Quartiles, Deciles, Percentiles, and Quantiles Quartiles, deciles, and percentiles divide a frequency distribution into a number of parts containing equal frequencies. The items are first put into order of increasing magnitude. Quartiles divide the range of values into four parts, each containing one quarter of the values. Again, if an item comes exactly on a dividing line, half of it is counted in the group above and half is counted below. Similarly, deciles divide into ten parts, each containing one tenth of the total frequency, and percentiles divide into a hundred parts, each containing one hundredth of the total frequency. If we think again about the median, it is the second or middle quartile, the fifth decile, and the fiftieth percentile. If a quartile, decile, or percentile falls between two items in order of size, for our purposes the value halfway between the two items will be used. Other conventions are also common, but the effect of different choices is usually not important. Remember that we are dealing with a quantity which varies randomly, so another sample would likely show a different quartile or decile or percentile. For example, if the items after being put in order are 1, 2, 2, 3, 5, 6, 6, 7, 8, a total of nine items, the first or lower quartile is (2 + 2)/2 = 2, the median is 5, and the upper or third quartile is (6 + 7)/2 = 6.5. Example 3.2 To start a program to improve the quality of production in a factory, all the products coming off a production line, under what we have reason to believe are normal operating conditions, are examined and classified as “good” products or “defective” products. The number of defective products in each successive group of six is counted. The results for 60 groups, so for 360 products, are shown in Table 3.1. Find the mean, median, mode, first quartile, third quartile, eighth decile, ninth decile, proportion defective in the sample, first estimate of probability that an item will be defective, sample variance, sample standard deviation, and coefficient of variation. Table 3.1: Numbers of Defectives in Groups of Six Items 100000000000 000010000010 010010000200 000020010010 10000100100Quartiles, Deciles, Percentiles, and Quantiles Quartiles, deciles, and percentiles divide a frequency distribution into a number of parts containing equal frequencies. The items are first put into order of increasing magnitude. Quartiles divide the range of values into four parts, each containing one quarter of the values. Again, if an item comes exactly on a dividing line, half of it is counted in the group above and half is counted below. Similarly, deciles divide into ten parts, each containing one tenth of the total frequency, and percentiles divide into a hundred parts, each containing one hundredth of the total frequency. If we think again about the median, it is the second or middle quartile, the fifth decile, and the fiftieth percentile. If a quartile, decile, or percentile falls between two items in order of size, for our purposes the value halfway between the two items will be used. Other conventions are also common, but the effect of different choices is usually not important. Remember that we are dealing with a quantity which varies randomly, so another sample would likely show a different quartile or decile or percentile. For example, if the items after being put in order are 1, 2, 2, 3, 5, 6, 6, 7, 8, a total of nine items, the first or lower quartile is (2 + 2)/2 = 2, the median is 5, and the upper or third quartile is (6 + 7)/2 = 6.5. Example 3.2 To start a program to improve the quality of production in a factory, all the products coming off a production line, under what we have reason to believe are normal operating conditions, are examined and classified as “good” products or “defective” products. The number of defective products in each successive group of six is counted. The results for 60 groups, so for 360 products, are shown in Table 3.1. Find the mean, median, mode, first quartile, third quartile, eighth decile, ninth decile, proportion defective in the sample, first estimate of probability that an item will be defective, sample variance, sample standard deviation, and coefficient of variation. Table 3.1: Numbers of Defectives in Groups of Six Items 100000000000 000010000010 010010000200 000020010010 10000100100
