{"id":81,"date":"2022-05-18T16:36:27","date_gmt":"2022-05-18T16:36:27","guid":{"rendered":"https:\/\/pressbooks.ccconline.org\/accintrostats\/chapter\/measures-of-the-location-of-the-data\/"},"modified":"2022-11-09T16:37:09","modified_gmt":"2022-11-09T16:37:09","slug":"measures-of-the-location-of-the-data","status":"publish","type":"chapter","link":"https:\/\/pressbooks.ccconline.org\/accintrostats\/chapter\/measures-of-the-location-of-the-data\/","title":{"raw":"Chapter 2.7: Measures of Position","rendered":"Chapter 2.7: Measures of Position"},"content":{"raw":" \r\n

The common measures of position or location are quartiles<\/span> and percentiles<\/span><\/p>\r\n

Quartiles are special percentiles. The first quartile, Q<\/em>1<\/sub>, is the same as the 25th<\/sup> percentile, and the third quartile, Q<\/em>3<\/sub>, is the same as the 75th<\/sup> percentile. The median, M<\/em>, is called both the second quartile and the 50th<\/sup> percentile.<\/p>\r\n

To calculate quartiles and percentiles, the data must be ordered from smallest to largest. Quartiles divide ordered data into quarters. Percentiles divide ordered data into hundredths. To score in the 90th<\/sup> percentile of an exam does not mean, necessarily, that you received 90% on a test. It means that 90% of test scores are the same or less than your score and 10% of the test scores are the same or greater than your test score.<\/p>\r\n

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 75th<\/sup> percentile. That translates into a score of at least 1220.<\/p>\r\n

Percentiles are mostly used with very large populations. Therefore, if you were to say that 90% 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.<\/p>\r\n

The median<\/span> is a number that measures the \"center\" of the data. You can think of the median as the \"middle value,\" but 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. \r\n<\/span>1;\u00a0 11.5;\u00a0 6;\u00a0 7.2;\u00a0 4;\u00a0 8;\u00a0 9;\u00a0 10;\u00a0 6.8;\u00a0 8.3;\u00a0 2;\u00a0 2;\u00a0 10;\u00a0 1 \r\n<\/span>Ordered from smallest to largest: \r\n<\/span>1;\u00a0 1;\u00a0 2;\u00a0 2;\u00a0 4;\u00a0 6;\u00a0 6. 8;\u00a0 7.2;\u00a0 8;\u00a0 8.3;\u00a0 9;\u00a0 10;\u00a0 10;\u00a0 11.5<\/p>\r\n

Since there are 14 observations, 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.<\/p>\r\n\r\n

\\(\\frac{6.8+7.2}{2}=7\\)<\/div>\r\n

The median is seven. Half of the values are smaller than seven and half of the values are larger than seven.<\/p>\r\n

Quartiles<\/span> 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, Q<\/em>1<\/sub>, is the middle value of the lower half of the data, and the third quartile, Q<\/em>3<\/sub>, is the middle value, or median, of the upper half of the data. To get the idea, consider the same data set: \r\n<\/span>1;\u00a0 1;\u00a0 2;\u00a0 2;\u00a0 4;\u00a0 6;\u00a0 6.8;\u00a0 7.2;\u00a0 8;\u00a0 8.3;\u00a0 9;\u00a0 10;\u00a0 10;\u00a0 11.5<\/p>\r\n

The median or second quartile<\/strong> is seven. The lower half of the data are 1,\u00a0 1,\u00a0 2,\u00a0 2,\u00a0 4,\u00a0 6,\u00a0 6.8. The middle value of the lower half is two. \r\n<\/span>1;\u00a0 1;\u00a0 2;\u00a0 2;\u00a0 4;\u00a0 6;\u00a0 6.8<\/p>\r\n

The number two, which is part of the data, is the first quartile<\/span>. One-fourth of the entire sets of values are the same as or less than two and three-fourths of the values are more than two.<\/p>\r\nThe upper half of the data is 7.2,\u00a0 8,\u00a0 8.3,\u00a0 9,\u00a0 10,\u00a0 10,\u00a0 11.5. The middle value of the upper half is nine.\r\n

The third quartile<\/span>, Q<\/em>3, is nine. Three-fourths (75%) of the ordered data set are less than nine. One-fourth (25%) of the ordered data set are greater than nine. The third quartile is part of the data set in this example.<\/p>\r\n

The interquartile range<\/span> is a number that indicates the spread of the middle half or the middle 50% of the data. It is the difference between the third quartile (Q<\/em>3<\/sub>) and the first quartile (Q<\/em>1<\/sub>).<\/p>\r\n

IQR<\/em> = Q<\/em>3<\/sub> \u2013 Q<\/em>1<\/sub><\/p>\r\nThe IQR<\/em> can help to determine potential outliers<\/strong>. A value is suspected to be a potential outlier if it is less than (1.5)(IQR<\/em>) below the first quartile or more than (1.5)(IQR<\/em>) above the third quartile<\/strong>. Potential outliers always require further investigation.\r\n

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NOTE<\/div>\r\n

A potential outlier is a data point that is significantly different from the other data points. These special data points may be errors or some kind of abnormality or they may be a key to understanding the data.<\/p>\r\n\r\n<\/div>\r\n

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For the following 13 real estate prices, calculate the IQR<\/em> and determine if any prices are potential outliers. Prices are in dollars. \r\n<\/span>389,950;\u00a0 230,500;\u00a0 158,000;\u00a0 479,000;\u00a0 639,000;\u00a0 114,950;\u00a0 5, 500,000;\u00a0 387,000;\u00a0 659,000;\u00a0 529,000;\u00a0 575,000;\u00a0 488,800;\u00a0 1,095,000<\/p>\r\n\r\n<\/div>\r\n

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Order the data from smallest to largest. \r\n<\/span>114,950;\u00a0 158,000;\u00a0 230,500;\u00a0 387,000;\u00a0 389,950;\u00a0 479,000;\u00a0 488,800;\u00a0 529,000;\u00a0 575,000; 639,000; 659,000; 1,095,000; 5,500,000<\/p>\r\n

M<\/em> = 488, 800<\/p>\r\nQ<\/em>1<\/sub> = \\(\\frac{\\text{230,500 + 387,000}}{2}\\) = 308,750\r\n\r\nQ<\/em>3<\/sub> = \\(\\frac{\\text{639,000 + 659,000}}{2}\\) = 649,000\r\n

IQR<\/em> = 649,000 \u2013 308,750 = 340,250<\/p>\r\n

(1.5)(IQR<\/em>) = (1.5)(340,250) = 510,375<\/p>\r\n

Q<\/em>1<\/sub> \u2013 (1.5)(IQR<\/em>) = 308,750 \u2013 510,375 = \u2013201,625<\/p>\r\n

Q<\/em>3<\/sub> + (1.5)(IQR<\/em>) = 649,000 + 510,375 = 1,159,375<\/p>\r\n

No house price is less than \u2013201,625. However, 5,500,000 is more than 1,159,375. Therefore, 5,500,000 is a potential outlier<\/span>.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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Try It<\/div>\r\n
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For the following 11 salaries, calculate the IQR<\/em> and determine if any salaries are outliers. The salaries are in dollars.<\/p>\r\n

\\$33,000\u00a0 \u00a0\\$<\/span>64,500\u00a0 \u00a0\\$<\/span>28,000\u00a0 \u00a0\\$<\/span>54,000\u00a0 \u00a0\\$<\/span>72,000\u00a0 \u00a0\\$<\/span>68,500\u00a0 \u00a0\\$<\/span>69,000\u00a0 \u00a0\\$<\/span>42,000\u00a0 \u00a0\\$<\/span>54,000\u00a0 \u00a0\\$<\/span>120,000\u00a0 \u00a0\\$<\/span>40,500<\/span><\/span><\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

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For the two data sets in the test scores example<\/a>, find the following:<\/p>\r\n\r\n

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  1. The interquartile range. Compare the two interquartile ranges.<\/li>\r\n \t
  2. Any outliers in either set.<\/li>\r\n<\/ol>\r\n<\/div>\r\n
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    The five number summary for the day and night classes is<\/p>\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
    <\/th>\r\nMinimum<\/th>\r\nQ<\/em>1<\/sub><\/th>\r\nMedian<\/th>\r\nQ<\/em>3<\/sub><\/th>\r\nMaximum<\/th>\r\n<\/tr>\r\n<\/thead>\r\n
    Day<\/strong><\/td>\r\n32<\/td>\r\n56<\/td>\r\n74.5<\/td>\r\n82.5<\/td>\r\n99<\/td>\r\n<\/tr>\r\n
    Night<\/strong><\/td>\r\n25.5<\/td>\r\n78<\/td>\r\n81<\/td>\r\n89<\/td>\r\n98<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n
      \r\n \t
    1. The IQR for the day group is Q<\/em>3<\/sub> \u2013 Q<\/em>1<\/sub> = 82.5 \u2013 56 = 26.5\r\n

      The IQR for the night group is Q<\/em>3<\/sub> \u2013 Q<\/em>1<\/sub> = 89 \u2013 78 = 11<\/p>\r\n

      The interquartile range (the spread or variability) for the day class is larger than the night class IQR<\/em>. This suggests more variation will be found in the day class\u2019s class test scores.<\/p>\r\n<\/li>\r\n \t

    2. Day class outliers are found using the IQR times 1.5 rule. So,\r\n