# 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!

• It might be helpful if you focused on one hypothesis in the question... I think for what you want, you will need the original counts for the figures, not just the percentages... A place to begin is to try to answer the questions without a hypothesis test. That is, calculate the cross-tabs for your data in a way that answers the question you are asking with descriptive statistics. E.g. Out of 100 female criminals, 25 committed crimes against children and 75 committed crimes against adults. So... (?) 75% of crimes by females are against adults. That's probably meaningfully more than 50%. Commented Apr 19, 2022 at 16:10

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.

• I notice you asked about expected counts. In a given cell take the row total times the column total and divide by the grand total to get the expected count for the cell. // In the second chi-sq test, the expected counts can be displayed as follows: chisq.test(TAB.1, cor=F)$exp returns first row$465, 135$and second row$465, 135.\$ For this table, the expected counts turn out to be integers, but if they are not integers, don't round them when you use them to find the chi-squared test statistic. Commented Apr 19, 2022 at 20:10