Social Science Reasoning Using Statistics

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Social Science Reasoning Using Statistics

• Psychology 138
• Spring 2004

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Project 1

• What
• Read and summarize a journal article
• See PIP packet for more details
• Durgin, Fox, Kim (2003). Not letting the left
  leg know what the right leg is doing
  Limb-specific locomotor adaptation to sensory-cue
  conflict
• Where class reserves, can download
• http//www.mlb.ilstu.edu
• Due Friday, Feb 27 by 430.

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Descriptive statistics

• In addition to pictures of the distribution,
  numerical summaries are also typically presented.
  
• Numeric Descriptive Statistics
• Shape skew and kurtosis
• Measures of Center mean, median, mode
• Measures of Variability (Spread) range,
  Inter-quartile range, standard deviation (
  variance)

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Variability of a distribution

• Variability provides a quantitative measure of
  the degree to which scores in a distribution are
  spread out or clustered together.
• In other words variabilility refers to the degree
  of differentness of the scores in the
  distribution.

• High variability means that the scores differ
  by a lot

• Low variability means that the scores are all
  similar (homogeneousness).

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Range

• The simplest measure of variability is the range,
  which weve already mentioned in our earlier
  discussions.
• Range Maximum value - minimum value
• there are some drawbacks of using the range as
  the description of the variability of a
  distribution
• the statistic is based solely on the two most
  extreme values in the distribution

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Interquartile range

• An alternative measure of variability is the
  inter-quartile range.
• Median (50tile) equals the point at which
  exactly half the distribution exists on one side
  and the other half on the other side.
• Considering the same logic
• What does the 25tile represent?
• The 75?

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Interquartile range

• The inter-quartile range is the distance between
  the first quartile and the third quartile. So
  this corresponds to the middle 50 of the scores
  of our distribution.

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Interquartile range

• Consider a frequency distribution table

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Interquartile range

• The inter-quartile range focuses on the middle
  half of all of the scores in the distribution.
• Thus it is more representative of the
  distribution as a whole compared to the range
• Extreme scores (i.e., outliers) will not
  influence the measure (sometimes referred to as
  being robust).
• However, this still means that all of the scores
  in the distribution are not represented in the
  measure (only the middle 50).

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Standard deviation

• The standard deviation is the most popular and
  most important measure of variability.
• In essence, the standard deviation measures how
  far off all of the individuals in the
  distribution are from a standard, where that
  standard is the mean of the distribution.
  Essentially, the average of the deviations.

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Computing standard deviation (population)

• Step 1 To get a measure of the deviation we need
  to subtract the population mean from every
  individual in our distribution.

Our population
2, 4, 6, 8
X - ? deviation scores
2 - 5 -3
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Computing standard deviation (population)

• Step 1 To get a measure of the deviation we need
  to subtract the population mean from every
  individual in our distribution.

Our population
2, 4, 6, 8
X - ? deviation scores
2 - 5 -3
4 - 5 -1
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Computing standard deviation (population)

• Step 1 To get a measure of the deviation we need
  to subtract the population mean from every
  individual in our distribution.

Our population
2, 4, 6, 8
X - ? deviation scores
2 - 5 -3
6 - 5 1
4 - 5 -1
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Computing standard deviation (population)

• Step 1 To get a measure of the deviation we need
  to subtract the population mean from every
  individual in our distribution.

Our population
2, 4, 6, 8
X - ? deviation scores
2 - 5 -3
6 - 5 1
Notice that if you add up all of the deviations
they must equal 0.
4 - 5 -1
8 - 5 3
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Computing standard deviation (population)

• Step 2 So what we have to do is get rid of the
  negative signs. We do this by squaring the
  deviations and then taking the square root of the
  sum of the squared deviations (SS).

SS ? (X - ?)2
(-3)2
(-1)2
(1)2
(3)2
9 1 1 9 20

• Note The above formula is sometimes referred to
  as the definitional formula for SS. There is
  another formula called the computational formula
  (see the book). The advantage of the
  computational formula is that it works with the X
  values directly.

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Computing standard deviation (population)

• Step 3 Now we have the sum of squares (SS), but
  to get the Variance which is simply the average
  of the squared deviations
• we want the population variance not just the SS,
  because the SS depends on the number of
  individuals in the population, so we want the
  mean
• So to get the mean, we need to divide by the
  number of individuals in the population.

variance ?2 SS/N
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Computing standard deviation (population)

• Step 4 However the population variance isnt
  exactly what we want, we want the standard
  deviation from the mean of the population. To
  get this we need to take the square root of the
  population variance.

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Computing standard deviation (population)

• To review
• Step 1 compute deviation scores
• Step 2 compute the SS
• either by using definitional formula or the
  computational formula
• Step 3 determine the variance
• take the average of the squared deviations
• divide the SS by the N
• Step 4 determine the standard deviation
• take the square root of the variance

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Computing standard deviation (sample)

• The basic procedure is the same.
• Step 1 compute deviation scores
• Step 2 compute the SS
• Step 3 determine the variance
• This step is different
• Step 4 determine the standard deviation

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Computing standard deviation (sample)

• Step 1 Compute the deviation scores
• subtract the sample mean from every individual in
  our distribution.

2 - 5 -3
6 - 5 1
4 - 5 -1
8 - 5 3
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Computing standard deviation (sample)

• Step 2 Determine the sum of the squared
  deviations (SS).

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Computing standard deviation (sample)

• Step 3 Determine the variance

Recall
Population variance ?2 SS/N
The variability of the samples is typically
smaller than the populations variability
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Computing standard deviation (sample)

• Step 3 Determine the variance

Recall
Population variance ?2 SS/N
The variability of the samples is typically
smaller than the populations variability
To correct for this we divide by (n-1) instead of
just n
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Computing standard deviation (sample)

• Step 4 Determine the standard deviation

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Characteristics of a standard deviation

• Change/add/delete a given score, then the
  standard deviation will change.

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Characteristics of a standard deviation

• Change/add/delete a given score, then the
  standard deviation will change.

• Add/subtract a constant to each score, then the
  standard deviation will NOT change.

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Characteristics of a standard deviation

• Change/add/delete a given score, then the
  standard deviation will change.

• Add/subtract a constant to each score, then the
  standard deviation will NOT change.

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Characteristics of a standard deviation

• Change/add/delete a given score, then the
  standard deviation will change.

• Add/subtract a constant to each score, then the
  standard deviation will NOT change.

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Characteristics of a standard deviation

• Change/add/delete a given score, then the
  standard deviation will change.

• Add/subtract a constant to each score, then the
  standard deviation will NOT change.

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Characteristics of a standard deviation

• Change/add/delete a given score, then the
  standard deviation will change.

• Add/subtract a constant to each score, then the
  standard deviation will NOT change.

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Characteristics of a standard deviation

• Change/add/delete a given score, then the
  standard deviation will change.

• Add/subtract a constant to each score, then the
  standard deviation will NOT change.

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Characteristics of a standard deviation

• Change/add/delete a given score, then the
  standard deviation will change.

• Add/subtract a constant to each score, then the
  standard deviation will NOT change.

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Characteristics of a standard deviation

• Change/add/delete a given score, then the
  standard deviation will change.

• Add/subtract a constant to each score, then the
  standard deviation will NOT change.

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Characteristics of a standard deviation

• Change/add/delete a given score, then the
  standard deviation will change.

• Add/subtract a constant to each score, then the
  standard deviation will NOT change.

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Characteristics of a standard deviation

• Change/add/delete a given score, then the
  standard deviation will change.

• Add/subtract a constant to each score, then the
  standard deviation will NOT change.

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Characteristics of a standard deviation

• Change/add/delete a given score, then the
  standard deviation will change.

• Add/subtract a constant to each score, then the
  standard deviation will NOT change.

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Characteristics of a standard deviation

• Change/add/delete a given score, then the
  standard deviation will change.

• Add/subtract a constant to each score, then the
  standard deviation will NOT change.

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Characteristics of a standard deviation

• Change/add/delete a given score, then the
  standard deviation will change.

• Add/subtract a constant to each score, then the
  standard deviation will NOT change.

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Characteristics of a standard deviation

• Change/add/delete a given score, then the
  standard deviation will change.

• Add/subtract a constant to each score, then the
  standard deviation will NOT change.

1 - 4 -3
5 - 4 1
3 - 4 -1
7 - 4 3
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Characteristics of a standard deviation

• Change/add/delete a given score, then the mean
  will change.

• Add/subtract a constant to each score, then the
  standard deviation will NOT change.

• Multiply (or divide) each score by a constant,
  then the standard deviation will change by being
  multiplied by that constant.

21 - 22 -1
(-1)2
23 - 22 1
(1)2
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Characteristics of a standard deviation

• Change/add/delete a given score, then the mean
  will change.

• Add/subtract a constant to each score, then the
  standard deviation will NOT change.

• Multiply (or divide) each score by a constant,
  then the standard deviation will change by being
  multiplied by that constant.

42 - 44 -2
(-2)2
46 - 44 2
(2)2
s
Sold1.41
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When to use which

• Extreme scores range is most affected, IQR is
  least affected
• Sample size range tends to increase as n
  increases, IQR s do not
• The range does not have stable values when you
  repeatedly sample from the same population, but
  the IQR S are stable and tend not to fluctuate.
• With open-ended distributions, one cannot even
  compute the range or S, so the IQR is the only
  option