On Q2, no, you probably don't want to do that. Why would you want to do that? (I mean that as an honest question, not a rhetorical one.)

On Q1, are you looking for standard correlation? For b and c, you might look at Pearson, Spearman, and Kendall correlation, and see which is most appropriate for your purpose and data. For a, if you have two genders, you can just code them as 0 and 1 and use a common form of correlation. If you have more than two genders, the approach is not so obvious. You could simply treat Conspiracy Score as a dependent variable, and use anova or Kruskal-Wallis or something similar, but then that would be out of sync with the correlation approach.

Sal Mangiafico

I am truly grateful for your help. It was very helpful. You asked why I would want to convert the Conspiracy Score back to ordinal. This is the honest reason:

First, I used Pearson Correlation Coefficient (Conspiracy Score Vs Knowledge Score that is Ratio vs Ratio). I checked many of the relationships and almost found no significant associations. In other words, most of my hypotheses turned out to be false. Then I converted both Conspiracy Score and Knowledge Score to ordinal data and used Gamma (for ordinal by ordinal relationships) and suddenly there were many significant associations.

What did I do wrong? Shouldn't I be getting the same kind of results either way?

Thank you very much.

Regarding Q2, if you convert averages to whole numbers, you are throwing away data. For examples, "0 < x 1" would treat .2 and .9 as the same (both 1). The two averages are not the same, of course, but you will lose that difference.

You could sum the conspiracy items for each participant to create a composite score. Then use multiple linear regression to analyze your data if the assumptions of ordinary least squares are met.

Shaho Sabbar , The discrepancy between the results for the different types of correlation may have to do with the relationship between the two variables. That is, try Spearman or Kendall correlation instead of Pearson correlation. But, always, plot your variables when you are doing correlation. Your eyes and your brain are always better at assessing things than using a statistical test blind.

Paul Jeffries Your comment was very helpful. Thank you very much.

I kindly suggest that you first calculate Cronbach's alpha (Cronbach, 1951) for the 15 Conspiracy items to ensure that your participants are responding to them as a unidimensional scale. If the alpha is sufficiently large (> 0.70) and cannot be increased substantially by removing any one of the items, then you are justified in using the average of all 15 of them items. (See Likert, 1932 for full description of formation of a Likert scale. Note that Likert used item-total instead of Cronbach's alpha.) As have others, I advise that you not convert the averages to whole numbers, either through truncating or through rounding. The purpose of using a Likert scale is to approximate an interval level of measurement (what SPSS calls "scale"). Converting the average into a whole number is defeating that purpose. For analysis I kindly suggest using multiple regression to assess the simultaneous relationships among your three predictors (gender, education, knowledge) and Conspiracy. Multiple regression will allow you to determine whether or not there is any shared variance among the predictors.

References

Cronbach, L. J. (1951). Coefficient alpha and the internal structure of tests. Psychometrika, 16(3), 297-334.

Likert, R. (1932). A technique for the measurement of attitudes. Archives of Psychology, 22(140), 5-55.