Variance: (dis)agreement
Although rare, research hypotheses may focus on the variation in scores rather than on score level. The hypothesis about polarization provides an example. Polarization means that we have scores well above the center and well below the center rather than all scores concentrated in the middle. If voters’ opinions about immigrants are strongly polarized, we have a lot of voters strongly in favour of admitting immigrants as well as many voters strongly opposed to admitting immigrants.
For a numeric variable, the variance or standard deviation—the latter is just the square root of the former—is the appropriate statistic to test a hypothesis about polarization. The research hypothesis concerns the variation of scores in two groups, for instance, young versus old voters. The statistical hypothesis would be that the variance in opinions in the population of young voters is different from the variance in the population of old voters.
Testing two variances in SPSS
Exercises
- Data set voters.sav contains information about the age and attitude towards immigration among a random sample of voters. Is the attitude towards immigrants equally polarized among young (under 30) and old (30+) voters? Justify your answer with a statistical test.
- Use the data of Exercise 1. Create a new variable grouping voter’s age with classes 18-35, 36-65, and 66+ years. Is the attitude towards immigrants equally polarized among these three age groups in the population? Justify your answer with a statistical test.
Answers
Answer to Exercise 1.
SPSS syntax:
* Check data.
FREQUENCIES VARIABLES=age_group immigrant
/ORDER=ANALYSIS.
* Independent-samples t test with Levene s test.
T-TEST GROUPS=age_group(1 2)
/MISSING=ANALYSIS
/VARIABLES=immigrant
/CRITERIA=CI(.95).
Check data:
There are no impossible values on the variables.
Check assumptions:
There are no assumptions that we have to check for
the Levene test.
Interpret the results:
The attitude towards immigrants is more polarized among older voters (SD = 2.06) than among young voters (SD = 1.26).The difference in variation is statistically significant, F = 4.99, p = .029.
Note that SPSS does not report the (two) degrees of freedom of the F test, so we cannot report them either.
SPSS, however, does report the degrees of freedom of Levene’s F test in a one-way analysis of variance. We could have used that approach here as well.
Answer to Exercise 2.
SPSS syntax:
* Check data.
FREQUENCIES VARIABLES=age immigrant
/ORDER=ANALYSIS.
* Group age.
RECODE age (Lowest thru 35=1) (36 thru 65=2)
(66 thru Highest=3) INTO age3.
VARIABLE LABELS age3 ‘Voter ages in three groups’.
EXECUTE.
* Define Variable Properties.
*age3.
VALUE LABELS age3
1.00 ‘18-35’
2.00 ‘36-65’
3.00 ‘66+’.
EXECUTE.
* ANOVA with descriptives.
ONEWAY immigrant BY age3
/STATISTICS DESCRIPTIVES HOMOGENEITY
/MISSING ANALYSIS.
Check data:
There are no impossible values on the variables.
Check assumptions:
There are no assumptions that we have to check for the Levene test.
Interpret the results:
The attitude towards immigrants seems to be more polarized among middle-aged (SD = 2.17) and aged voters (SD = 2.06) than among young voters (SD = 1.37).
The difference in variation, however, is not statistically significant, F (2, 63) = 2.77, p = .070.
This result seems to contradict the statistically significant test result that we found in Exercise 1, comparing only young to old voters. There is no contradiction, however. We merely see that the classification into groups can matter to the significance of results.