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                                   Chapter 12

                               Descriptive Statistics

 

Outline

 

Concepts

 

I.  Statistics in
    Communication
    Research
descriptive statistics: numbers that characterize some information
inferential statistics: tools that help researchers draw conclusions about the probable populations from which samples did or did not belong
    A.  Measures of
         Central Tendency
         (Arithmetic Mean,
         Median, and Mode)
measures of central tendency: measures that describe what is going  on within sample groups or populations on the average
arithmetic mean: (the number most people call "the average") the sum of a set of scores divided by the number of scores
--unbiased estimator: a sample statistic that
  is likely to approximate the population
  parameter
median: a score that appears in the middle of an ordered list of scores
mode: the most commonly occurring score (bimodal: the condition that occurs when data have two modes)
    B.  Measures of Variability
          or Dispersion        


         1.   Range range: the difference between the highest and lowest scores (range is greatly affected by extreme scores)
         2.   Variance variance: the average of squared differences of scores from the mean (abbreviated for sample variance s2 and for the population variance s2 [sigma squared])
         3.   Standard Deviation standard deviation: a measure of the average of how far the scores deviate from the mean (abbreviated s for the sample standard deviation and s for the population standard deviation)
II.  Distributions
    A.  Nonnormal and Skewed
          Distributions
          1.   Types of Skew
                (negative or
                leftward skew;
                positive or rightward
                skew)
skew: a measure of centeredness (skewness reveals the side of the distribution in which the longer "tail" lies)
           2.  Peakedness of
                Distributions
kurtosis: a measure of peakedness (in a perfect normal distribution, the distribution is as high as three standard deviations is wide)
platykurtic: a very flat distribution
mesokurtic: a peaked distribution that is neither very high nor very low
leptokurtic: an extremely peaked distribution
    B.  Standard Normal
          Distribution
the standard normal curve: a probability distribution that tells the expected value that would be obtained by sampling at random
          1. The Gaussian Curve
               --characteristics of
                  the curve: median,
                  mean, and mode
                  are all at the same
                  place on the
                  distribution, marked
                  as 0 and symbolized
                  as mu (m).
                  Skewness is 0 and
                  kurtosis is 3 since
                  the distribution is
                  perfectly centered
                  and peaked.
                  Tails never touch
                  bottom.   A standard
                  deviation equals 1.
          2.  Interpreting Areas
               Under the Normal
               Curve
               --approximately 2/3rds
                  (68.2%) of the
                  distribution exists
                  from 1s below the
                  mean to 1s above
                  the mean. The
                  standard normal
                  curve can help
                  identify long run
                  expectations we
                  might have for
                  samples we take.
          3.  Using z Scores
               --researchers can use
                  the standard normal
                  curve to make
                  decisions by
                  changing their
                  sample data into "z
                  scores" (also called
                  standard scores).
                  Z scores permit us
                  to represent data
                  scores as units
                  under the standard
                  normal curve.
probability distributions: a distribution of the expected value that would be obtained by sampling an random
data distributions: data collected from actual samples of events
III.  Measures of Association correlation: a measure of the coincidence of variables
--correlation coefficients can range from
   -1.00 to 1.00
     A.  Interpreting Correlations
          --direct and inverse
             relationships

 

 

 

direct relationship: a correlation indicating that as one variable increases,  the other variable also
increases
--in a scatterplot, researchers often add a line of
  "best fit" through the data (sometimes it is
   called a "line of regression")
--a correlation between .80 to 1.00 is a highly
   dependable relationship;  between .60 to .79
   is a moderate to marked relationship;
   between .40 to .59  is a fair degree of
   relationship; between .20 to .39 is a slight
   relationship; between .00 to .19 is a
   negligible or chance relationship
inverse relationship: a correlation coefficient indicating that an increase in one variable corresponds to a decrease in the other (identified by negative signs before correlation coefficients)

         --calculating proportions
            of variance explained
coefficient of determination: the percentage of variation in one variable that can be explained by a knowledge of the other variable alone (computed by squaring a correlation coefficient or computing eta if nonlinear patterns are to be examined)
     B.  Major Forms of
           Correlations
           --though causal
              relationships should
              produce high
              correlations, a
              correlation cannot
              show causation by
              itself (researchers
              must use the
              experimental
              method or wait for
              the method of
              history to resolve
              matters).
          1.   Pearson Product
               Moment Correlation
Pearson product moment correlation: a correlation method is suitable for situations in which both the independent and dependent variables (identified as X and Y respectively in most notation) are interval or ratio level measures
          2.   Spearman Rank
               Order Correlation
Spearman rank order correlation: a correlation method is suitable for situations in which both the independent and dependent variables are ordinal measures