About Percentages
& Percentiles
TABLE OF CONTENTS
About Percentages & Percentiles........................................................................................ 1
What is a PERCENTAGE?............................................................................................. 1
What is a PERCENTILE?............................................................................................... 1
Percentages ......................................................................................................................... 1
Percentages ..................................................................................................................... 1
Percentiles ........................................................................................................................... 1
Percentiles ....................................................................................................................... 1
The Problem with Percentiles ......................................................................................... 2
Percentages vs. Percentiles ................................................................................................. 3
Percentages vs. Percentiles ............................................................................................. 3
Glossary .............................................................................................................................. 4
References........................................................................................................................... 5
About Percentages & Percentiles
What is a PERCENTAGE?
●
A percentage is a proportion of 100 that represents how large one
quantity is in relation to another quantity, where the first quantity
expresses a part of, or a change in, the second quantity.
What is a PERCENTILE?
●
A percentile is a score at or below which a certain percentage of the
distribution lies.
Percentages
Percentages
●
The formula for calculating percentages is:
where n is the quantity of interest and N is the total quantity.
●
Percentages are very commonly used in everyday life because they are
easily calculated and easily convey information about proportions.
Furthermore, they allow us to make comparisons.
●
Percentages can also be used to indicate an increase or a decrease in a
specific quantity.
Percentiles
Percentiles
●
Percentiles are useful in that they divide the distribution into 100
equal parts and show us the relative placement of a score in the
distribution.
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●
In such a distribution, the xth percentile is equal to the score at or
below which x% of the scores fall.
The Problem with Percentiles
●
The problem with using percentiles with normally distributed scores is
that the real differences between the raw scores may be minimized
near the ends of the distribution and exaggerated in the middle of the
distribution.
This is due to the fact that the majority of the scores occur in the
middle of the distribution. While these differences might be quite
small, they will appear as differences in percentiles.
This distortion may be even worse for highly skewed data, where the
differences between the raw scores might be large, but will appear as
relatively small differences in percentiles.
●
As can be seen in the graph below, the percentiles in the middle of the
distribution are relatively close together; whereas, the percentiles at
the extreme ends of the distribution (i.e. 1st and 99th) are relatively
farther from their neighbouring percentiles.
Another way to look at it is that in the middle of the distribution, there
are fewer scores between each percentile; whereas, at the extreme
ends of the distribution, there are more scores between each
percentile.
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Percentages vs. Percentiles
Percentages vs. Percentiles
●
Be careful not to confuse percentage correct with percentiles.
●
Percentiles are converted scores that refer to a percentage of test
takers.
●
Percentage correct refers to the percentage of correct responses an
individual obtained on a test.
●
In other words, percentage correct gives us information about how an
individual performed on a test; whereas, percentiles give us
information about how that score compares to the scores of other test
takers.
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Glossary
Normal distribution:
a smooth, perfectly symmetrical, bell-shaped
curve that has the highest part in the center.
Both sides taper from the center and
approach but never touch the x-axis. The
ends theoretically stretch to negative and
positive infinity.
Percentage:
is a proportion of 100 that represents how
large one quantity is in relation to another
quantity, where the first quantity expresses a
part of, or a change in, the second quantity.
Percentage correct:
the percentage of correct responses an
individual obtained on a test.
Percentile:
a score at or below which a certain
percentage of the distribution lies.
Proportion:
a comparison of things in relation to size,
number, quantity, etc.
Raw score:
this refers to the actual scored obtained on
the test or measure.
Skewed distribution:
when most of the scores fall either in the
positive or negative end of the distribution
of the test or measure.
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References
Cohen, R. J., & Swerdlik, M. E. (2005). Psychological testing and
assessment: An introduction to tests and measurements. (6th ed.) NY:
McGraw-Hill
Percentage. (2007). Retrieved May 2, 2007 from
http://en.wikipedia.org/wiki/Percentage.
Statistics Canada. (2007). Follow-up survey of graduates. Ottawa. Author.
Retrieved on May 2, 2007 from
http://www.statcan.ca/Daily/English/070502/d070502c.htm.
Wilson, J. H. (2005). Essential Statistics. New Jersey: Pearson Prentice Hall
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