Which statistical test to compare frequencies across groups? I am running an analysis in which I want identify if there are significant differences in the features that make up two groups: people that commit crimes against children and people that commit crimes against adults – the group sizes are not equal.
My data is all frequency counts, and has been added up to create percentages
For example –
80% of offenders that commit crimes against children are male 20% are female
75% of offenders that commit crimes against adults are male 25% are female
50% of offenders that commit crimes against children plead guilty, 20% plead not guilty, 30% are acquitted
20% of offenders that commit crimes against adults plead guilty, 60% plead not guilty, 20% are acquitted
I want to know:
Is there a significant difference in the number of females that commit crimes against adults vs. females that commit crimes against children?
Is there a significant difference in the number of males that commit crimes against adults vs. males that commit crimes against children?
Is there significant difference in the number of guilty pleas/ non guilty pleas/ acquittals/ from offenders that commit crimes against children vs. offenders that commit crimes against adults?
Is there significant difference in the number of guilty pleas/non-guilty pleas / acquittals between those commit offences against adults vs. those that commit offences against children?
I am thinking it should be a chi-squared test – but this seems to be for answering questions such as ‘is there a significant difference between the number of men vs. the number of women that commit crimes against children?’ rather than answering ‘is there a significant difference in the number of women that commit crimes against children vs. the number of women that commit crimes against adults?’
Also, in a chi squared test, would my ‘expected value’ just be what I observed in one group and my ‘observed value’ be what I observed in the other group? I,e., I expect the groups to be the same.
I also thought about a t-test where male and female could be coded as 0 and 1 but that would give no standard deviations so would not be feasible.
I would greatly appreciate any help or advice with this or on which test would be appropriate, thank you!
 A: For the first part, you might have the following $2 \times 2$ matrix of counts. In TAB you have data for 100 crimes against children and 100 against adults.
 Criminal:      M   F
 Victim 
   C           80  20
   A           75  25

In R:
TAB = rbind(c(80, 20), c(75, 25));  TAB
     [,1] [,2]
[1,]   80   20
[2,]   75   25

Pearson's Chi-squared test without Yates' correction (on account of moderately large
sample sizes) shows no significant difference
between Child and Adult victims; the P-value exceeds 5%.
chisq.test(TAB,  cor=F)

        Pearson's Chi-squared test

data:  TAB
X-squared = 0.71685, df = 1, p-value = 0.3972

    Pearson's Chi-squared test

data:  TAB
X-squared = 0.71685, df = 1, p-value = 0.3972

By contrast, if you had about 600 victims of each type with (about) the same percentages, then you would have enough data to find a significant difference at the 5% level.
TAB.1 = 6*TAB;  TAB.1
     [,1] [,2]
[1,]  480  120
[2,]  450  150

chisq.test(TAB.1, cor=F)

        Pearson's Chi-squared test

data:  TAB.1
X-squared = 4.3011, df = 1, p-value = 0.03809

Whether the difference between $80\%:20\%$
and $75\%:25\%$ is meaningful in society would be
an issue in criminology, not statistics.
