Question

The following frequency table summarizes 60 data values. What is the 80th percentile of the data? The table is read from left to right.

1	1	1	1	2
2	2	2	2	2
2	3	3	3	3
6	6	6	7	7
8	8	9	9	9
9	10	11	11	11
11	11	12	12	12
12	13	13	13	14
14	14	14	15	15
15	16	17	17	17
17	17	18	19	19
20	20	20	20	20

Answer 1

The p^th percentile is a value such that at least p percent of the observations is less than or equal to this value and at least (100−p) percent of the observations is greater than or equal to this value.

The first step is to sort the values.

The sorted values are 1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,6,6,6,7,7,8,8,9,9,9,9,10,11,11,11,11,11,12,12,12,12,13,13,13,14,14,14,14,15,15,15,16,17,17,17,17,17,18,19,19,20,20,20,20,20

Since there are 60 values, then n=60.

Now, calculate the index

Since the index i is an integer, the 80^th percentile is the average of the values at the positions i and i+1.

The value at the position i=48 is 17 and at the position i+1=49 is 17.

Their average is the percentile:

Answer: the 80^th percentile is 17.