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.](https://tex.z-dn.net/?f=18%2C14%2C31%2C34%2C14%2C29%2C50%2C43%2C29%2C20%2C23%2C25%2C22%2C15%2C23%2C18%2C21%2C24%2C19.)
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.](https://tex.z-dn.net/?f=14%2C14%2C15%2C18%2C18%2C19%2C20%2C21%2C22%2C23%2C23%2C24%2C25%2C29%2C29%2C31%2C34%2C43%2C50.)
2) We calculate the rank r for the percentile p that we want to find.
![r=\frac{p}{100}\cdot(n-1)+1.](https://tex.z-dn.net/?f=r%3D%5Cfrac%7Bp%7D%7B100%7D%5Ccdot%28n-1%29%2B1.)
• 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{.}](https://tex.z-dn.net/?f=P%3Dx_%7Bri%7D%2Br_f%5Ccdot%28x_%7Bri%2B1%7D-x_%7Bri%7D%29%5Ctext%7B.%7D)
(a) for the 40th percentile, p = 40,
![r=\frac{40}{100}\cdot(19-1)+1=8.2.](https://tex.z-dn.net/?f=r%3D%5Cfrac%7B40%7D%7B100%7D%5Ccdot%2819-1%29%2B1%3D8.2.)
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.](https://tex.z-dn.net/?f=P_%7B40%7D%3D21%2B0.2%5Ccdot%2822-21%29%3D21.2.)
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.](https://tex.z-dn.net/?f=r%3D%5Cfrac%7B75%7D%7B100%7D%5Ccdot%2819-1%29%2B1%3D14.5.)
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
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• x_(ri + 1) = x_15 = 29
![P_{75}=29+0.5\cdot(29-29)=29.](https://tex.z-dn.net/?f=P_%7B75%7D%3D29%2B0.5%5Ccdot%2829-29%29%3D29.)
Answers
(a) The 40th percentile: 21 (rounded to the nearest integer)
(b) The 75th percentile: 29