Today authorities announced that the day before yesterday was the coldest 11 March ever recorded in the Netherlands. This newsflash may not be very remarkable, if only these same authorities, weeronline.nl, hadn’t announced on 5 March that it was the warmest day ever recorded in the Netherlands!
So, what’s going on here? A maximum and a minimum temperature record within 7 days?! Is the weather turning sick?
Maybe, but probably not. Let’s have a look at the odds, assuming that the climate has not significantly changed over the past century, and that temperature levels have been recorded for about 100 years (which is the case). Every day, there generally are two chances for records to be broken: during the day, for the maximum temperature level, and during the night, for the minimum temperature level.
If climate has not changed, it is a process of “randomness” that decides whether 11 March 2013 will be any higher than 11 March 2012, since they are drawn from the same distribution. This holds for any preceding 11 March for which temperature levels have been recorded. If temperature levels truly come from the same distribution, each (say) 11 March of the past century has an equal chance of being the hottest 11 March ever recorded (1 out of 100 chance) or the coldest 11 March ever recorded (also, 1 out of 100 chance). Thus, the probability for a random day or night to break a temperature record equals about 0.01, or 1%.
Now, from 5 March to 11 March there are 7 days, and hence 14 opportunities to break a record. The chance of breaking 2 records or more is equal to 1 minus the chance that no records are broken or that 1 record was broken. Or:
P(2 or more records) = 1 – P(no records) – P(1 record) = 1 – (0.99^14) – (1/100)*(99/100)^13*14 ≈ 0.84%
Thus, a likelihood of less than 1% – that seems pretty extreme.
Extending the period
Looking only at these 6 days is probably not a good idea, since it enforces some sort of a selection bias. (To explain the success of FC Barcelona you should not only look at FC Barcelona, but also at other football clubs, and perhaps especially at those who failed to achieve similar successes. Or, to explain why someone got sick you should not restrict yourself to studying this person, but also consider similar persons who did not get sick.)
Thus, let’s extend the period a bit more, to make sure we’re not only studying a remarkable event to derive conclusions for the entire distribution. Searching the news archives of the same authorities, the previous temperature record seems to have been broken at 30 January 2013. Let’s calculate the odds that 3 or more records would have been broken between 30 January 2013 and 11 March 2013 – a period of 41 days.
P(3 or more records) = 1 – P(no records) – P(1 record) – P(2 records) = 1 – 0.99^82 – (1/100)*(99/100)^81*82 – (1/100)^2*(99/100)^80*3321 = 1 – 0.439 – 0.363 – 0.149 ≈ 5%
So, extending the period a bit more results in a likelihood of 5%, which happens to be a standard boundary for deciding whether something is statistically significant or not. Wondering how the 3321-figure got into the equation? It is the number of ways in which 2 days can be picked out of 82 days. To calculate this number, you calculate “82 choose 2”. This equals 82!/(80!*2!) = 82*81/2 = 3321.
Extending the period even further
But even now we suffer from a selection bias, because the period under investigation starts and ends with an extreme observation – no wonder that the average number of extreme observations tends to be high.
Again considering the archives of weeronline.nl it seems that no other records were broken in November and December 2012 and in the remaining days of January 2013. This adds 90 days to the calculation:
P(3 or more records) = 1 – P(no records) – P(1 record) – P(2 records) = 1 – 0.99^82 – (1/100)*(99/100)^261*262 – (1/100)^2*(99/100)^260*34191 = 1 – 0.072 – 0.190 – 0.251 ≈ 49%
The likelihood of 3 or more temperature records being broken in a period of 4 months is substantial. As a matter of fact, it is almost equally likely that this would not happen (51% versus 49%). So, relax – the weather seems to be doing just fine.