First, the figure gives the answer on CO2 and temperature over the last 800,000 years:
Here we have each value of CO2 plotted against the temperature deviation from reference values -- with the temperature being for 1000 years before the corresponding CO2 value. The correlation for this is about 0.89 (R), meaning that you could explain 79 percent (0.89*0.89) of the variation (R^2) in CO2 by looking at the temperature. As is mentioned in my post Does CO2 correlate with temperatures, where we saw equally high correlation between temperature and CO2 (even higher if you give CO2 a 20-30 year lead on temperature), and quite a few times in the comments, correlation is not causation. Could be that both temperature and CO2 are pushed around by something else. More about that in a moment.
It's common to see the claim that temperature 'leads' CO2. Loosely speaking, this means that temperatures generally change before CO2 does, and that there is a consistent pattern to the connection -- if temperature rises, CO2 does as well. This is true; it leads by about 800 years [Caillon and others, 2003]. The amount of lead also seems to depend on whether you're in a glacial period or (as we are now) an interglacial. The correlation between temperature and CO2 is lower (by a very small amount) if you take their values for the same time (drops to 0.88). And it drops by more if we take the temperatures for 2000 years before the CO2 value, to 0.87, indicating from this very simple approach that the lead is between 0 and 1000 years, probably closer to 1000 -- as is found from the more serious approach in the reference.
So there's some interesting science to do, to understand why that lead exists, why it is many hundred years (rather than a few dozen years, or a few thousand), and why the lead time depends on whether you're in a glacial period or an interglacial period. Oddly, to me, most of the time that people mention this lead relationship, they are not referring to any of this.
Rather, they want to conclude, or for you to conclude, that because in the ice age record temperature changes before CO2, that a) the current rise in CO2 is also caused by temperatures (sometimes citing the medieval warm period as the warm time that is causing the current rise in temperature) and b) that CO2 doesn't have any affect on temperatures.
Have another look at the figure. See that point sitting way the heck away from all others? That's the most recent value.
You don't need a lot of scientific background to see that whatever it is that produced that CO2 value, it certainly was not the relationship that held for the other 798,000 years of the temperature - CO2 record. We can be more precise about it. The best fit line (the one that gives that nice high correlation) through the data points is CO2 = 266 + 8*T. So, for temperatures at the reference value, we expect a CO2 level of 266 parts per million. For temperatures 10 C below reference, we expect CO2 levels of 186 parts per million. Both of these accord fairly well with what we do see in the record -- except for the most recent CO2 value. As you can see from the plot as well, there is indeed scatter around the best fit line. The standard deviation is about 11 ppm. That means we're not surprised to see values 22 ppm away from the line (2 standard deviations), and, given 799 data points, we expect a few to be 33 ppm away. On the other hand, a value 9 standard deviations away -- the case for the current CO2 levels -- is ludicrous. Something must have changed.
We can turn this around, and ask: "If the relationship that held for the previous 798,000 years still did, what would the temperature need to have been to give the observed CO2 level -- maybe CO2 is so sensitive to temperature that we simply have a one-time temperature observing problem?" Certainly the ice sheet temperatures do have their own observing issues. As I've mentioned, all data have problems. Still, same as we know that, we also know something about the size of the problem. So, turn the equation around and solve for the temperature that would correspond to the observed (modern) CO2 level. That's a temperature anomaly of about 13 C (20 F) -- equal to the entire range from the very warmest interglacial to the very coldest glacial! And that had to have occurred 600-1000 years ago, by the lead relationship. Meaning there had to have been an absolutely enormous warming (many times larger than even the highest medieval warm period values) and nobody noticed it ... or else the previous relationship between temperature and CO2 broke down.
Since we know what did cause the CO2 rise -- human activity -- we're not really surprised to see this answer. The old relationship did get broken. Rather than CO2 rising because the oceans released CO2 to the atmosphere, it is because humans have been burning fossil fuels and making cement. But we do like to be able to arrive at our conclusions from different directions. Here, we need only to look at the temperature and CO2 values themselves to see that the relationship that used to hold has broken down in the modern day. Don't need to know the first thing about isotope geochemistry (which is explained nicely in that faq by Jan Schloerer) to see this.
Let's return, though, to the fact that correlation is not causation. There are generally 4 options when a correlation is seen: it's chance, A causes B, B causes A, both A and B are caused by something else. We use statistical tests to decide whether chance is plausible (in this case, as with Does CO2 correlate with temperature?, the answer is a clear no). So one of the other 3 is involved. We can reject CO2 being the (sole) cause of temperature changes (it can certainly feed back on temperatures) because it is the temperatures that changes first. So it's one of two now -- either temperatures drive CO2 (with feedback from CO2 back to temperatures), or something else drives changes in both temperature and CO2 (and perhaps manages to change temperature sooner than CO2).
Oddly, most of the comments about temperature leading CO2 ignore the last option -- in other words, they assume that correlation is indeed causation. Now ... is there anything in the universe that could affect temperatures? Could any of those things also affect CO2 levels? That will be a separate post. But start thinking about it yourself. Richard Alley gives away the answer in his talk, to which I referred you Tuesday.
I've labelled this a 'project folder' post in part for the questions I mentioned inside it -- why the lead is what it is (or at least was what it was), and so forth. But also because the things I did you can do yourself. The data I used were from a BBC news article on ice and CO2. You can get it yourself and experiment with the lead relationships, find the CO2 response (slope of CO2 versus temperature in the regressions), and the like.
You can go farther as well. For instance, are the correlation and slope different in different subsections of the record? Is it different for different temperature ranges? (compute the slope for only temperatures from -10 to -5, -5 to 0, and so forth) Does the correlation improve if you use the logarithm of CO2 rather than CO2 levels themselves? (if the only thing involved were the radiative properties of CO2 and its connection to temperatures, we would expect a 'yes' here)