Download Normal Distribution

Statistical Distributions
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Describe the Normal Curve and its
characteristics
Describe the characteristics of skewed
distributions.
Define and differentiate among
measures of relative position, including
percentile ranks and standard scores.
Educational Research: Competencies for
Analysis and Application, 9th edition.
Gay, Mills, & Airasian
1
© 2009 Pearson Education, Inc.
All rights reserved.
The Normal Curve
Educational Research: Competencies for
Analysis and Application, 9th edition.
Gay, Mills, & Airasian
2
© 2009 Pearson Education, Inc.
All rights reserved.
The Normal Curve
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If a variable is normally distributed then
several things are true about the distribution
of the variable.
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Fifty percent of the scores are above the mean (µ)
and 50% are below the mean.
The mean, median, and mode have the same
value.
Most scores are near the mean.
Educational Research: Competencies for
Analysis and Application, 9th edition.
Gay, Mills, & Airasian
3
© 2009 Pearson Education, Inc.
All rights reserved.
The Normal Curve
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68.3% of all the scores fall within ± one
standard deviation of the mean.
More than 99% of the scores fall within
three standard deviations above and
below the mean.
Educational Research: Competencies for
Analysis and Application, 9th edition.
Gay, Mills, & Airasian
4
© 2009 Pearson Education, Inc.
All rights reserved.
The Normal Curve
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Skewed distributions
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When a distribution is not normally
distributed, it is said to be skewed.
A skewed distribution is not symmetrical.
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The mean, median, and mode are not the
same value.
The farther apart the mean and the median,
the more skewed the distribution.
Educational Research: Competencies for
Analysis and Application, 9th edition.
Gay, Mills, & Airasian
5
© 2009 Pearson Education, Inc.
All rights reserved.
The Normal Curve

Skewed distributions
Educational Research: Competencies for
Analysis and Application, 9th edition.
Gay, Mills, & Airasian
6
© 2009 Pearson Education, Inc.
All rights reserved.
The Normal Curve

Skewed distributions
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A negatively skewed distribution has
extreme scores at the lower end of the
distribution.
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Mean<Median<Mode
A positively skewed distribution has extreme
scores at the higher end of the distribution.
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Mean>Median>Mode
Educational Research: Competencies for
Analysis and Application, 9th edition.
Gay, Mills, & Airasian
7
© 2009 Pearson Education, Inc.
All rights reserved.
Measures of Relative Position
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Measures of relative position indicate where
a score falls in the distribution relative to all
the other scores.
Measures of relative position indicate how
well an individual has scored in comparison
to others in the distribution.
Measures of relative position express
different scores on a common scale.
Educational Research: Competencies for
Analysis and Application, 9th edition.
Gay, Mills, & Airasian
8
© 2009 Pearson Education, Inc.
All rights reserved.
Measures of Relative Position
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The two most frequently used measures
of relative position are percentile
ranks and standard scores.
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Percentile ranks indicate the percentage of
scores that fall at or below a given score.
Percentile ranks are appropriate for ordinal
data and are also used for interval data.
A percentile rank of 50 indicates the median
score.
Educational Research: Competencies for
Analysis and Application, 9th edition.
Gay, Mills, & Airasian
9
© 2009 Pearson Education, Inc.
All rights reserved.
Measures of Relative Position

A standard score uses standard deviation
units to express how far an individual
student’s test score is from the mean.
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i.e., a standard score reports how many
standard deviations a given score is from the
mean of a distribution.
Standard scores allow scores from
different tests to be compared on a
common scale and therefore allow for
calculations on those data.
Educational Research: Competencies for
Analysis and Application, 9th edition.
Gay, Mills, & Airasian
10
© 2009 Pearson Education, Inc.
All rights reserved.
Measures of Relative Position
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A z-score is the most basic and most
often used standard score.
A z-score is directly tied to the standard
deviation.
A score that represents the mean has a
z-score of 0.
A score at 1 standard deviation above the
mean has a z-score of 1.
Educational Research: Competencies for
Analysis and Application, 9th edition.
Gay, Mills, & Airasian
11
© 2009 Pearson Education, Inc.
All rights reserved.
Z Score
Educational Research: Competencies for
Analysis and Application, 9th edition.
Gay, Mills, & Airasian
12
© 2009 Pearson Education, Inc.
All rights reserved.
Measures of Relative Position
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To convert a raw score to a z-score we
use the following formula where X is
the raw score.
Z=
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The characteristics of the normal
distribution can be used to approximate
where a score falls based upon a
standard score.
Educational Research: Competencies for
Analysis and Application, 9th edition.
Gay, Mills, & Airasian
X-X
SD
13
© 2009 Pearson Education, Inc.
All rights reserved.
Measures of Relative Position
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A T-score is a standard score
sometimes used instead of a z-score.
A T-score is calculated by multiplying a
z-score by 10 and adding 50.
T-scores transform the scores such that
there are no negative values.
Educational Research: Competencies for
Analysis and Application, 9th edition.
Gay, Mills, & Airasian
14
© 2009 Pearson Education, Inc.
All rights reserved.