In earlier posts I have shown plots of all possibly linear trends in about a century of surface temperature data from various sources. In

this post I modified the plot by fading regions where the trend was not significantly different from zero.

But as commenter Frank noted, this is not the only significant difference that might be of interest. So in this post, I'll show some plots that help to bound the range of signficance.

The first plot shows, for each start and end month, the lowest trend that the observed trend would be significantly less than. That is useful for testing predictions that might be failing on the cool side. For example, if you think that there is a claim that the trend over a period should have been 2°C/century, you can see where the actual trend was

*significantly* (at 95%) below that.

The second plot, below the jump, shows the converse. What is the highest trend that was significantly (95%) below the observed. This is probably mainly of interest in establishing whether a trend was significantly above zero, but you might be interested in other values - if you think a theory significantly under-predicts.

The final plot is of the t-statistic - the trend normalized by its standard error. This lets you look at other degrees of significance - mainly relative to zero, but in conjunction with the other plots, you can work out other trend comparisons too.

I've retained the apparatus whereby you can check each point (by clicking) against its plot, echoing the numerical trend value (and period). I've changed from earlier plots of this kind by allowing trend periods down to 1 year.

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Purpose

I should at this stage say that my purpose here is to show how the much invoked arithmetic of trend fitting works out. I'm not saying that it is always a good thing to do, and there are cautions about what significance means.

Saying that a trend is significantly different from a base trend is saying that the on the null hypothesis that the data is formed from the base trend plus random noise the observed result is improbable. Cautions:

- Lack of significance does not mean the base trend is right. It just means it is consistent with this data. Many other possibilities would also be consistent.
- Significance does not mean that any physics, say AGW, is disproved. It just means that there may be something more than random variation plus trend.
- And it may not even mean that. It says the result, on the null hypothesis, is improbable. But improbable things happen. There are about half a million dots on the longest plots. If they were independent, and each had a 5% chance of being in a certain range, then that means 25000 significant dots. Even with correlation, you might still expect something like 5% of the area to show as significant.

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Plots

So here is the first plot, showing which trends the observed trend would be significantly less than. To use it, pick a trend (color) you want to test from the legend. That and

~~bluer~~ redder colors indicate regions where the trend observed was significantly less than the test (color). I have reversed the coloring custom of earlier posts, marking zero with dark brown, and 1.7°C/Century with gray. For the associated plot and mode of operation, see

this post. But note that you can click anywhere on the plot to show the real trend there (shown in text and also by the time series plot).

And here is the second plot, showing trends where the observed would be significantly greater. The brown marks the edge of the area where the trend is significantly greater than zero.

And the third plot, which shows the t-statistic, or ratio of the trend to its standard error. For the number of degrees of freedom here, this is distributed normally, and 1.96, marked in brown, is the level of 95% significance. 1.64 is 90%, and 2.58 is 99%. One observation here is that there is only a small fringe region where a choice of a different test level would alter the result..