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337 lines
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<title>Grouped Frequency Distribution</title>
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<meta name="Description" content="By counting frequencies we can make a Frequency Distribution table. It is also possible to group the values." />
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<article id="content" role="main"><!-- #BeginEditable "Body" -->
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<h1 align="center">Grouped Frequency Distribution</h1>
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<h2>Frequency</h2>
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<p><b>Frequency</b> is how often something occurs.</p>
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<div class="example">
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<p style="float:right; margin: 0 0 5px 10px;"><img src="../images/soccer-teams.jpg" width="209" height="86" alt="soccer teams" /></p>
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<h3>Example: Sam played football on:</h3>
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<ul>
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<li>Saturday Morning, </li>
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<li>Saturday Afternoon</li>
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<li>Thursday Afternoon </li>
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</ul>
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<p>The frequency was <span class="large">2</span> on Saturday, <span class="large">1</span> on Thursday and <span class="large">3</span> for the whole week.</p>
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</div>
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<h2>Frequency Distribution</h2>
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<p>By counting frequencies we can make a <a href="frequency-distribution.html">Frequency Distribution</a> table.</p>
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<div class="example">
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<h3>Example: Newspapers</h3>
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<p>These are the numbers of newspapers sold at a local shop over the last 10 days:</p>
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<p align="center" class="larger">22, 20, 18, 23, 20, 25, 22, 20, 18, 20</p>
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<p>Let us count how many of each number there is:</p>
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<table border="0" align="center">
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<tr>
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<th width="100" align="center">Papers Sold</th>
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<th width="100" align="center">Frequency</th>
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</tr>
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<tr>
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<td width="100" align="center">18</td>
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<td width="100" align="center">2</td>
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</tr>
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<tr>
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<td width="100" align="center">19</td>
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<td width="100" align="center">0</td>
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</tr>
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<tr>
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<td width="100" align="center">20</td>
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<td width="100" align="center">4</td>
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</tr>
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<tr>
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<td width="100" align="center">21</td>
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<td width="100" align="center">0</td>
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</tr>
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<tr>
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<td width="100" align="center">22</td>
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<td width="100" align="center">2</td>
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</tr>
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<tr>
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<td width="100" align="center">23</td>
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<td width="100" align="center">1</td>
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</tr>
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<tr>
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<td align="center">24</td>
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<td align="center">0</td>
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</tr>
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<tr>
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<td align="center">25</td>
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<td align="center">1</td>
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</tr>
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</table>
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<p>It is also possible to <b>group</b> the values. Here they are grouped in 5s:</p>
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<table border="0" align="center">
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<tr>
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<th width="100" align="center">Papers Sold</th>
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<th width="100" align="center">Frequency</th>
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</tr>
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<tr>
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<td width="100" align="center">15-19</td>
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<td width="100" align="center">2</td>
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</tr>
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<tr>
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<td width="100" align="center">20-24</td>
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<td width="100" align="center">7</td>
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</tr>
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<tr>
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<td width="100" align="center">25-29</td>
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<td width="100" align="center">1</td>
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</tr>
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</table>
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</div>
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<h2>Grouped Frequency Distribution</h2>
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<p>We just saw how we can group frequencies. It is very useful when the scores have many different values.</p>
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<div class="example">
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<p style="float:right; margin: 0 0 5px 10px;"><img src="images/oak-leaves.jpg" width="200" height="123" alt="oak leaves" /></p>
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<h3>Example: Leaves</h3>
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<p>Alex measured the lengths of leaves on the oak tree (to the nearest cm):</p>
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<p class="center">9,16,13,7,8,4,18,10,17,18,9,12,5,9,9,16,1,8,17,1,<br>
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10,5,9,11,15,6,14,9,1,12,5,16,4,16,8,15,14,17 </p>
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</div>
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<p>Let's try to group them, but what groups should we use? </p>
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<div class="center80">
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<p>To get started, put the numbers <b>in order</b>, then find the <b>smallest</b> and <b>largest</b> values in your data, and calculate the <b>range</b> (range = largest - smallest).</p>
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</div>
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<div class="example">
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<h3>Example: Leaves (continued)</h3>
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<p>In order the lengths are:</p>
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<p class="center">1,1,1,4,4,5,5,5,6,7,8,8,8,9,9,9,9,9,9,10,10,11,12,12,<br>
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13,14,14,15,15,16,16,16,16,17,17,17,18,18 </p>
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<p>The smallest value (the "minimum") is <b>1 cm</b></p>
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<p>The largest value (the "maximum") is <b>18 cm</b></p>
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<p>The range is 18−1 = <b>17 cm</b></p>
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</div>
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<div class="center80">
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<h3>Group Size</h3>
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<p>Now calculate an approximate group size, by dividing the <b>range</b> by how many groups you would like. </p>
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<p>Then round that group size <b>up</b> to some <b>simple value</b> (like 2 instead of 1.83 or <b>5</b> instead of 4.26).</p>
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</div>
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<div class="example">
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<h3>Example: Leaves (continued)</h3>
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<p>Let us say we want about 5 groups.</p>
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<p>Divide the range by 5: </p>
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<div class="so">
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17/5 = 3.4
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</div>
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<p>Then round that up to <b>4</b></p>
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</div>
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<div class="center80">
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<h3>Start Value</h3>
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<p>Pick a starting value that is less than or equal to the smallest value. Try to make it a multiple of the group size if you can.</p>
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</div>
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<p>In our case a start value of <b>0</b> makes the most sense.</p>
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<div class="center80">
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<h3>Groups</h3>
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<p>Now calculate the list of groups. (We must go up to or past the largest value).</p>
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</div>
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<div class="example">
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<h3>Example: Leaves (continued)</h3>
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<p>Starting at 0 and with a group size of 4 we get: <b>0, 4, 8, 12, 16</b></p>
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<p>Write down the groups.</p>
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<p>Include the <b>end value</b> of each group that must be<b> less than the next group</b>:</p>
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<table border="1" align="center">
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<tr>
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<th width="100" align="center">Length (cm)</th>
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<th width="100" align="center">Frequency</th>
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</tr>
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<tr>
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<td width="100" align="center">0-3</td>
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<td width="100" align="center"> </td>
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</tr>
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<tr>
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<td width="100" align="center">4-7</td>
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<td width="100" align="center"> </td>
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</tr>
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<tr>
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<td width="100" align="center">8-11</td>
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<td width="100" align="center"> </td>
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</tr>
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<tr>
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<td width="100" align="center">12-15</td>
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<td width="100" align="center"> </td>
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</tr>
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<tr>
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<td width="100" align="center">16-19</td>
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<td width="100" align="center"> </td>
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</tr>
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</table>
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<p>The last group goes to 19 which is greater than the largest value. That is OK: the main thing is that it must include the largest value.</p>
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</div>
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<p>(Note: If you don't like the groups, then go back and change the group size or starting value and try again.) </p>
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<div class="center80">
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<h3>Upper and Lower Values For Each Group</h3>
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<p>Even though Alex only measured in whole numbers, the data is <a href="data-discrete-continuous.html">continuous</a>, so "4 cm" means the actual value could have been anywhere from 3.5 cm to 4.5 cm. Alex just rounded the numbers to whole centimeters.</p>
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</div>
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<div class="example">
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<h3>Example: Leaves (continued) </h3>
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<p>Here are the groups with the Lower and Upper limits shown:</p>
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<table border="1" align="center">
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<tr>
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<th width="100" align="center">Length</th>
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<th width="100" align="center">Lower/Upper</th>
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<th width="100" align="center">Frequency</th>
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</tr>
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<tr>
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<td width="100" align="center">0-3 cm</td>
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<td width="100" align="center">0-3.5</td>
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<td width="100" align="center"> </td>
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</tr>
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<tr>
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<td width="100" align="center">4-7 cm</td>
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<td width="100" align="center">3.5-7.5</td>
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<td width="100" align="center"> </td>
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</tr>
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<tr>
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<td width="100" align="center">8-11 cm</td>
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<td width="100" align="center">7.5-11.5</td>
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<td width="100" align="center"> </td>
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</tr>
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<tr>
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<td width="100" align="center">12-15 cm</td>
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<td width="100" align="center">11.5-15.5</td>
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<td width="100" align="center"> </td>
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</tr>
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<tr>
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<td width="100" align="center">16-19 cm</td>
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<td width="100" align="center">15.5-19.5</td>
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<td width="100" align="center"> </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="center80">
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<h3>Tally and Total</h3>
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<p>Now tally the results to find the frequencies. And do a total. </p>
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</div>
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<div class="example">
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<h3>Example: Leaves (continued) </h3>
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<p class="center">1,1,1,4,4,5,5,5,6,7,8,8,8,9,9,9,9,9,9,10,10,11,12,12,<br>
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13,14,14,15,15,16,16,16,16,17,17,17,18,18:</p>
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<table border="1" align="center">
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<tr>
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<th width="100" align="center">Length</th>
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<th width="100" align="center">Lower/Upper</th>
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<th width="100" align="center">Frequency</th>
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</tr>
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<tr>
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<td width="100" align="center">0-3 cm</td>
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<td width="100" align="center">0-3.5</td>
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<td width="100" align="center">3</td>
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</tr>
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<tr>
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<td width="100" align="center">4-7 cm</td>
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<td width="100" align="center">3.5-7.5</td>
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<td width="100" align="center">7</td>
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</tr>
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<tr>
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<td width="100" align="center">8-11 cm</td>
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<td width="100" align="center">7.5-11.5</td>
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<td width="100" align="center">12</td>
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</tr>
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<tr>
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<td width="100" align="center">12-15 cm</td>
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<td width="100" align="center">11.5-15.5</td>
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<td width="100" align="center">7</td>
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</tr>
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<tr>
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<td width="100" align="center">16-19 cm</td>
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<td width="100" align="center">15.5-19.5</td>
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<td width="100" align="center">9</td>
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</tr>
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<tr>
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<td align="center"> </td>
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<td align="right"><b>Total:</b></td>
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<td align="center"><b>38</b></td>
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</tr>
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</table>
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</div>
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<p><b>Done!</b></p>
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<h2>Histogram</h2>
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<p>You might also like to make a <a href="histograms.html">Histogram</a> of your data. </p>
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<p> </p>
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<div class="related"><a href="frequency-distribution.html">Frequency Distribution</a> <a href="index.html">Data Index</a></div>
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