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The Statistical Imagination
• Chapter 5. Measuring
Dispersion or Spread in a
Distribution of Scores
© 2008 McGraw-Hill Higher Education
Measuring Dispersion in a
Score Distribution
• Dispersion: How the scores of an
interval/ratio variable are spread out
from lowest to highest and the shape
of the distribution in between
• The most commonly used dispersion
statistics are the range and standard
deviation
© 2008 McGraw-Hill Higher Education
The Range
• The range is an expression of
how the scores of an interval/ratio
variable are distributed from
lowest to highest
• It is the distance between the
minimum and maximum scores in
a sample
© 2008 McGraw-Hill Higher Education
Calculating the Range
• 1. Rank the scores from lowest to
highest
• 2. Identify the minimum and maximum
scores
• 3. Identify the value of the rounding unit
• Range = (Maximum score - Minimum
score) + the value of the rounding unit
© 2008 McGraw-Hill Higher Education
Limitations of the Range
• The range is greatly affected by
outliers
• The range has a narrow informational
scope. It provides the width of a
distribution of scores, but tells us
nothing about how they are spread
between the maximum and minimum
scores
© 2008 McGraw-Hill Higher Education
The Standard Deviation
• The standard deviation describes how scores
are spread across the distribution in relation
to the mean score
• It provides a standard unit of comparison – a
common unit of measure for comparing
variables with very different observed units of
measure
• Its computation centers on how far each score
is from the mean – how far it “deviates”
© 2008 McGraw-Hill Higher Education
Calculating the Standard
Deviation
• Sort givens and calculate the mean and
deviation scores
• Sum the deviation scores and verify a
result of zero
• Square the deviation scores and sum them
to obtain the variation or “sum of squares”
• Divide the variation by n - 1 to get the
variance
• Take the square root of the variance to get
the standard deviation
© 2008 McGraw-Hill Higher Education
The Elements of the Standard
Deviation (See Table 5-1)
• We square deviation scores to remove
negative signs and to obtain a sum other
than zero
• We divide the sum of squares by n - 1 to
adjust for sample size and sampling error
• We take the square root of the variance to
obtain directly interpretable units of
measure (units instead of squared units)
© 2008 McGraw-Hill Higher Education
Limitations of the Standard
Deviation
• The standard deviation is greatly
inflated by outliers
• It can be misleading if the
distribution is skewed
© 2008 McGraw-Hill Higher Education
Three Ways to Express the
Value of a Score, X
1. As a raw score – the observed value
of X in its original units of measure
such as inches
2. As a deviation score – the difference
between a raw score and its mean,
also in original units of measure
3. As a standardized score (Z-score) –
as a number of standard deviations
(SD) from the mean
© 2008 McGraw-Hill Higher Education
Standardized Scores or
Z-scores
• Z-scores express a raw score as
a number of standard deviations
(SD) from the mean score
• Divide the deviation score by the
standard deviation to produce a
measure of X in standard
deviation units
© 2008 McGraw-Hill Higher Education
The Standard Deviation Is a
Part of the Normal Curve
• For any normally distributed variable:
• 99.7% of cases fall within 3 SD of the
mean in both directions
• About 95%, within 2 SD of the mean
in both directions
• About 68%, within 1 SD of the mean
in both directions
© 2008 McGraw-Hill Higher Education
Using the Normal Curve to
Partition Areas
• If a variable is distributed
normally, we can use sample
statistics and what we know
about the normal curve to
estimate how many scores in a
population fall within a certain
range
© 2008 McGraw-Hill Higher Education
Statistical Follies
• Comparing the relative sizes of the
mean and standard deviation is a
good way to detect skews
• When the calculated standard
deviation is larger than the mean
for the variable, the distribution is
skewed or otherwise oddly shaped
© 2008 McGraw-Hill Higher Education