'infer' vs 'extrapolate'

The two examples are taken from OALD:

We have extrapolated these results from research done in other countries.

The results cannot be extrapolated to other patient groups.

Could I substitute infer for extrapolate here? I think they are synonyms. Any nuances implied?

• Putting syntactically issues aside, extrapolate is something numerical to me, whereas infer is more about something logical. The distinction is not that clear-cut, though. Jul 17, 2014 at 16:47
• Not necessarily. One could extrapolate truth, I suppose. @DamkerngT. Jul 17, 2014 at 16:50
• That's why I think the difference is not clear-cut. However, I think I tend to allow myself to say "extrapolate" something (like an opinion or a conclusion) only when it's an extrapolation from numerical data. Jul 17, 2014 at 16:57
• @DamkerngT. nails it. Interpolation and extrapolation are mathematical operations, predicting unknown values from the curve defined by known values. Inference is a logical operation, drawing conclusions from established premises. Jul 17, 2014 at 17:05
• @ZhanlongZheng Extrapolate is used a little more loosely in this sense (from M-W): "extend the application of (a method or conclusion, especially one based on statistics) to an unknown situation by assuming that existing trends will continue or similar methods will be applicable." But use to mean infer is simply a mistake by writers who don't know what the word means. Jul 17, 2014 at 17:25

It might help to see "extrapolate" as a sub-type of "infer."

"Infer" can refer to small abductions that aren't always as well supported. One can infer emotions in the speaker, but not extrapolate emotions. It's not uncommon for a heated discussion to feature an accusation of "You implied (X)!" and response of "I did not! You inferred that!"

"Extrapolate" generally refers to statistical or logical inferences and tends to indicate a more scientific and controlled context. The product of extrapolation (the extrapolated information) is generally larger quantitatively or qualitatively, whereas "infer" can be is more limited. The etymology of "extrapolate" points to looking at patterns within some set and reproducing those patterns outside the set as a prediction, which should reinforce that "extrapolate" is a term used in more logical and quantitative speech.

I would advise against in the first case. Infer does fit in this sentence properly, but in research contexts extrapolate is the correct word to use if you want to say you used results from other research and applied it to the situation in question.

In the second case, infer is not acceptable at all. "Infer to" is not used as the opposite of "infer from".

Extrapolation is a "weak" form of inference. So, in the second sentence, something that cannot be extrapolated also cannot be inferred.

Extrapolation means, we have seen such-and-such results, at such and such intervals in the past (without knowing why). Based on past "intervals," we are "due" for another occurrence of these results.

Inference means, we have seen such-and-such results in the past following presence of such-and-such causal variables (the why). (E.g., rain has usually followed the gathering of dark clouds in the sky). Based on the return of these causal variables (dark clouds), we predict the return of the consequent results (rain).