Question

The following are the P/E ratios (price of stock divided by projected earnings per share) for 19 banks.

Answer 1

In this problem, we have to compute some percentiles for a data sample. The data sample is:

$18,14,31,34,14,29,50,43,29,20,23,25,22,15,23,18,21,24,19.$

n = number of values = 19.

1) First, we order the data in ascending order:

$14,14,15,18,18,19,20,21,22,23,23,24,25,29,29,31,34,43,50.$

2) We calculate the rank r for the percentile p that we want to find.

$r=\frac{p}{100}\cdot(n-1)+1.$

• 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:

$P=x_{ri}+r_f\cdot(x_{ri+1}-x_{ri})\text{.}$

(a) for the 40th percentile, p = 40,

$r=\frac{40}{100}\cdot(19-1)+1=8.2.$

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.

$P_{40}=21+0.2\cdot(22-21)=21.2.$

So the 40th percentile is P = 21.2.

(b) for the 75th percentile, p = 75,

$r=\frac{75}{100}\cdot(19-1)+1=14.5.$

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

$P_{75}=29+0.5\cdot(29-29)=29.$

Answers

(a) The 40th percentile: 21 (rounded to the nearest integer)

(b) The 75th percentile: 29