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Chi-square Basics. The Chi-square distribution. Positively skewed but becomes symmetrical with increasing degrees of freedom Mean = k where k = degrees of freedom Variance = 2k Assuming a normally distributed dataset and sampling a single z 2 value at a time 2 (1) = z 2

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The Chi-square distribution

- Positively skewed but becomes symmetrical with increasing degrees of freedom
- Mean = k where k = degrees of freedom
- Variance = 2k
- Assuming a normally distributed dataset and sampling a single z2 value at a time
- 2(1) = z2
- If more than one… 2(N) =

Why used?

- Chi-square analysis is primarily used to deal with categorical (frequency) data
- We measure the “goodness of fit” between our observed outcome and the expected outcome for some variable
- With two variables, we test in particular whether they are independent of one another using the same basic approach.

One-dimensional

- Suppose we want to know how people in a particular area will vote in general and go around asking them.
- How will we go about seeing what’s really going on?

- Hypothesis: Dems should win district
- Solution: chi-square analysis to determine if our outcome is different from what would be expected if there was no preference

- Reject H0
- The district will probably vote democratic
- However…

Conclusion

- Note that all we really can conclude is that our data is different from the expected outcome given a situation
- Although it would appear that the district will vote democratic, really we can only conclude they were not responding by chance
- Regardless of the position of the frequencies we’d have come up with the same result
- In other words, it is a non-directional test regardless of the prediction

More complex

- What do stats kids do with their free time?

- Is there a relationship between gender and what the stats kids do with their free time?
- Expected = (Ri*Cj)/N
- Example for males TV: (100*50)/200 = 25

- df = (R-1)(C-1) kids do with their free time?
- R = number of rows
- C = number of columns

Interpretation kids do with their free time?

- Reject H0, there is some relationship between gender and how stats students spend their free time

Other kids do with their free time?

- Important point about the non-directional nature of the test, the chi-square test by itself cannot speak to specific hypotheses about the way the results would come out
- Not useful for ordinal data because of this

Assumptions kids do with their free time?

- Normality
- Rule of thumb is that we need at least 5 for our expected frequencies value

- Inclusion of non-occurences
- Must include all responses, not just those positive ones

- Independence
- Not that the variables are independent or related (that’s what the test can be used for), but rather as with our t-tests, the observations (data points) don’t have any bearing on one another.

- To help with the last two, make sure that your N equals the total number of people who responded

Measures of Association kids do with their free time?

- Contingency coefficient
- Phi
- Cramer’s Phi
- Odds Ratios
- Kappa
- These were discussed in 5700

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