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I'll add references when I have a moment.

To give a direct answer to the question, whether the warming is significant or not depends on the dataset you look at, how you choose the period in question, your statistical assumptions (e.g. taking into account autocorrelation and multiple hypothesis testing due to choosing the period after looking at the data etc.). Even then, it doesn't necessarily mean much unless you also look at the statistical power of the test.

The way frequentist hypothesis tests work is broadly as follows: Say you have a hypothesis (H1) that you wish to support using a set of observations (X). Next you define a "null hypothesis" that is basically what you need to show to be false in order for your H1 to be true. For example, if you want to show that there has been some warming, then the obvious choice for H0 is that there has been no warming at all, i.e. the rate of warming is zero. You then calculate the p-value, which is the probability of observing a trend at least as large as that observed IF H0 is true. If the p-value is sufficiently small, say p < 0.05, this is taken as sufficient evidence that H0 is false so we say that "we reject the null hypothesis" or equivalently "the rate of warming is statistically significant" and otherwise "we fail to reject the null hypothesis" or "the rate of warming is not statistically significant".

Now the first point to notice here is that H0 should be the hypothesis you don't want to be true. So for mainstream science, which suggests there will be warming due to the greenhouse effect, the natural H0 is that there is no warming. The "Skeptics" on the other hand want to argue that there is no warming, yet they are using that as their null hypothesis as well. This is a grave statistical error as it means that hypothesis testing no longer functions as a sanity check, as the skeptics are assuming that they are right and requiring evidence to prove them wrong. Mainstream science on the other hand are assuming that they are wrong (H0 is true) and asking if the observations refute H0 (implying, but not proving that H1 is true).

The way frequentist hypothesis tests work is broadly as follows: Say you have a hypothesis (H1) that you wish to support using a set of observations (X). Next you define a "null hypothesis" that is basically what you need to show to be false in order for your H1 to be true. For example, if you hypothesise that there has been some warming, then the obvious choice for H0 is that there has been no warming at all, i.e. the rate of warming is zero. You then calculate the p-value, which is the probability of observing a trend at least as large as that observed IF H0 is true. If the p-value is sufficiently small, say p < 0.05, this is taken as sufficient evidence that H0 is false so we say that "we reject the null hypothesis" or equivalently "the rate of warming is statistically significant" and otherwise "we fail to reject the null hypothesis" or "the rate of warming is not statistically significant".

Now the first point to notice here is that H0 should be the hypothesis you are arguing against. So for mainstream science, which suggests there will be warming due to the greenhouse effect, the natural H0 is that there is no warming. The "Skeptics" on the other hand hypothesise there is no warming, yet they are using that as their null hypothesis as well. This is a grave statistical error as it means that hypothesis testing no longer functions as a sanity check, as the skeptics are assuming that they are right and requiring evidence to prove them wrong. Mainstream science on the other hand are assuming that they are wrong (H0 is true) and asking if the observations refute H0 (implying, but not proving that H1 is true).

This is the case for the "not statistically significant" observed trend we have now, given the expected size of the anthropogenic trend and the noise in the data (weather), the power of the test is so low that it is not at all surprising that the result is not statistically significant. Easterling and Wehner have demonstrated that the climate will occasionally show decadal (or more) periods with little or no trend, and that this is also found in model simulations.

The pause in warming is interesting, it is well explained as a result of the effects of ENSO (see the paper by Foster and Rahmsdorf), and it is providing an interesting area for research in climate variability. This does not however mean that the underlying rate of warming has changed, or that carbon dioxide is not a greenhouse gas etc. So the two views are not actually contradictory.

This is the case for the "not statistically significant" observed trend we have now, given the expected size of the anthropogenic trend and the noise in the data (weather), the power of the test is so low that it is not at all surprising that the result is not statistically significant. Easterling and Wehner have demonstrated that the climate will occasionally show decadal (or more) periods with little or no trend, and that this is also found in model simulations.

The pause in warming is interesting, it is well explained as a result of the effects of ENSO (see the paper by Foster and Rahmsdorf), and it is providing an interesting area for research in climate variability. This does not however mean that the underlying rate of warming has changed, or that carbon dioxide is not a greenhouse gas etc. So the two views are not actually contradictory.