Can somebody tell me the difference between margins of error and confidence intervals? On the Internet I see these two meanings getting used interchangeably.
Is it right to say,
"Confidence intervals are shown as 1.96 and displayed on the graphs as error margins"?
The Internet is full of garbage, as all of us know. It helps to find authoritative sources and focus on them to help resolve such issues. A pamphlet published by the American Statistical Association (attributed to Fritz Scheuren and "thoroughly updated circa 1997") defines the margin of error as a 95% confidence interval (p. 64, at right).
In light of this, it is surprising that the Wikipedia article on margin of error uses a different definition, even though it references this pamphlet! Wikipedia writes,
The margin of error is usually defined as the "radius" (or half the width) of a confidence interval for a particular statistic from a survey. ... When a single, global margin of error is reported for a survey, it refers to the maximum margin of error for all reported percentages using the full sample from the survey.
In other words, to Wikipedia the MoE is one-half the maximum width of a set of confidence intervals (which might have coverages differing from 95%).
We have discussed this confusion (or, at least, lack of standardization) in comments elsewhere on this site. Our conclusion was that you need to be clear what you mean by "margin of error" whenever you use that term.
There is no universally followed convention on what a "margin of error" is but I think (as you have observed) it is most often used as meaning the radius of a confidence interval, either in the original scale of the estimate or as a percentage of an estimate. Sometimes it is used as synonymous with the "standard error", so you need to be careful that others understand what you mean when you use it.
A "confidence interval" does have universal convention on its meaning. It basically is the range of possible estimates generated by an estimating process that would, X% of the time (95% being the most commonly used) contain the true value of the parameter being estimated. This concept of a "process" that would produce the true value X% of the time is a bit counter-intuitive and not to be mixed up with a "credibility interval" from Bayesian inference, which has a much more intuitive definition, but is not the same thing as the widely used confidence interval.
Your actual quote is a little messy and needs some minor fixing as described. I would avoid this additional use of the word "margin" and favour "error bars". So:
"Confidence intervals are estimated as 1.96 multiplied by the relevant standard errors and shown on the graphs as error bars."
(This puts aside the question whether this is a good way to calculate confidence intervals, which depends on your model etc and isn't relevant).
Final comment on terminology - I don't like "standard error", which just means "the standard deviation of the estimate"; or "sampling error" in general - I prefer to think in terms of randomness, and the variance of statistics, rather than "errors". But I slipped into using the term "standard error" above because it is so widely used I guess.
