Transcript weiten6_PPTAppB

Appendix B
Statistical Methods
Statistical Methods: Graphing Data
 Frequency
distribution
 Histogram
 Frequency
polygon
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B.1 - Graphing data. (a) Our raw data are tallied into a frequency distribution. (b)
The same data are portrayed in a bar graph called a histogram. (c) A frequency
polygon is plotted over the histogram. (d) The resultant frequency polygon is shown
by itself.
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Descriptive Statistics
 Measures
of Central Tendency
 Mean
 Median
 Mode
 Skewed
Distributions
 Negative/Positive
 Measuring Variability
 Standard Deviation
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B.2 - Measures of
central tendency.
The mean, median,
and mode usually
converge, as in the
case of our TV
viewing data.
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B.3 - Measures of central tendency in skewed distributions. In a
symmetrical distribution (a), the three measures of central tendency converge.
However, in a negatively skewed distribution (b) or in a positively skewed
distribution (c), the mean, median, and mode are pulled apart as shown here.
Typically, in these situations the median provides the best index of central
tendency.
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B.4 - The standard
deviation and dispersion
of data. Although both
these distributions of golf
scores have the same
mean, their standard
deviations will be different.
In (a) the scores are
bunched together and there
is less variability than in (b),
yielding a lower standard
deviation for the data in
distribution (a).
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B.5 - Steps in calculating the
standard deviation. (1) Add the
scores (X) and divide by the
number of scores (N) to calculate
the mean (which comes out to 3.0
in this case). (2) Calculate each
score’s deviation from the mean
by subtracting the mean from
each score (the results are shown
in the second column). (3) Square
these deviations from the mean
and total the results to obtain
(d2) as shown in the third
column. (4) Insert the numbers for
N and d2 into the formula for the
standard deviation and compute
the results.
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The Normal Distribution
 Psychological
tests
– Relative measures
– Standard deviation the unit of measure
 Conversion
to percentile scores
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B.6 - The normal distribution. Many characteristics are distributed in a pattern represented by this bellshaped curve (each dot represents a case). The horizontal axis shows how far above or below the mean a
score is (measured in plus or minus standard deviations). The vertical axis shows the number of cases
obtaining each score. In a normal distribution, most cases fall near the center of the distribution, so that
68.26% of the cases fall within plus or minus 1 standard deviation of the mean. The number of cases
gradually declines as one moves away from the mean in either direction, so that only 13.59% of the cases
fall between 1 and 2 standard deviations above or below the mean, and even fewer
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cases (2.14%) fall between 2 and 3 standard deviations above or below the mean.
B.7- The normal distribution and SAT scores. The normal distribution is the basis for the
scoring system on many standardized tests. For example, on the Scholastic Aptitude Test
(SAT), the mean is set at 500 and the standard deviation at 100. Hence, an SAT score tells
you how many standard deviations above or below the mean you scored. For example, a
score of 700 means you scored 2 standard deviations above the mean.
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Measuring Correlation
 Correlation
coefficient
– Positive = direct relationship
– Negative = inverse relationship
 Magnitude:
0 to plus/minus 1
 Scatter diagrams
 Correlation of determination
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B.8 - Scatter diagrams of positive and negative correlations. Scatter
diagrams plot paired X and Y scores as single points. Score plots slanted in the
opposite direction result from positive (top row) as opposed to negative (bottom
row) correlations. Moving across both rows (to the right), you can see that
progressively weaker correlations result in more and more scattered plots of data
points.
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B.9 - Scatter diagram of the correlation between TV viewing and SAT scores. Our
hypothetical data relating TV viewing to SAT scores are plotted in this scatter diagram.
Compare it to the scatter diagrams seen in Figure B.8 and see whether you can estimate
the correlation between TV viewing and SAT scores in our data (see the text for the
answer).
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B.10 - Computing a
correlation
coefficient. The
calculations required
to compute the
Pearson productmoment coefficient of
correlation are shown
here. The formula
looks intimidating, but
it’s just a matter of
filling in the figures
taken from the sums
of the columns shown
above the formula.
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B.11 - Correlation and the coefficient of determination. The coefficient of
determination is an index of a correlation’s predictive power. As you can see,
whether positive or negative, stronger correlations yield greater predictive
power.
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Hypothesis Testing
 Inferential
statistics
 Sample
 Population
 Null
hypothesis vs. research hypothesis
 Statistical significance
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B.12 - The relationship
between the population
and the sample. In
research, we are usually
interested in a broad
population, but we can
observe only a small
sample from the
population. After making
observations of our
sample, we draw
inferences about the
population, based on the
sample. This inferential
process works well as
long as the sample is
reasonably representative of the population.
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