The extent to which recent heat waves can be shown to be caused by global warming depends in part on how one frames the question.

As humans continue to increase the concentrations of atmospheric greenhouse gases, heat waves are becoming more frequent.1 Here in Durham, North Carolina, where I live, March 2012 was one of the warmest months over the past several decades, relative to the average temperature for each month. Let’s imagine that three scientists looked into this heat wave, assessed the contribution from global warming, and reported their findings in news stories using three different headlines.

Headline A: Recent heat wave was due 71% to natural variability and 29% to global warming.

Headline B: Global warming increased the odds of the recent heat wave by only 0.25%.

Headline C: Global warming has made heat waves like the recent one occur 22 times as often as they would have otherwise.

A reader might think the three scientists fundamentally disagree on global warming’s role in the heat wave. But the headlines could all be technically correct. The different sounding conclusions can all be drawn from looking at the same data from slightly different directions.

For headline A, the scientist needs to calculate global warming’s contribution to the magnitude of the heat wave. Of the three, it is probably the most straightforward way to summarize the data. The left panel of figure 1 shows the monthly temperature anomaly2—the difference between the observed mean temperature for each month and the long-term average for that month of the year—for Durham from 1900 to 2013. The red line in the left panel is an estimate of long-term global warming in Durham, calculated from the average results of 27 independent simulations of physics-based numerical climate models.3

Figure 1. Monthly temperature anomalies in Durham, North Carolina, with and without global warming, 1900–2013. The black data in the panels show the anomalies. (a) The red curve gives the contribution due to global warming, as estimated by climate models. (b) When the global warming contribution is subtracted out, the remaining anomalies show the effects of natural climate variability. The horizontal blue line corresponds to zero anomaly. In March 2012, the +6 °C total anomaly shown in panel a (red dot) received the +4.25 °C natural contribution displayed in panel b (blue dot) and a +1.75 °C contribution from global warming.

Figure 1. Monthly temperature anomalies in Durham, North Carolina, with and without global warming, 1900–2013. The black data in the panels show the anomalies. (a) The red curve gives the contribution due to global warming, as estimated by climate models. (b) When the global warming contribution is subtracted out, the remaining anomalies show the effects of natural climate variability. The horizontal blue line corresponds to zero anomaly. In March 2012, the +6 °C total anomaly shown in panel a (red dot) received the +4.25 °C natural contribution displayed in panel b (blue dot) and a +1.75 °C contribution from global warming.

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What would the temperature in Durham have looked like if there had been no global warming? We can calculate that by subtracting the estimate of global warming (red line) from each month’s temperature anomaly (black). The result is shown in the right panel of figure 1. March 2012 would still have been a hot month even without global warming, but it would not have been as hot.

The result presented in headline A was thus calculated as follows: If the total anomaly with global warming in March 2012 was +6 °C and the contribution from natural variability was +4.25 °C, then global warming contributed +1.75 °C, or 29% of the anomaly, and the natural variability contribution was 71%.

Headline A quantifies how much global warming contributed to the magnitude of the heat wave, but let’s now consider how much global warming contributed to the likelihood that the heat wave would occur in the first place.

Headlines B and C require the scientists to calculate global warming’s influence on the change in the heat wave’s likelihood. The conclusions of headlines B and C sound very different from one another, but arriving at those numbers requires similar calculations. Climate scientists often assume that in the absence of global warming, temperature anomalies follow some kind of probability distribution. There is precedent for thinking of surface-temperature anomalies as being normally distributed,4 as is illustrated in figure 2. However, the quantitative results of this article, though not the qualitative point, are sensitive to the type of distribution assumed.5

Figure 2. Normal distribution of monthly temperature anomalies in Durham, North Carolina, without global warming (blue) and with global warming (red). The shift due to global warming produces a small change in the absolute probability of a large temperature anomaly (top) but a big relative change (bottom).

Figure 2. Normal distribution of monthly temperature anomalies in Durham, North Carolina, without global warming (blue) and with global warming (red). The shift due to global warming produces a small change in the absolute probability of a large temperature anomaly (top) but a big relative change (bottom).

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Once a probability distribution is assumed, the next step is to notice how global warming has shifted the distribution over time. The top panel of figure 2 shows how the +1.75 °C change in the baseline temperature from global warming has affected the probability of all temperature anomalies. So the result used in headline B was obtained as follows: Without global warming, an anomaly of +6 °C or warmer was very unlikely; its chance of occurring in any given month was about 0.0117%. Even considering that global warming shifted the mean of the distribution by +1.75 °C, an anomaly of +6 °C or greater was still very unlikely; its chance of occurring in any given month was about 0.26%. So global warming increased the chance of the March 2012 heat wave by approximately 0.25%.

The difference in probability with and without global warming may not seem big; however, the small shift in absolute probability translates into a big change in the return time (the time it takes on average to see a heat wave of a specified magnitude). A probability of 0.0117% for a +6 °C anomaly indicates that without global warming such an anomaly would be expected to occur once in 8550 months. However, a probability of 0.26% for a +6 °C anomaly indicates that with global warming the expectation is once in 380 months. Now the thought behind headline C becomes obvious: Global warming caused a heat wave of this magnitude or warmer to occur 22 times more often than would be the case without global warming.

All three headlines are technically justifiable; they are simply different ways to present the data. Headline A says what proportion of the +6 °C anomaly is due to global warming. Headline B shows how much global warming changed the absolute probability of a heat wave of +6 °C or warmer. And headline C addresses how much global warming changed the expected frequency of a heat wave of +6 °C or warmer.

In my judgment, only headline B is fundamentally misleading. Since extremes have small probabilities by definition, a large relative change in the probability of an extreme will always seem small when expressed in terms of the absolute change in probability. Headlines A and C, on the other hand, quantify different pieces of information that can both be valuable when considering global warming’s role in a heat wave. One of the lessons here is that writers with agendas have a great deal of latitude to craft headlines advancing the particular narrative they favor.

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