PLS 101
American National Government
Roger C. Lowery, Ph.D.

Two Basic
Summary
Statistics

• measures the strength (or weakness) of the relationship between two variables
  • for scatter plots in this course, we will always use the Pearson's r correlation coefficient (can vary between -1 to 0 to +1)
  • for cross tabulations, we will always use the Cramer's V correlation coefficient (can vary between 0 to +1)
  • a positive (or direct) correlation indicates that as the values of one variable increase, so do the values of the other variable
  • a negative (or inverse) correlation indicates that as the values of one variable increase, the values of the other variable decrease
  • the closer either correlation coefficient is to zero, the weaker the relationship
• in social-science research, the following classification of both correlation coefficients is commonly used:
  • a weak relationship is present if either the Pearson's r or Cramer's V is less than plus or minus 0.10
  • a moderate relationship is present if either the Pearson's r or Cramer's V is between plus or minus 0.10 and 0.25
  • a strong relationship is present if either the Pearson's r or Cramer's V is greater than plus or minus 0.25
• statistical association is not necessarily the same thing as causation
  • i.e., just because there is present a weak, moderate, or strong level of statistical association between two variables does not necessarily mean that changes in one variable cause changes observed in the other variable
  • statistical association just means that the values of one variable change consistently with changes in the values of the other variable
  • it may be a third variable that accounts for (or causes) the changes in the two variables that you are measuring

• measures the probability of random-sampling error in survey data
• random-sampling error causes the observed patterns that we see in sample data to be unrepresentative of the actual pattern that we would see if we had looked at the whole population from which the sample was randomly drawn
  • the good news with random sampling is that we can accurately estimate population patterns by looking at only a relatively small sample randomly drawn from a much larger population
  • the bad news is that any randomly-drawn sample can be unrepresentative of the population from which it is drawn
  • however, the good news is that the probability of having an unrepresentative sample can be calculated (this is what the Chi-square statistic tells us)
• in this course, we will use the Chi-square statistic to determine the probability of random-sampling error
• in social-science research, the following classification of Chi-square probability is commonly used:
  • if the Chi-square probability of random-sample error is less than 0.05, then the sample results are assumed to be statistically significant (i.e., there is less than a 5% chance that our observed sample results are unrepresentative of the population results)
  • if the Chi-square probability of random-sample error is equal to or greater than 0.05, then the sample results are assumed to be statistically not significant (i.e., there is a 5% chance or more that our observed sample results are misleading us)
• the Chi-square probability is reported one of two ways:
  • sometimes an asterisk is used
    • if there is an asterisk next to the correlation coefficient, then the sample results are considered statistically significant
    • if there is no asterisk, then the sample results are considered statistically not significant
  • or, sometimes the actual Chi-square probability is given
    • if the Chi-square probability is less than 0.05, then the sample results are considered statistically significant,
    • if the Chi-square probability is equal to or greater than 0.05, then the sample results are considered statistically not significant,
• statistical significance is not the same thing as substantive significance:
  • for example, national-sample surveys repeatedly show that there is no statistically significant difference between Protestants (as a group) and Catholics (as a group) in their attitudes toward legalizing abortion
    • as we can see in the 2004 NES survey results in the table below, roughly the same proportion of Catholics as Protestants support legalizing abortion);
    • the relationship is weak (the Cramer's V is less than 0.10)
    • and any sample differences are not statistically significant (there is no asterisk, indicating that the Chi-square probability of random sampling error is equal to or greater than 5% or 0.05 -- the Chi-square probability for this cross tabulation is actually 0.06)
  • such a finding (a weak correlation that is not statistically significant) is nevertheless substantively significant, given the strong anti-abortion stand taken by the Catholic Church's leadership.

   Missing                              88                 36                41                  156

   TOTAL                               584                256               239                  840
                                     100.0%            100.0%                                 100.0%