data and arranges the data in rank order and that a box-and-
whisker plot emphasises the detail contained in the tails of a distribution.</p></li>
</ol>]]></text>
<file name="36.2.3 1 stem and leaf.svg" path="/" encoding="base64">[encoded data, 23940 characters]</file>
<file name="36.2.3 Skewness.svg" path="/" encoding="base64">[encoded data, 12612 characters]</file>
<file name="36.2.3.2 stem-and-leaf diagram.svg" path="/" encoding="base64">[encoded data, 17680 characters]</file>
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