Percentiles
& Percentile Ranks
-
In order to define what we mean by percentiles
and percentile ranks, we first begin by defining cumulative
frequencies and cumulative percentages.
- Cumulative Frequencies
- The cumulative frequency of a particular category in
a frequency table or distribution is the number of observations
in or below that category.
- Cumulative Percentages
- The cumulative percentage of a particular category in
a frequency table or distribution is the percentage of
observations in or below that category.
In the following table, the "cf" column presents
cumulative frequencies and the "c%" column presents
cumulative percentages.
_________________
X f cf c%
5 1 20 100%
4 5 19 95%
3 8 14 70%
2 4 6 30%
1 2 2 10%
A "cf" value of a category is the sum of
the frequencies in the categories at and below the category
in question.
A "c%" value of a category is 100 times the
cumulative frequency of the category, divided by the
total frequency. That is
c%=100*(cf/N)
- Percentile Rank
- The percentile rank of a particular score is defined
as the percentage of the scores in a distribution which
are at or below the particular score.
It is possible to determine some percentile ranks directly
from the frequency distribution table, provided that
the percentile ranks are percentages that appear in
the table.
For example, the percentile rank of X=3.5 is 70% (note
that 3.5 is the upper real limit of the category where
X=3).
- Percentile
- When a score is identified by its percentile rank, the
score is called a percentile.
It is also possible to determine some percentiles directly
from the frequency distribution table, provided that
the percentiles are upper real limits of a category.
Thus, the 95th percentile is X=4.5, the upper real
limit of the X=4 category.
- Interpolation
- The process known as interpolation must be used
to estimate percentile ranks and percentile for values
not given in the table.
The interpolation process is explained in the textbook
on pages 54-57.
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