<p style="text-align:justify"><span style="font-size:14px"><span style="color:black">The common measures of location are </span><strong><span style="color:black">quartiles </span></strong><span style="color:black">and </span><strong><span style="color:black">percentiles</span></strong><span style="color:black">.</span></span></p>

<p style="text-align:justify">&nbsp;</p>

<p style="text-align:justify"><span style="font-size:14px"><span style="color:black">Quartiles are special percentiles.</span></span></p>

<ul>
	<li style="text-align:justify">
	<p><span style="font-size:14px"><span style="color:black">The </span><strong><span style="color:black">first quartile</span></strong><span style="color:black">, </span><em><span style="color:black">Q</span></em><span style="color:black">1, is the same as the 25th percentile, </span></span></p>
	</li>
	<li style="text-align:justify">
	<p><span style="font-size:14px"><span style="background-color:white"><span style="color:#373d3f">The <strong>second quartile</strong>,&nbsp;<em>Q</em></span></span><em><sub><span style="color:#373d3f">2</span></sub></em><span style="background-color:white"><span style="color:#373d3f">, is the&nbsp;same as the 50th percentile / median.</span></span></span></p>
	</li>
	<li style="text-align:justify">
	<p><span style="font-size:14px"><span style="color:black">The <strong>third quartile</strong>, </span><em><span style="color:black">Q</span></em><span style="color:black">3, is the same as the 75th percentile. </span></span></p>
	</li>
</ul>

<p style="margin-left:52px; text-align:justify"><span style="font-size:14px"><span style="color:black">The median, </span><em><span style="color:black">M</span></em><span style="color:black">, is called both the second quartile and the 50th percentile.</span></span></p>

<p style="margin-left:52px; text-align:justify">&nbsp;</p>

<p style="text-align:justify"><span style="font-size:14px"><span style="color:black">To calculate quartiles and percentiles, you must order the data from smallest to largest. Quartiles divide ordered data into quarters. Percentiles divide ordered data into hundredths. Recall that a percent means one-hundredth. So, percentiles mean the data is divided into 100 sections. To score in the 90th percentile of an exam does not mean, necessarily, that you received 90 percent on a test. It means that 90 percent of test scores are the same as or less than your score and that 10 percent of the </span>test scores are the same as or greater than your test score.</span></p>

<p style="text-align:justify">&nbsp;</p>

<p style="text-align:justify"><span style="font-size:14px">Percentiles are useful for comparing values. For this reason, universities and colleges use percentiles extensively. One instance in which colleges and universities use percentiles is when SAT results are used to determine a minimum testing score that will be used as an acceptance factor. For example, suppose Duke accepts SAT scores at or above the 75<sup>th</sup> percentile. That translates into a score of at least 1220.</span></p>

<p style="text-align:justify">&nbsp;</p>

<p style="text-align:justify"><span style="font-size:14px">Percentiles are mostly used with very large populations. Therefore, if you were to say that 90 percent of the test scores are less, and not the same or less, than your score, it would be acceptable because removing one particular data value is not significant.</span></p>

<p style="text-align:justify">&nbsp;</p>

<p style="text-align:justify"><span style="font-size:14px">The <strong>median </strong>is a number that measures the <em>center </em>of the data. You can think of the median as the <em>middle value</em>, bu t it does not actually have to be one of the observed values. It is a number that separates ordered data into halves. Half the values are the same number or smaller than the median, and half the values are the same number or larger. For example, consider the following data:</span></p>

<p style="text-align:justify">&nbsp;</p>

<p style="text-align:justify"><span style="font-size:14px">1, 11.5, 6, 7.2, 4, 8, 9, 10, 6.8, 8.3, 2, 2, 10, 1</span></p>

<p style="text-align:justify">&nbsp;</p>

<p style="text-align:justify"><span style="font-size:14px">Ordered from smallest to largest:</span></p>

<p style="text-align:justify">&nbsp;</p>

<p style="text-align:justify"><span style="font-size:14px">1, 1, 2, 2, 4, 6, 6.8, 7.2, 8, 8.3, 9, 10, 10, 11.5</span></p>

<p style="text-align:justify">&nbsp;</p>

<p style="text-align:justify"><span style="font-size:14px">When a data set has an even number of data values, the median is equal to the average of the two middle values when the data are arranged in ascending order (least to greatest). When a data set has an odd number of data values, the median is equal to the middle value when the data are arranged in ascending order.</span></p>

<p style="text-align:justify">&nbsp;</p>

<p style="text-align:justify"><span style="font-size:14px">Since there are 14 observations (an even number of data values), the median is between the seventh value, 6.8, and the eighth value, 7.2. To find the median, add the two values together and divide by two.</span></p>

<p style="text-align:justify"><span style="font-size:14px"><span class="math-tex">\(\frac{6.8+7.2}{2}=7\)</span></span></p>

<p style="text-align:justify">&nbsp;</p>

<p style="text-align:justify"><span style="font-size:14px">The median is seven. Half of the values are smaller than seven and half of the values are larger than seven.</span></p>

<p style="text-align:justify">&nbsp;</p>

<p style="text-align:justify"><span style="font-size:14px">Quartiles are numbers that separate the data into quarters. Quartiles may or may not be part of the data. To find the quartiles, first find the median, or second, quartile. The first quartile, Q1, is the middle value of the lower half of the data, and the third quartile, Q3, is<span style="color:black"> the middle value, or median, of the upper half of the data. To get the idea, consider the same data set:</span></span></p>

<p style="text-align:justify">&nbsp;</p>

<p style="text-align:justify"><span style="font-size:14px"><span style="color:black">1, 1, 2, 2, 4, 6, 6.8, 7.2, 8, 8.3, 9, 10, 10, 11.5</span></span></p>

<p style="text-align:justify">&nbsp;</p>

<p style="text-align:justify"><span style="font-size:14px">The data set has an even number of values (14 data values), so the median will be the average of the two middle values (the average of 6.8 and 7.2), which is calculated as (6.8 + 7.2)/2 and equals 7.</span></p>

<p style="text-align:justify">&nbsp;</p>

<p style="text-align:justify"><span style="font-size:14px">So, the median, or second quartile ( Q2 ), is 7.</span></p>

<p style="text-align:justify">&nbsp;</p>

<p style="text-align:justify"><span style="font-size:14px">The first quartile is the median of the lower half of the data, so if we divide the data into seven values in the lower half and seven values in the upper half, we can see that we have an odd number of values in the lower half. Thus, the median of the lower half, or the first quartile ( Q1 ) will be the middle value, or 2. Using the same procedure, we can see that the media&nbsp; of the upper half, or the third quartile ( Q3 ) will be the middle value of the upper half, or 9.</span></p>

<p style="text-align:justify">&nbsp;</p>

<p style="text-align:justify"><span style="font-size:14px">The quartiles are illustrated below:</span></p>

<p style="text-align:justify">&nbsp;</p>

<p style="text-align:justify"><span style="font-size:14px"><img alt="" src="https://edupmquestionhelp.s3.ap-south-1.amazonaws.com/hint/uploads/images/16238318861623831886.jpg" style="height:182px; width:800px" /></span></p>

<p style="text-align:center"><span style="font-size:14px"><span style="background-color:white"><strong><span style="color:#002060">Figure&nbsp;2.11</span></strong></span></span></p>

<p>&nbsp;</p>

<p>&nbsp;</p>

<p style="text-align:justify"><span style="font-size:14px">The <strong>interquartile range </strong>is a number that indicates the spread of the middle half, or the middle 50 percent of the data. It is the difference between the third quartile (<em>Q</em>3) and the first quartile (<em>Q</em>1)</span></p>

<p style="text-align:justify">&nbsp;</p>

<p style="text-align:justify"><span style="font-size:14px"><em>IQR </em>= <em>Q</em>3 &ndash; <em>Q</em>1. The <em>IQR </em>for this data set is calculated as 9 minus 2, or 7.</span></p>

<p style="text-align:justify">&nbsp;</p>

<p style="text-align:justify"><span style="font-size:14px">The <em>IQR </em>can help to determine potential <strong>outliers</strong>. <strong>A value is suspected to be a potential outlier if it is less than 1.5 &times;</strong><strong><em>IQR </em></strong><strong>below the first quartile or more than 1.5 &times; </strong><strong><em>IQR </em></strong><strong>above the third quartile</strong>. Potential outliers always require further investigation.</span></p>