Download Measures of Variability

Chapter 3
Statistical Concepts
Assessment Procedures for Counselors and Helping Professionals, 7e
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Statistical Terms
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Statistics
Descriptive Statistics
Inferential Statistics
Sample
Population
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Scales of Measurement
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Nominal
Ordinal
Interval
Ratio
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Nominal Scale
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The nominal scale of measurement
describes qualitative (word) rather than
quantitative (number) differences.
Any item being measured can only fit into
one category.
Categories imply no level of ranking or
quantitative value.
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Ordinal Scale
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Example: 1st, 2nd, 3rd
Ordinal scales are similar to nominal scales
in that they classify discrete variables.
Unlike nominal scales, ordinal scales rank
variables in terms of magnitude.
Because ordinal scales are not quantitative
in nature, no assumption of equal intervals
between values is made.
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Interval and/or Ratio Scale
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Ranks variables by order of magnitude.
Unlike ordinal scales, interval scales
provide equally spaced levels between
variables.
Main difference between interval and ratio:
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Interval has no absolute zero
Ratio has absolute zero
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Examples of Scales of Measurement
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Describing Test Scores
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Distributions
Graphs
Measures of Central Tendency
Measures of Variability
The Normal Curve
Measures of Relationship
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Distributions
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Distribution – a set of scores
Frequency Distribution – a distribution
ranked by the number of responses for each
variable.
Graphs – Distributions can often be
represented on a graph.
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Graphical Presentations
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Histogram
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Frequency Polygon
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Smoothed Frequency Polygon
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Graphical Example of a Normal Curve
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Descriptors of Curves
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Symmetry
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Symmetrical – one side of the curve
mirrors the other
Asymmetrical – skew exists in the curve
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Skewness – the degree to which the distribution of a
curve is asymmetrical.
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Positive Skew - a distribution with an asymmetrical “tail”
extending out to the right.
Negative Skew - a distribution with an asymmetrical “tail”
extending out to the left.
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Kurtosis – a statistic that reflects the
peakedness or flatness of a distribution
relative to a normal distribution.
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Measures of Central Tendency
Measures of Central Tendency describe
distributions based on the average
performance of a test score.
Measures of Central Tendency are typically
represented through the mean, median,
and mode.
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Mean
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Average value for the distribution of scores.
Most common measure of central tendency
Used with interval and ratio scales
Calculated by totaling test scores and dividing
the sum by the number of individuals who
took the test.
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Median
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The middle score that divides a distribution
in half.
Used with ordinal, interval, or ratio scales
Helpful with highly skewed distributions.
Calculated by determining the center score in
a distribution. When there is an even number
of scores, the two middle scores are
averaged.
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Mode
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The most common score or value that appears
in a set of scores.
Used with nominal, ordinal, interval, and
ratio variables.
Calculated by counting the score that appears
most often in a distribution.
It is possible to have a bimodal (2 most
common scores) or multimodal (multiple
most common scores) distribution.
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Distribution Curves and Measures of
Central Tendency
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In a symmetrical (normal) curve, the values for the mean,
mode, and median are identical.
The mean can be impacted by outlying scores.
In asymmetrical distributions, the median may be the best
measure of central tendency.
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Measures of Variability
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Variability – the degree to which scores differ from one another.
Measures of Variability – the degree to which scores differ
from the mean. There are several methods for measuring
variability.
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Standard Deviation
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Most common measure of variability.
Provides an average distance of test scores
from the mean.
Larger standard deviations indicate greater
variance from the mean and greater variance
between scores.
Standard Deviation provides a sense of where
an individual score stands in relation to the
mean.
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The Normal Curve
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The normal curve, often referred to as the
normal distribution, represents the theoretical
distribution of any set of scores.
The normal curve is applied to many constructs
in counseling, such as memory and
intelligence.
Some areas in counseling are not normally
distributed, such as depression.
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Properties of the Normal Curve
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It is bell shaped.
It is bilaterally symmetrical, which means its two halves are identical.
The mean, median and mode are equal to one another.
The tails are asymptotic, meaning they approach but never touch the
baseline.
It is unimodal, which means that it has a single point of maximum
frequency or maximum height.
100% of the scores fall between -3 and +3 standard deviations from the
mean with
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approximately 68% of the scores falling between -1 and +1 standard deviations,
approximately 95% of the scores falling between -2 and +2 standard deviations,
and
approximately 99.5% of the scores falling between -3 and +3 standard
deviations.
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Illustration of a Normal Curve
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Graphical Example of a Normal Curve
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Standard Scores and T Scores with
Normal Curve
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Measures of Relationships Between
Variables
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Measures of Relationship
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Another important statistical measure in
testing involves the measurement of the
relationship between two variables.
For example, counselors may be interested
in the relationship between depression and
hours of sleep each evening.
Often this is measured with a correlation
coefficient
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Correlation Coefficient
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Correlation Coefficients range from -1.00 to
+1.00 and indicate the relationship between
two variables.
The direction of a correlation coefficient is
either positive, indicating that when a score in
one variable goes up that the score in the other
variable will also go up, or negative, indicating
that when a score on one variable goes up that
a score in the other variable will go down.
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Correlation Coefficient, Cont.
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The strength of a correlation is indicated by
the numeric value of the coefficient. A 1.00
correlation, negative or positive, indicates a
perfect relationship between two variables. A
zero indicates no relationship between two
variables.
Correlation does not imply causation!
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