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I'm not really minded to average over a few days because the random element that needs averaging gets taken care of in using a long timeseries - .
The measurement technique isn't perfect. You don't remove any of the measurement uncertainty in the June 2014 measurement, by using an average over previous years.
The same applies to how the weather is actually moving the ice about in June 2014. Its similar, but not quite the same as measurement uncertainty. There is no reason to expect that sort of randomness to correlate perfectly between June and September. By using the minimum, rather than a particular September day, you reduce the error associated with random stuff happening on a particular day in September 2014, but in cherrypicking June 23rd, I think you are overfitting.
Detrend the data around the time you are considering. Check the standard deviation on that. That's a source of error that you can either treat as negligible (because the standard deviation is small) or you can diminish by smoothing the June curve in 2014 rather than picking the value on a particular date. You can't get rid of it by cherrypicking which day in June over the past however mnay years was least affected by it. Using a long time series protects you against a severely overoptimistic error bound from your cherry pick, but doesn't change that its a cherry pick giving you an overoptimistic error bound.
I think daily CT area anomaly is bouncy enough that a simple smoothing would be a modest but worthwhile improvement, but I haven't calculated the variance to prove it.
Its one thing to do the calculation and keep it to yourself. Its another to put it out there when everybody reckons you are wrong and stick by it.