In this problem, we have to compute some percentiles for a data sample. The data sample is:
[tex]18,14,31,34,14,29,50,43,29,20,23,25,22,15,23,18,21,24,19.[/tex]n = number of values = 19.
1) First, we order the data in ascending order:
[tex]14,14,15,18,18,19,20,21,22,23,23,24,25,29,29,31,34,43,50.[/tex]2) We calculate the rank r for the percentile p that we want to find.
[tex]r=\frac{p}{100}\cdot(n-1)+1.[/tex]• If r is an integer then the data value at location r, x_r, is the percentile p: p = x_r.
,• If r is not an integer, p is interpolated using ,ri,, the integer part of r, and, rf,, the fractional part of r:
[tex]P=x_{ri}+r_f\cdot(x_{ri+1}-x_{ri})\text{.}[/tex](a) for the 40th percentile, p = 40,
[tex]r=\frac{40}{100}\cdot(19-1)+1=8.2.[/tex]We have r = 8.2, which is not an integer, so we interpolate p using:
• ri = 8,
,• rf = 0.2,
,• x_ri = x_8 = 21,
,• x_(ri + 1) = x_9 = 22.
[tex]P_{40}=21+0.2\cdot(22-21)=21.2.[/tex]So the 40th percentile is P = 21.2.
(b) for the 75th percentile, p = 75,
[tex]r=\frac{75}{100}\cdot(19-1)+1=14.5.[/tex]We have r = 14.5, which is not an integer, so we interpolate p using:
• ri = 14
,• rf = 0.5
,• x_ri = x_14 = 29
,• x_(ri + 1) = x_15 = 29
[tex]P_{75}=29+0.5\cdot(29-29)=29.[/tex]Answers
(a) The 40th percentile: 21 (rounded to the nearest integer)
(b) The 75th percentile: 29
