2.1 Measures of the location of the data  (Page 3/15)

Fifty statistics students were asked how much sleep they get per school night (rounded to the nearest hour). The results were:

Find the 28 ^th percentile . Notice the 0.28 in the "cumulative relative frequency" column. Twenty-eight percent of 50 data values is 14 values. There are 14 values less than the 28 ^th percentile. They include the two 4s, the five 5s, and the seven 6s. The 28 ^th percentile is between the last six and the first seven. The 28 ^th percentile is 6.5.

Find the median . Look again at the "cumulative relative frequency" column and find 0.52. The median is the 50 ^th percentile or the second quartile. 50% of 50 is 25. There are 25 values less than the median. They include the two 4s, the five 5s, the seven 6s, and eleven of the 7s. The median or 50 ^th percentile is between the 25 ^th , or seven, and 26 ^th , or seven, values. The median is seven.

Find the third quartile . The third quartile is the same as the 75 ^th percentile. You can "eyeball" this answer. If you look at the "cumulative relative frequency" column, you find 0.52 and 0.80. When you have all the fours, fives, sixes and sevens, you have 52% of the data. When you include all the 8s, you have 80% of the data. The 75 ^th percentile, then, must be an eight . Another way to look at the problem is to find 75% of 50, which is 37.5, and round up to 38. The third quartile, Q ₃ , is the 38 ^th value, which is an eight. You can check this answer by counting the values. (There are 37 values below the third quartile and 12 values above.)