Measures of Dispersion

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Measures of Dispersion
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Measures of central tendency and relative
position tell only part of the story about
distributions.
Another perspective is the idea of variability,
or how much things differ from one another.
Viva la difference
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Comparison of Income Data for Two Cities
The average income for the two municipalities
is identical.
Centerville
Spreadsville
How would you describe these two cities in terms
of their differences?
But the distribution of wealth paints a
picture of two very different cities.
In one city, the Range in income indicates wide
variability while in the other, the spread is
much more narrow, indicating smaller Variance.
Hence, their names
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f Scores
Joe Bill 10 9 8 7 6 5 4 3
1 2 4 6 2 3 1 1
5 8 5 2
Lets play darts
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Terminology, AKA statistical jargon
Words like spread, scatter and dispersion have
been used to describe the tendency scores to
vary from person to person and to range from high
to low.
The standard term is variability, represented by
the statistic, variance, which tells us the size
of scores differ from the mean in the
distribution.
Within subjects variability is an indicator of
the tendency of a persons performance to
fluctuate from time to time - an indicator of
consistency. Joe is less consistent than Bill.
Between groups variability is an indication of
how samples from a common population differ from
one another, as in Spreadsville to Centerville.
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A statistical indicator of variability is the
Range, defined as the number of scale points in a
distribution from the highest to the lowest,
inclusive.
The range can be found by subtracting the
lowest score from the highest and adding 1.
Range hi - lo 1
The dart game scores ranged from three to 10,
which is a difference of 7, but from 3 thru 10,
there are 8 scale points, so the range is 8
But the range has limited value for
interpretation. It allowed us to only see that
the range in income between the two cities was
different.
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The best measure of dispersion would tie every
score to the same origin. If we could find the
average distance of every score from the mean, we
would have a standard yardstick for dispersion
for a given distribution of scores.
To do this, we compute the difference for each
score from the mean, sum those differences, and
divide by the number of scores to find the
average.
That would be a standard value for dispersion of
scores for the distribution, the standard
deviation
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_______ 9.8711 1.3059 .7348
8.1635 97.1645 26.1483 17.1627 160.8573
00.00
This is to illustrate the concept. It is not a
formula for computing standard deviation.
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Definitional Formula
Computational Formula
Raw Score Formula
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8.14 3.34
12.9