Using SPSS to calculate summary statistics

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Using SPSS to calculate summary statistics
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SPSS - summary statistics

• In the data view
• Enter the scores from problem 4, page 66
• Analyze-gtDescriptive Statistics-gtFrequencies
• Select VAR00001 for analysis
• Click Statistics
• Choose the Central Tendencies you want
• Continue
• OK
• Use SPSS to check your work on problem 6 (except
  for SS and mean deviation).
• Do you get the same answers?

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Chapter 3B
More on summary statistics
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Other ways to look at the mean and the variance

• Weighted average (mean)
• Efficient formula for the variance

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The weighted mean

• Suppose that one had repeating scores in a sample
  to be averaged
• Example - a students grades in a 4.0 grading
  system
• 2,3,3,3,3,3,4,4
• One could find the mean directly, but this would
  be somewhat redundant
• Instead, group like score together and weight
  each by the frequency of each score

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The weighted mean

• An even better motivation for using the weighted
  mean is that one might want to combine means from
  different studies
• We cant just average the means
• The means from larger samples must have a greater
  weight
• How much weight?
• Answer the sample size
• Suppose we have 3 studies with means
• And sample sizes
• The resulting combined mean is then

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Computationally efficient formula for the
variance

• Not absolutely necessary
• But, saves time in hand calculations
• Is used by the author
• Most of the work is in computing SS.
• We will focus there.

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Deriving the shortcut SS

• Sum of squares is the numerator of the variance.
• We will often have reason to handle it
  separately.

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Deriving the shortcut SS

• Start with
• .
• .
• .
• .
• .
• End with

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Using the shortcutSS to find the variance and
standard deviation

• Recall that the variance is SS/N for the
  population
• And the unbiased variance is SS/(N-1) for the
  sample
• And the standard deviation is just the square
  root of the variance
• When calculating for the sample, we will almost
  always be using the unbiased expression

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Properties of the meanDevelop your intuition
about the mean

• If a constant C is added to every score in a
  sample, the new mean is C the original mean
• If every score is multiplied by a constant C,
  then the new mean will be C the original mean
• This can be deduced by the properties of
  summation
• The sum of the deviations from the mean will
  always be zero
• The sum of the squared deviations from the mean
  (SS) will be less than the sum of the squared
  deviations around any other point in the
  distribution
• Called least squared property
• Important when fitting a straight line to a cloud
  of points

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Properties of the standard deviationDevelop your
intuition about standard deviation

• If a constant C is added to every score in a
  sample, the new standard deviation is the same as
  the original mean
• If every score is multiplied by a constant C,
  then the new standard deviation will be C the
  original standard deviation
• This can be deduced by the properties of
  summation
• The standard deviation from the mean will be
  smaller than the standard deviation from any
  other point in the distribution
• Follows from the related property of the mean

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Exercises

• Page 76 1-4, 5 a b, 6-10

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Chapter 4

• Standardized scores and the normal distribution

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Evaluating a single score within a distribution

• The numerical distance between a score and the
  mean may not be very meaningful
• However, if that distance can be expressed in
  units of standard deviation, then we have a much
  better understanding of the relationship between
  the score and the rest of the data

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Z scores

• Such a scaled distance is called the z score
• We can replace our X scores with the z scores
• Computed like so.

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Properties of the z scores

• Mean of the z scores is 0.
• From the properties of the mean
• Subtracting a constant from each score shifts the
  mean by the constant
• Multiplying each score by a constant multiplies
  the mean by that constant.

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This gives us a simplified formula for the
standard deviation
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The standard deviation of a population of z
scores is always 1!

• From properties of standard deviation
• Adding a constant to each score does not change
  the standard deviation
• Multiplying each score by a constant multiplies
  the standard deviation by that constant.

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Normal distributions, standard normal
distributions and z scores revisited

• What means are zero?
• Only the mean of a standardized distribution is
  guaranteed to be zero
• Normal distributions can have non-zero means
• What is the purpose of a z score?
• To permit the score to be inserted into a
  standard normal distribution for comparison
• Why do we want to insert our score into a
  standard normal distribution?
• If we dont use the standard distribution, we
  need lots of normal distribution tables (an
  infinite number, actually)
• What would happen if we had a distribution of z
  scores?
• It would be the standard normal distribution

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Limitations of z scores

• By translating the set of scores by the mean
• And scaling by the standard deviation
• We can compare two different distributions
• But not if they are skewed differently

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The normal distribution

• Comparing scores in distributions from the same
  family overcomes this problem
• The normal distribution is such a family of
  distributions
• Why is it a family of distributions?

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The standard normal distribution

• Generic parameters
• Mean set to zero
• Standard deviation set to 1

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Probability that a score falls between any two
values

• Recall that a probability distribution represents
  the relative probability of various scores
• And that the total area under a probability
  distribution is 1
• Hence the probability that a score falls in any
  interval is the area under the corresponding
  part of the curve
• The table of the standard normal deviation
  contains the cumulative area under the curve,
  from the mean outward

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Distribution of sample means

• What if we sample means rather than individuals?
• Take a sample
• Find the mean
• Use that as a score
• Form a distribution of such scores
• This is called a Distribution of sample means

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Properties of the distribution of sample means

• If the underlying distribution is normal, the
  sampling distribution will be normal
• The mean of the sampling distribution will tend
  to be the same as the mean of the population (as
  the number of means approaches infinity)
• Groups vary less than individuals
• Therefore the standard deviation of the sampling
  distribution is less than the standard deviation
  of the population
• This value is called the standard error

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Why do we care about the distribution of sample
means?

• Because when we take the mean of a sample, we
  want to know how good of an estimate it is of the
  population mean
• If the means vary a lot from sample to sample,
  then the estimate is not very good
• If the means vary little, then the estimate is
  good
• We may never actually do repeated sampling, we
  just wanted to come up with this equation(!),
  which tells us how the sample size improves our
  estimation of the mean

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Exercises

• Page 103
• 1,7,8,10,12

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Exercises

• Page 103
• 1,7,8,10,12