# How do I present averages from different sample sizes across years?

I have 20 years' worth of observations that either say YES or NO to the question of whether breeding was observed. I want to present the averages of the YES observations across the years, demonstrating what the average breeding effort looks like across a 20-year period. My issue is that early on a lot fewer observations (n=100) were made compared to present day (n=1000). I believe this might skew the results and result in a graph that is misleading in presenting a genuine trend over the years.

My question is whether I should weight my averages, log transform them or disregard certain data based on confidence intervals?

Any help would be greatly appreciated.

This is what the data looks like at the moment:

• Why don't you plot two standard errors with each sample's mean to indicate how variable the estimate might be? Commented May 31, 2023 at 4:31

The usual thing to do here would be to include "error bars" around your sample average giving a confidence interval for the true average from the sampled data each year. For binary data you can get a good confidence interval estimator from the Wilson score interval. This estimator will take account of the sample size for each year, and years with a smaller sample size will tend to have a wider interval (subject to some other factors). Your reader will then be able to see that there is more uncertainty in the true average in the earlier years than in the later years.

• Hi Ben, thank you for the answer! I'll look into getting these done in R and presented within the graph if possible.
– SamR
Commented May 31, 2023 at 5:00
• "Years with a smaller sample size will have a wider interval" - a decent rule of thumb, but not always true. Extreme proportions near 0% or 100% get narrower confidence intervals than ones near 50%, even with identical sample size. The Wilson 95% CI around a point estimate of 50% with N=100 is five times wider than the 95% CI around a point estimate of 0% with a smaller sample size of N=99! A smaller N is associated with a wider interval for a particular proportion, but not necessarily when the proportion changes in addition to the sample size. Commented May 31, 2023 at 16:19