Figure 4.6 (page 119) Typical ways of presenting frequency graphs and descriptive statistics

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Figure 4.6 (page 119)Typical ways of presenting
frequency graphs and descriptive statistics
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Sample variance as an unbiased statistic

• Sample variance s2 is an unbiased estimator for
  the population variance s2 of the underlying
• Average value of the sample variance will equal
  the population value
• Sample standard deviation is biased but is still
  a good estimate of population

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Table 4.1 (p. 120) The set of all the possible
samples for n 2 selected from the population
described in Example 4.7. The mean is computed
for each sample, and the variance is computed two
different ways (1) dividing by n, which is
incorrect and produces a biased statistic and
(2) dividing by n 1, which is correct and
produces an unbiased statistic. From a population
of scores 0,0,3,3,9,9 with m 4 and s2 14.
36/9 4 63/9 7.0 126/9 14
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Factors that Affect Variability

1. Extreme scores
2. Sample size
3. Stability under sampling
4. Open-ended distributions

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Mean and Standard Deviation as Descriptive
Statistics

• If you are given numerical values for the mean
  and the standard deviation, you should be able to
  construct a visual image (or a sketch) of the
  distribution of scores.
• As a general rule, about 70 of the scores will
  be within one standard deviation of the mean, and
  about 95 of the scores will be within a distance
  of two standard deviations of the mean.
• It is common to talk about descriptive statistics
  as the mean plus or minus the standard deviation.
• For example 36 or - 4 as shown in figure
  4.7.

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Figure 4.7 (page 122) Caution the data
distribution needs to be symmetrical for this to
work.Imagine what this looks like with extremely
skewed data.
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Normal Curve with Standard Deviation
? or - one s.d. ?
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Properties of the Standard Deviation

• If a constant is added to every score in a
  distribution, the standard deviation will not be
  changed.
• If you visualize the scores in a frequency
  distribution histogram, then adding a constant
  will move each score so that the entire
  distribution is shifted to a new location.
• The center of the distribution (the mean)
  changes, but the standard deviation remains the
  same.

Add 5 points to each score
M 2.5 s 1.05
M 7.5 s 1.05
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Properties of the Standard Deviation

• If each score is multiplied by a constant, the
  standard deviation will be multiplied by the same
  constant.
• Multiplying by a constant will multiply the
  distance between scores, and because the standard
  deviation is a measure of distance, it will also
  be multiplied.

Multiply each score by 5
M 2.5 s 1.05
M 12.5 s 5.24
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Properties of the Standard Deviation

• Transformation of scale
• Can change the scale of a set of score by adding
  a constant
• Exam with a mean of 87
• Add 13 to every score
• Exam mean changes to 100
• Can change the scale by multiplying by a constant
• Exam with standard deviation of 5
• Multiply each score by 2
• Exam mean changes to 10

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Table 4.2 (p. 124) The number of aggressive
responses of male and female children after
viewing cartoons. Example of APA tale format when
reporting descriptive data. Did type of cartoon
make a difference on aggressive responses? How do
males and females differ?
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Figure 4-8 (p. 125) Using variance in
inferential statisticsGraphs showing the results
from two experiments. In Experiment A, the
variability within samples is small and it is
easy to see the 5-point mean difference between
the two samples. In Experiment B, however, the
5-point mean difference between samples is
obscured by the large variability within samples.
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Chapter 5 z-scores Location of Scores and
Standardized Distributions
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2014 Boston Marathon, Wheelchair Race

• Ernst Van Dyk (RSA) and Tatyana McFadden (USA)
  won the mens and womens titles at the 118th
  B.A.A. Boston Marathon. Capturing an
  unprecedented tenth Boston Marathon title, Van
  Dyk led from wire-to-wire. McFadden celebrated
  her 25th birthday by defending her title in the
  womens race, just eight days after winning and
  breaking her own course record at the London
  Marathon.
• http//www.baa.org/news-and-press/news-listing/201
  4/april/2014-boston-marathon-push-rim-wheelchair-r
  ace.aspx
• Using SPSS to get standardized scores

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z-Scores and Location

• By itself, a raw score or X value provides very
  little information about how that particular
  score compares with other values in the
  distribution.
• To interrupt a score of X 76 on an exam need
  to know
• mean for the distribution
• standard deviation for the distribution
• Transform the raw score into a z-score,
• value of the z-score tells exactly where the
  score is located
• process of changing an X value into a z-score
• sign of the z-score ( or ) identifies whether
  the X value is located above the mean (positive)
  or below the mean (negative)
• numerical value of the z-score corresponds to the
  number of standard deviations between X and the
  mean of the distribution.

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Two distributions of exam scores. For both
distributions, µ 70, but for one distribution,
s 12. The position of X 76 is very different
for these two distributions.
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Figure 5.2 page 140 With a box for individual
scores. Two distributions of exam scores. For
both distributions, µ 70, but for one
distribution, s 12. The position of X 76 is
very different for these two distributions.
A
B
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z-Scores and Location

• The process of changing an X value into a z-score
  involves creating a signed number, called a
  z-score, such that
• The sign of the z-score ( or ) identifies
  whether the X value is located above the mean
  (positive) or below the mean (negative).
• The numerical value of the z-score corresponds
  to the number of standard deviations between X
  and the mean of the distribution.

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Z-scores and Locations

• Visualize z-scores as locations in a
  distribution.
• z 0 is in the center (at the mean)
• extreme tails
• 2.00 on the left
• 2.00 on the right
• most of the distribution is contained between z
  2.00 and z 2.00
• z-scores identify exact locations within a
  distribution
• z-scores can be used as
• descriptive statistics
• inferential statistics

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Figure 5-3 (p. 141)The relationship between
z-score values and locations in a population
distribution.
One S.D.
Two S.D.