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I discussed how even a small change in a normal curve increases the likelihood of an extreme occurrence in Normal Density Tail Differences. I'll try to contextualize that thought from the perspective of a warming environment in today's post. The outcomes of two simulations are shown in Figure 1. We consider a fictional city with a mean summertime temperature of 85°F and a standard deviation of 5°F throughout the course of 60 days, such as July and August. The stable scenario has the mean temperature remaining constant over 100 years. We pick 1901–2000 for our example. We choose a temperature at random from a normal distribution for every summer day. We keep track of the summer's maximum temperature as well as the average temperature over the course of the season's 60 days for each summer. The blue points are these. In scenario 2, referred to as increasing, we raise the normal distribution's mean temperature by 0.01°F annually. This is an increase of 4°F during the simulated 100 years. The red points are these. Remember that Figure 1 only represents one simulation.

Observe how much less variable the mean temperatures are (the lighter colors) than the maximum temperatures. The mean temperature in the stable scenario remains at or near 85°F, while it rises to 89°F in the increasing scenario. There are times when there is little to no increase or even a reduction in temperature, even if the increasing scenario predicts an overall increase in temperature. For instance, the number of lighter red dots decreased between 1950 and 1957. Someone in 1957 might have used this as evidence that climate change is not happening. Random events will do that. In the lighter red dots, one can easily pick out time periods when the means were steady or declining, even though the overall trend is increasing.

Now let's focus on the maximum temperatures. Three standard deviations from the mean for the steady scenario correspond to 100°F. In 99.7% of cases, the data is within three standard deviations of the mean, as you may remember from statistics. Even though the chance is small, there are a few times when the maximum temperature passes 100°F and approaches 105°F. This is due to the fact that we have 6,000 possibilities, or 60 days per summer multiplied by 100 years. In other words, there are enough opportunities for this to occur. For example, if you roll a 100-sided die enough times, a 1 will come up. Nevertheless, we only reached over 100°F eight times and never exceeded 105°F. Highly livable.

What about the increasing scenario? Once more, the maximum temperature shows a great deal of variation and rises gradually, with a few occurrences approaching 110°F. The highest temperature in the increasing scenario had repeatedly soared above 100°F by the time we reached 1975. How then do we measure these variations in the highest temperature? First of all, remember that this is only one simulation, which is insufficient to draw any firm conclusions. Six of these simulations are shown in Figure 2, so the interested reader can compare what might or might not change from simulation to simulation.

We perform 10,000 simulations and analyze the frequency with which the temperature exceeds 100°F and 105°F throughout the course of a summer in order to quantify the changes in maximum temperatures. The mean or expected number of days for each temperature for the first 50 and second 50 years is then determined. The expected, or mean, number of days with temperatures above 100°F is shown in Figure 3 for the various scenarios and time periods.

The projected number of days for the stable scenario is the same for the first and second 50 years, as can be seen in Figure 3. In other words, we anticipate about 4 summers with temperatures above 100°F during the 50 periods. Given that the temperature isn't changing, the two 50-year periods should be the same. We expect a roughly 50% increase in summers with temperatures over 100°F over the first 50 years of the rising temperature scenario, followed by a 5-fold rise during the next 50 years. One important thing to note is that, compared to no change, the slow rise in temperature won't be felt as much in the first 50 years. In the scenario of rising temperatures, even the difference between the first 50 and the second 50 is tripled. In essence, nothing is noticeably bad until it is.

The difference between Figure 4 and Figure 3 is that we now use a 105°F threshold. We predict 0.1 summers over 105°F for the stable scenario over a 50-year period. In other words, if we perform Figure 1's simulation ten times, only one of those will exceed 105 degrees once over the 50 years. To see a year above 105°F in the first 50 years of the rising temperature scenario, we would need to conduct around five simulations, but in the second 50 years, we will typically see a year over 105°F in every simulation. Be aware that there is a 10-fold rise from the stable scenario to the second 50 years of the rising temperature scenario.

The real temperatures shouldn't be taken too seriously. The key is to pay attention to how each scenario differs from the last. The increase in extreme temperatures, or maximum temperatures in this case, will be significant by the time we reach the second half of the century, despite the 0.01°F increase every year for a century. Also take note of how, during the first 50 years of the increasing scenario, the changes to the stable scenario are not particularly significant, but that everything changes dramatically during the next 50 years. In fact, in the scenario of rising temperatures, even the difference between the first 50 and the second 50 is tripled. In essence, nothing is truly horrible until it is.

Technically speaking, the simulation takes into account the independence of each day's temperature. In the real world, this is certainly false. Again, the key here is to recognize and gain insights into the differences between each circumstance and to avoid taking the projections too literally.

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#### Disagreeing and using comments

I'd rather know the truth and understand the world than always be right. I'm not writing to upset or antagonize anyone on purpose, though I guess that could happen. I welcome dissent and disagreement in the comments. We all should be forced to articulate our viewpoints and change our minds when we need to, but we should also know that we can respectfully disagree and move on. So, if you think something said is wrong or misrepresented, then please share your viewpoint in the comments.