# Testing if year-to-year change is significant?

I have Two related questions:

1) I have raw counts for voting vs. non-voting for the years 2018 and 2020.

2018 -> 11,000 voters and 3,000 non-voters

2020 -> 10,000 voters and 3,500 non-voters

How can I calculate that the decrease in voters from 2018 to 2020 is significant or not? What test would I use?

2) I also have the voter turnout rate which is 67% in 2018 and 66% in 2020. How can I check whether this 1% drop in voter turnout rate is significant?

what tests could I use in each case?

Thank you in advance!

• I should add that--I have the total voting population number as well (if that is helpful) Commented Feb 6, 2022 at 2:21
• Commented Feb 6, 2022 at 2:30
• If that is what actually happened in the election, then any change is significant in the sense that you can be sure that there was a change. If this is based on random sampling of individuals from a much larger population and asking them whether they were voters or non-voters or something else (e.g. children or non-citizens) then you can apply usual techniques Commented Feb 6, 2022 at 2:43
• It sounds like the total population for 2018 is 14,000 people. Is this true? If so, there's probably no need to conduct a hypothesis test (like a chi-squared test), since you have data on the whole population. If this is not true, wouldn't you be able to get the data for the total potential voters in the election and the total number of actual voters? Commented Feb 6, 2022 at 16:03
• @SalMangiafico Thanks for the question. No, the total population is a bit bigger because it includes those who are not allowed to vote (I do have the total population size as well). But my problem is that I wanted to know whether the year-to-year change is significant. The total population size and sample sizes across both years are slightly different in size. Commented Feb 6, 2022 at 18:48

## 1 Answer

(1) In 2018, you have 11,000 voters out of 14,000 potentially available, and in 2020, you have 10,000 voters out of 13,500.

In R, assuming randomness in willingness and ability to vote, you could compare the two proportions of voters using prop.test as follows (declining continuity correction with parameter cor=F on account of large sample sizes):

prop.test(c(11000,10000), c(14000,13500), cor=F)

2-sample test for equality
of proportions without
continuity correction

data:  c(11000, 10000) out of c(14000, 13500)
X-squared = 77.015, df = 1, p-value < 2.2e-16
alternative hypothesis: two.sided
95 percent confidence interval:
0.03493139 0.05501570
sample estimates:
prop 1    prop 2
0.7857143 0.7407407


The proportions $$.786$$ and $$.741$$ are significantly different on account of the P-value near $$0.$$

Alternatively, you could do a chi-squared test on a $$2\times 2$$ table:

TBL = rbind(c(11000, 10000), c(3000, 3500)); TBL
[,1]  [,2]
[1,] 11000 10000
[2,]  3000  3500

chisq.test(TBL, cor=F)

Pearson's Chi-squared test

data:  TBL
X-squared = 77.015, df = 1, p-value < 2.2e-16


Except for the input syntax, the two tests are essentially equivalent.

(2) You need to use counts (as above) instead of percentages.

• Thank you so much! I will use the chi-squared test approach Commented Feb 6, 2022 at 5:40