Page 92 - DECO504_STATISTICAL_METHODS_IN_ECONOMICS_ENGLISH

Statistical Methods in Economics

Notes The value v of the P-th percentile may now be calculated as follows:
If P < p or P > p , then we take v = v or v = v , respectively.
1 N 1 N
If there is some integer k for which P = p , then we take v = v .
k k
P − p
Otherwise, we find the integer k for which p < P < p , and take v = v k ( + v + v k =
) k+1
k k + 1 k p k+1 − p k
P − p k
k +v ( × N +1 − v v k ) k .
100
th
Using the list of numbers above, the 40 percentile would be found by linearly interpolating between
the 30 percentile, 20, and the 50 , 35. Specifically:
th
th
40 − 30
v = 20 +× ( − )35 20 = 27.5
5
100
This is halfway between 20 and 35, which one would expect since the rank was calculated above as
2.5.
It is readily confirmed that the 50 percentile of any list of values according to this definition of the P-
th
th percentile is just the sample median.
Moreover, when N is even the 25 percentile according to this definition of the P-th percentile is the
th
N
median of the first values (i.e., the median of the lower half of the data).
2
Weighted percentile
In addition to the percentile function, there is also a weighted percentile, where the percentage in the
total weight is counted instead of the total number. There is no standard function for a weighted
percentile. One method extends the above approach in a natural way.
Suppose we have positive weights w , w , w , ..., w associated, respectively, with our N sorted sample
2
3
N
1
values. Let
n
S = ∑ w k ,
n
k= 1
the n-th partial sum of the weights. Then the formulas above are generalized by taking
100 ⎛ w ⎞
p = ⎜ S − n ⎟ n
n S N ⎝ 2 ⎠
and
−
pp k
v
v = k ( + v + v k .
) k+1
p k+1 − p k
Alternative methods
Some software packages, including Microsoft Excel (up to the version 2007) use the following method,
noted as an alternative by NIST to estimate the value, v , of the P-th percentile of an ascending
P
ordered dataset containing N elements with values v 1 ≤ 2 ≤ v . . . ≤ v .
N
The rank is calculated:
P
n = ( − )N1 + 1
100

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