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Will someone please explain what the difference is between standard error and margin of error and when to use which? I have not had luck finding a simple explanation on the differences anywhere on the online. The questions was posed on this site in 2011 by Adhesh Josh: What is the difference between "margin of error" and "standard error"?, but the answers given did little to clear up my ignorance.

From what I understand, SE and MOE both measure accuracy with which a sample represents a population, are both population estimates based on sample statistics (SD and n), and are both can be used to show +/- significant differences between groups.

The only difference I can see between the two is that MOE is SE multiplied by critical z, which of course makes the MOE larger than the SE.

Why are there two different ways to measure what seems like the same thing?

When is it appropriate to take the extra step to compute MOE instead of just reporting the SE?

An example or two would be great..

Update: According to "Survey Methodology" (Groves et al., 2009), "The [margin of error] computation is relevant only for statistics that are percentages; it gives no information for statistics that are means (other than percentages), differences of means, or totals."

That certainly helps... does anyone have something else to add?

Another Update: According to online course materials of Prof. John Matthews of the School of Statistics and Mathematics at the University of Newcastle, "It is commonplace to indicate the ‘error’ in a quantity by quoting the value plus or minus the estimated error... Certainly, this is often the impression given in the medical literature, where it is common to find table headings of ‘Mean (± SE)’ and graphs with bars that are a standard error in length stretching out above and below some mean. However, this is a misleading practice that should be discouraged... The problem can be understood by recalling that for a Normal distribution most values (in fact about 95% of them) lie between μ-2σ and μ+2σ. As the distribution of sample means has mean μ and standard deviation equal to the SE, there is a 95% chance that the sample mean, m, is between μ±SE2, which amounts to saying that there is a 95% chance that μ lies between m±2SE. Consequently it is much more accurate to assert that the population mean lies in the interval than that implied by headings such as mean±2SE."

I think the above quoted is relevant to the SE vs. MOE issue at hand. MOE is equal to SE multiplied by a critical z value--usually 1.96 for the 95% confidence level. Because 1.96 is ~ 2, then 2SE is a rough estimate of MOE, which Prof. Matthews says is the more accurate way to test the null hypothesis at p < .05.

Am I on track with this interpretation? Any other info to add?

Update the 3rd: Upon further reflection, I'm not satisfied with my interpretation of Prof. Matthew's explanation"

"I think the above quoted is relevant to the SE vs. MOE issue at hand. MOE is equal to SE multiplied by a critical z value--usually 1.96 for the 95% confidence level. Because 1.96 is ~ 2, then 2SE is a rough estimate of MOE, which Prof. Matthews says is the more accurate way to test the null hypothesis at p < .05."

The problem I'm having is as follows. Lets say I run an ANOVA/GLM/regression in SPSS and I get a significant effect, p < .05. Lets say I get a p < .05 effect, and I graph the group means in a bar chart. If I were to then add error bars according to group-relevant standard errors, then I'd find that my error bars align with the null hypothesis test, i.e., if p < .05 then the SE-based error bars will not overlap, but is p > .05, the SE-based error bars do overlap. Importantly, SPSS default is two-tailed test for these analyses. So, if plotting error bars according to SE is wrong in the case of p < .05 null hypothesis testing--as should be the case given that SE is associated with only 68% of the area under the normal curve--then why do SE-derived error bars perfectly associate with differences, p < .05?

I feel like I'm making progress here even though I've derailed my own post into a different discussion. It would be great if others had something to add to either topic because I'll have to keep talking to myself if not. Haha.

FINAL Update: I reached out to Prof. Matthews with the aforementioned question, and he was kind enough to reply. Please note, I know for a fact that he's extremely busy, so please don't contact him with personal stats questions (unless you're a U. of Newcastle student(-:). His comments are as follows:

"The point of the passage you quote is that people often add error bars to denote uncertainty and my suspicion is that bars of length +/- SE are often chosen because it gives an impression of greater accuracy (SE being less than SD or 2 x SE etc. ). Also, the idea is that SE means the 'error' might be misinterpreted as the whole of the error. I realise that you appreciate this but this was my sole intention. The business of whether error bars should overlap when making comparisons based on hypothesis testing is rather more subtle. The appropriate length with which to compare a difference is the SED , i.e. the SE of the difference, and this quantity is not simply visualised from the individual SEs. It is interesting to note that in the statistical package Genstat SEDs are encountered much often in the output than are SEs. While Genstat is not that easy to use and has not really taken off, it started life in Rothamsted, the agricultural experimental station where Fisher & Yates developed much of practical frequentist statistics and has impeccable intellectual credentials (or a least it had, I haven't used if for some time now)."

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As you've discovered, MoE and SE are very similar, which makes sense, after all MoE is directly calculated from SE. Which you use for your own understanding is entirely a personal choice.

Which one you use for publication depends almost entirely on what the standards of your field are. "Hard" sciences, for instance, tend to report SE, partly for historical reasons, partly because p-values are less of a focus than in "soft" sciences. As long as you are explicit about which one you are using, the two can be considered interchangeable; pretty much anyone reading an academic journal article knows that to get a p < .05 confidence interval you just double SE.

If you're writing for a more generalist audience, then the MoE tends to be more popular, and in my opinion, more appropriate. Lay audiences tend to assume that the $\pm$ around a measurement are hard limite, so MoE gives a more accurate representation of the uncertainty around a measurement or calculation than SE in that situation.

Ultimately, which to report is a matter of style and audience.

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