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Unit 6: Dispersion: Meaning and Characteristics, Absolute and Relative Measures of Dispersion including Range...

Definition Notes

There is no standard definition of percentile, however all definitions yield similar results when the
number of observations is very large.
Nearest rank

One definition of percentile, often given in texts, is that the P-th percentile ( < P 100 of N ordered
) ≤ 0
values (arranged from least to greatest) is obtained by first calculating the (ordinal) rank

P 1
n = × +N
100 2
rounding the result to the nearest integer, and then taking the value that corresponds to that rank.

P
(Note that the rounded value of n is just the least integer which exceeds × N .)
100
For example, by this definition, given the numbers
15, 20, 35, 40, 50
th
the rank of the 30 percentile would be
30 1
5
n = ×+ = 2.
100 2
th
Thus the 30 percentile is the second number in the sorted list, 20.
The 35 percentile would have rank
th
35 1
5
n = ×+ = 2.25,
100 2
so the 35 percentile would be the second number again (since 2.25 rounds down to 2) or 20
th
The 40 percentile would have rank
th
40 1
5
n = ×+ = 2.5,
100 2
so the 40 percentile would be the third number (since 2.5 rounds up to 3), or 35.
th
th
The 100 percentile is defined to be the largest value. (In this case we do not use the above definition
with P = 100, because the rank n would be greater than the number N of values in the original list.)
In lists with fewer than 100 values the same number can occupy more than one percentile group.
Linear interpolation between closest ranks
An alternative to rounding used in many applications is to use linear interpolation between the two
nearest ranks.
In particular, given the N sorted values v 1 ≤ 2 ≤ v 3 ≤ v ... ≤ v , we define the percent rank corresponding
N
to the n value as:
th
⎛ 100 1 ⎞
p = ⎜ n − ⎟ .
n N ⎝ 2 ⎠
th
In this way, for example, if N = 5 percent rank corresponding to the third value is
⎛ 100 1 ⎞
p = ⎜ 3 − ⎟ = 50.
3 5 ⎝ 2 ⎠

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