Percentiles
Percentiles (Measures of Positions)
Introduction
Percentiles are measure of relative standing of an observation
within a data. Percentiles divides a set of observations into 100
equal parts, and percentile scores are frequently used to report
results from national standardized tests such as NAT, GAT etc.
The pth percentile is the value Y(p) in order statistic such that
p percent of the values are less than the value Y(p) and (100 − p)
percent of the values are greater Y(p) . The 5th percentile is
denoted by P5 , the 10th by P10 and 95th by P95 .
Percentiles for the ungrouped data
To calculate percentiles for the ungrouped data, adopt the
following procedure
1. Order the observation
2. For the mth percentile, determine the product m.n 100
. If m.n
100
is not an integer, round it up and nd the corresponding
ordered value and if m.n
100
is an integer, say k , then calculate
the mean of the K th and (k + 1)th ordered observations.
Example: For the following height data collected from
students nd the 10th and 95th percentiles. 91, 89, 88, 87, 89,
91, 87, 92, 90, 98, 95, 97, 96, 100, 101, 96, 98, 99, 98, 100, 102,
99, 101, 105, 103, 107, 105, 106, 107, 112.
Solution: The ordered observations of the data are 87, 87, 88,
89, 89, 90, 91, 91, 92, 95, 96, 96, 97, 98, 98, 98, 99, 99, 100, 100,
101, 101, 102, 103, 105, 105, 106, 107, 107, 112.
10 × 30
P10 = =3
100
So the 10th percentile i.e P10 is 3rd observation in sorted data
is 88, means that 10 percent of the observations in data set are
less than 88.
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Percentiles
95 × 30
P95 = = 28.5
100
29th observation is our 95th percentile i.e. P95 = 107.
Percentiles for the Grouped data
The mth percentile for grouped data is
h m.n
Pm = l + −c
f 100
Like median, m.n
100
is used to locate the mth percentile group.
l is the lower class boundary of the class containing the mth
percentile
h is the width of the class containing Pm
f is the frequency of the class containing
n is the total number of frequencies Pm
c is the cumulative frequency of the class immediately
preceding to the class containing Pm
Note that 50th percentile is the median by de nition as half
of the values in the data are smaller than the median and half
of the values are larger than the median. Similarly 25th and
75th percentiles are the lower (Q1 ) and upper quartiles (Q3 )
respectively. The quartiles, deciles and percentiles are also called
quantiles or fractiles.
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Percentiles
Example: For the following grouped data compute
P10 , P25 , P50 , and P95 given below.Deciles, Percentiles for
Grouped data.
Solution:
1. Locate the 10th percentile (lower Deciles i.e. D1 )by
10×n
100
= 10×3o
100
= 3 observation. so, P10 group is 85.590.5
containing the 3rd observation
h 10n
P10 = l + −c
f 100
5
= 85.5 + (3 − 0)
6
= 85.5 + 2.5 = 88
2. Locate the 25th percentile (lower Quartiles i.e. Q1 ) by
10×n
100
= 25×3o
100
= 7.5 observation. so, P25 group is 90.595.5
containing the 7.5th observation
h 25n
P25 = l + −c
f 100
5
= 90.5 + (7.5 − 6)
4
= 90.5 + 1.875 = 92.375
3. Locate the 50th percentile (Median i.e. 2nd quartiles, 5th
deciles) by 50×n
100
= 50×3o
100
= 15 observation. so, P50 group
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Percentiles
is 95.5100.5 containing the 15th observation
h 50n
P50 = l + −c
f 100
5
= 95.5 + (15 − 10)
10
= 95.5 + 2.5 = 98
4. Locate the 95th percentile by 95×n
100
= 95×3o
100
= 28.5th
observation. so, P95 group is 105.5110.5 containing the
3rd observation
h 95n
P95 = l + −c
f 100
5
= 105.5 + (28.5 − 26)
3
= 105.5 + 4.1667 = 109.6667
The percentiles and quartiles may be read directly from the
graphs of cumulative frequency function.
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