Seth M' Noar, Ph'D'

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DistributionsPart 1

• Seth M. Noar, Ph.D.
• Department of Communication
• University of Kentucky

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Measurement

• Last week we focused primarily on measurement
• But once we have measured variables, what is the
  best way to summarize / depict the data?

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Distributions

• Data can be presented in various ways, and one
  way to summarize / discuss data is in terms of
  their distribution
• Distribution a collection of measurements
  usually viewed in terms of the frequency with
  which observations are assigned to each category
  or point on a measurement scale (W M, p. 31)

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Distributions and Measurement

• Measurement related to distributions
• Categorical vs. continuous data will present
  different distributions and different options for
  presenting data
• (See W M, p. 34, for examples of tables and
  figures)

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Summarizing Data / Distributions

• Measures of Central Tendency tell us about the
  central tendency of scores in a distribution
• Measures of Dispersion (Variability) tell us
  about the scatter or dispersion of scores

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Measures of Central Tendency

• Mean arithmetic average
• 5362 16 / 4 4
• Median midpoint
• 3 4 5 6 7
• Mode most frequent score
• 2 2 4 4 5
• Mode is 2

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Measures of Central Tendency

• Very handy for describing how scores cluster
  toward the center.
• Case of normal distribution is simple.
• However, MCT can be misleading.
• Why is this?
• Lying with statistics example

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MCT Your Income

• 28,000, 57,000, 59,000, 178,000
• Mean income 80,500
• Median income 58,000
• Conclusion Mean is sensitive to extreme scores
• What other measures can help us describe
  distributions?

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Measures of Dispersion

• Dispersion (variability) the degree to which
  scores are distributed around the mean
• Range highest minus lowest score
• 178,000-28,000
• 150,000
• Variance the mean of the squared deviation
  scores about the mean of a distribution
• Standard Deviation (SD) square root of the
  variance

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SD Example

• Exam scores 75, 80, 85, 90, 90, 95
• What is mean, median, mode?
• What is range?
• In order to calculate SD, we must calculate
• Deviation scores
• Sum of squares
• Variance

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Example (contd)

• Deviation scores
• 75 85.8 -10.8
• 80 85.8 -5.8
• 85 85.8 -.8
• 90 85.8 4.2
• 90 85.8 4.2
• 95 85.8 9.2
• Why not just average the deviation scores?

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Example (contd)

• Squared deviation scores
• (-10.8)2 116.6
• (-5.8)2 33.6
• (-.8)2 .64
• (4.2)2 17.6
• (4.2)2 17.6
• (9.2)2 84.6
• Sum of squares 270.6

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Example (contd)

• The variance
• Sum of squares / N-1
• 270.6 / 5 54.1
• Is 54.1 useful?
• Standard deviation
• Square root of variance
• SD 7.4

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More about SD

• We can think of it (loosely) as the average
  deviation from the mean
• Comparing SDs between distributions is useful.
  A SD of 7.4 versus 21.8.
• With normal distribution, 2/3 of scores tend to
  be within 1 SD of the mean
• 85.8 ( or ) 7.4
• Scores 75, 80, 85, 90, 90, 95
• Concept of variability / variance very important
  in statistics. We will see it again and again

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Skewness and Kurtosis

• Skewness the degree to which a distribution is
  asymmetrical
• Kurtosis the degree of peakedness (versus
  flatness) of a distribution
• Both of these are examined in relation to a
  normal curve
• If we know something about either of these, it
  can help us to understand a distribution

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Skewness and Kurtosis

• We can calculate skew and kurtosis values
• Distributions that are skewed or kurtotic can
  cause problems for varying statistical procedures
• Many statistics we use assume normality of
  distribution
• Skewness and Kurtosis should be examined before
  statistics are run

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18
DistributionsPart 2
Seth M. Noar, Ph.D. Department of Communication
University of Kentucky
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Populations and Samples

• Parameter characteristic of a population
• Statistic characteristic of a sample
• Often, we are trying to infer parameters from
  statistics (statistical inference)
• Related to this, we can examine the
• Population distribution
• Sample distribution
• Sampling distribution frequency with which
  values of a statistic (e.g, M) would be expected
  when sampling randomly from a population

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Distributions

• INSERT figure 5.1 (page 51)
• Talk about population and sample distributions.

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Sampling Distribution

• INSERT figure 5.2 (page 53)
• Talk about sampling distributions.

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Normal Curve

• Definition of a particular functional
  relationship between deviations about the mean of
  a distribution and the probability of these
  different deviations occurring.
• It is a theoretical curve
• However, we often assume that many populations
  have this distribution

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Normal Curve

• INSERT figure 5.4 (page 55)
• Talk about normal curve probabilities.

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Standard Error

• By understanding these distributions, we can
  estimate various types of error (of the mean, SD,
  etc.)
• Standard error of the mean standard deviation of
  a distribution of sample means

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Normal Curve as Sampling Distribution

• INSERT figure 5.6 (page 57)
• Talk about normal curve, sampling distribution,
  and standard error of the mean.

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Practical Situations

• In standard research, we do not know what the
  actual population parameters are (if we did we
  wouldnt need samples!)
• Thus, we sometimes take the sample statistics as
  our best guess of the population parameters
• However, based on the logic of distributions, we
  can extrapolate more information related to
  sampling error.
• Sampling error estimate of how sample statistics
  differ from population parameters

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Sample Example

• INSERT figure 5.7 (page 59)
• Talk about what we can learn from this example
  going from sample to population.

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Summary

• Sample statistics are always estimates of
  population parameters.
• We can accept sample statistics as the best
  representation of population parameters.
• However, because of sampling error, it is
  unlikely that our statistics are exact.
• The procedures outlined here give us some
  guidance as to how we can estimate the likelihood
  of error related to our sampling statistics.