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      <text>36.2.1.1 Exploring data: introduction</text>
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      <text><![CDATA[<h3>Exploratory data analysis</h3>


<h4>The basics of EDA</h4>
<p>The basic principles followed in EDA are:</p>
<ul>
<li>To measure the location and spread of a distribution we use statistics which are <b>resistant</b> to departures from normality;</li>
<li>To summarise shape location and spread we use several statistics rather than just two;</li>
<li>Visual displays as well as numerical displays are used to summarise information obtained about shape, location and spread.</li>
</ul><p>You can see these principles illustrated below.</p>
<p>Traditionally, the location and spread of a distribution are measured by calculating its mean and standard deviation. The problem with these statistics is that is sensitive to the influence of extreme values. For example, the data set</p>
<p style="text-align: center;">\(1, 2, 2, 3, 3, 3, 4, 4, 4, 5, 5, 6\)</p>
<p>has mean \(\bar{x}=3.5\) and standard deviation \(s_{n-1}=1.45\). These values are quite acceptable since the distribution is symmetrical about its mean of \(3.5\). The symmetry is easily seen simply by inspecting the data although the bar chart below might make the symmetry more obvious.</p>
<br><div style="text-align: center;"><img src="@@PLUGINFILE@@/36.2.1%20symmetrical%20distribution.svg" alt="It is a data bar chart, in which any coloumn represents how many times (namely the frequency) the numbers of the data set appear" role="presentation" class="img-responsive atto_image_button_text-bottom" width="435" height="203"><br></div>


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<p>The shape of the distribution may also be shown by the <b>stem-and-leaf</b> diagram below. Notice that the<i> stem</i> consists of the numbers \(1\) to \(6\) and the <i> leaves</i> are just the members of each class.</p>
<br><p style="text-align: center;"><img src="@@PLUGINFILE@@/36.2.2%20stem-and-leaf%20diagram.svg" alt="Stem and leaf diagram of the data set (1,2,2,3,3,3,4,4,4,5,5,6)" role="presentation" class="img-responsive atto_image_button_text-bottom" width="71" height="136"><br></p><p><br></p>



<p>You will study the stem-and-leaf diagram in more detail later in this quiz.</p><p>The effects of changes in extreme values are easily illustrated by looking at what happens if we take the last number to be 60 instead of 6. This destroys the symmetry of the distribution and gives mean \(\bar{x}=8\) and standard deviation \(s_{n-1}=16.42\). Clearly, these values do not describe the distribution very well at all; a mean which is higher than \(92%\) of the members of the distribution can hardly be described as a representative!
</p>
<p>The simplest and most common examples of <b>resistant statistics</b> are those based on the idea of rank order - we simply order a distribution starting at the highest value and ending at the lowest value (or lower to highest).</p>
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<div class="HELM_keypoint">
<h4>Key Point</h4>
<p>The five essential statistics based on rank order are illustrated in the diagram below. </p>
<br><div style="text-align: center;"><img src="@@PLUGINFILE@@/36.2.3%20Essential%20statistics%20%281%29.svg" alt="the five essential statistics are representes on a vertical line. In order from the bottom to the top we find: lowest value, lower hinge, median, upper hinge, highest value" role="presentation" class="atto_image_button_text-bottom" width="250" height="214"><br></div>
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<div class="HELM_keypoint">
<h4>Key Point</h4>
<p>Using the values in the previews Key Point other statistics which represent the shape or spread of the distribution may be defined. These statistics are known as the Mid-Spread, High-Spread and Low-Spread and their definition is indicated in the diagram below.</p>
<br><div style="text-align: center;"><img src="@@PLUGINFILE@@/36.2.4%20Other%20statistics.svg" alt="The five essential statistics are represented on a vertical line as in the previous picture. In addition on the right we find vertical rows indicating the mid-spread (difference between the lower hinge and the upper hinge), the high-spread (difference between the highest value and the median) and the low spread (difference between the median and the lowest value)" role="presentation" class="img-responsive atto_image_button_text-bottom" width="420" height="199"><br></div>


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<p>Elementary EDA recommends the use of a <b>five-number summary</b> to characterise a distribution consisting of:</p>
<ol>
<li>the lowest value;</li>
<li>the lower hinge;</li>
<li>the median;</li>
<li>the upper hinge;</li>
<li>the highest value.</li>
</ol>
<p>You will find that the five-number summary, especially when used in conjunction with the three spreads shown in the diagram above gives an adequate representation of a non-symmetrical distribution.</p><br>
<p>Notice that</p>
<ul>
<li>the spreads shown in the diagram above are easily calculated once the five-number summary is known;</li>
<li>the median and the hinges are unaffected by changes in extreme values.</li>
</ul>]]></text>
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