Timothy -

One should care about the standard error of a statistic of interest, or standard deviation if only the distribution is of interest, if the study results are to be used for anything where accuracy matters; and if accuracy does not matter, then why care about the numbers/results at all?  It could be anything.

Having spent many years helping to provide Official Statistics on the energy industry, I know that many readers see a number and assume it is "the" exact "answer" down to the last digit reported, so I spent a lot of effort trying to convince management and data report consumers alike that accuracy was important to understand, but that is a surprisingly difficult struggle, even when what they were looking for was a difference in a number that may be smaller than a standard error. 

So I consider relative standard errors to be very important as a part of the information one may acquire regarding accuracy. 

If I see that hydroelectric generation went down 1%, but the relative standard error for that estimate is 2% (which won't even account for all of the nonsampling error), then I know that claiming hydroelectric generation has declined, is not a good idea. If a leading economic indicator goes up by 0.1%, and there is no information on accuracy, I suspect that jumping to conclusions based on that number is a bad idea.  And I suspect that for laboratory experiments, where standard error may be the most important piece of information you will know about the accuracy of your result (but beware systematic bias), that if I don't want to know your standard error, or standard deviation of a set of data, then unless it is otherwise obvious that your results are meaningful in some way, I would not much care about your results at all. 

Is the above relevant to your question?

Thanks - Jim 

PS -  Hypothetically, I guess I'm only interested in a standard deviation and other information about your data distribution if say I am intimately interested in your experiment, or say the standard deviation and skewness of the plant sizes for a set of industrial plants, due to usefulness in decision making.  If I am only interested in the accuracy of some "bottom line," then I would more likely be interested in standard errors.

What do you do with a standard deviation or standard error?  assess the data as meaningful or meaningless, assess the paper as something to consider or to use at garbage disposal 

Dear Yannis and James, 

I could add this: In whatever data analysis we should become exploring it, because standard deviation includes all the data and  if you have outliers , you should apply robust measures insteadt. usually we could do a sensibility analysis of the results with and without those outliers.

But the problem with std and std error is the same as with mean, because it is affected by all observations..and the mean should be near mode to be relevant and the distributions should be at least simetric.

Have a nice time

Helena

A good question.  I am almost always interested in the standard deviation - for 2 reasons:

1)  I want to know the spread of the data.  I think this is almost always what is of interest to most readers.  Of course, the SE can be derived from the SD if SE is needed.

2)  SD is a direct measure of data variation.   By just doubling the SD the reader can easily see the spread of the data (or 95% of it).  The same information can be obtained from the SE, but it is more work.  The reader must multiply the SE by the sqrt(n) to first obtain the SD.

I think one reason that SE is more often used is because SE error bars are smaller, hence the data look "better".  When reviewing a paper I commonly suggest to the authors that they consider what they want to show - the variation in their data, or how precisely the mean was estimated.  They almost always opt for SD.

Good discussion! Another good thing about reporting the standard deviation is that it allows us to assess the coefficient of variation, i.e., SD / mean.

 -Bob Vadas, Jr.

Thank all of you for your response.

@Yannis: From your response you want to see one or the other but don't care which one?

@Randall: I think one needs to be careful about point #1. Say that there are four treatments, and 20, 34, 17, and 23 replicates in treatments A, B, C, and D respectively. One could calculate a SD for each treatment, or one could pool all the data and calculate a pooled SD. At least in SAS it looks like SAS will report the pooled SD divided by the square root of the number of replicates in each treatment as the SE. In contrast, one probably would provide the SD for each treatment. I haven't been able to convert one to the other.

The spread of the data as described by the SD and the precision of the statistic as described by the SE both have their uses and (as others have pointed out) one can be calculated from the other if the author supplies the n for each. But, in many experimental cases the standard error is appropriate because the design of the experiment introduces sources of variation other than the variation among experimental units within the same group. For example in a randomized complete block design the variation from block-to-block would be partitioned out before calculating the mean square for error. The mean square for error would then be used for calculating the standard error of each group mean. The assumption of equal variances is being made to perform the associated testing and the variation within each each group is assumed to be equal already. For any design with two or more random effects, and for pretty much any error control structure more complex than a completely randomized design, I wouldn't know what SD means... but I would know what SEM means.

@Daniel,

Interesting, and than you for your response.

One clarification: you are using two definitions for SE that are not equivalent. The first is the mean divided by the square root of the sample size. The other is a least squares estimate. The former is easy to deal with by just reporting the sample size. The latter is not as easy.

My problem with this is that a blocking factor is trying to account for variability due to one or more unmeasured factors. So if I run an experiment where apple yield = block + nitrogen +block* nitrogen and then I run another experiment where I measure apple yield = soil water + nitrogen + water*nitrogen then I should get the same SE if soil moisture was the primary source of variability accounted for by blocking. I don't think this would work. In general, the SE is not comparable unless you have identical models because the SE is corrected for different factors.

Another scenario: I run two trials where I have apple yield = block + nitrogen + block*nitrogen. In the first trial the blocking factor is highly significant due to one part of the field being flooded. In the other trial the blocking factor is highly significant due to a shading effect from a hedgerow. Is it valid to compare the SE from these two models? Since the cause for a significant blocking effect is seldom (if ever) enumerated we will never catch this problem.

SD refers to the spread of a variable.

SE refers to the spread of an estimated parameter.

SD is a widely-used measure of (constant) risk in financial returns.

Risk varies over time, even if very high frequency (such as seconds or minutes) tick data are used.

Although it is very popular in empirical finance, SD is not a very useful measure of risk, especially if it is time varying.

SD, like the mean, is an inherent property of the data set. In contrast, SE is not b/c it's N-dependent. So SD has more biological meaning than SE, whereas the latter is relevant to statistical significance (w/ confid. intervals often reported as 2*SE). But we must always be cautious b/c statistical & biological significance aren't necessarily equivalent.

-Bob

Robert L. Vadas and Michael John McAleer; So when you present data to an audience (written or aural), you use SD or SE, or either one depending on circumstance? Do you get manuscript reviews back suggesting that you always use SD or SE, or clients that want one or the other? Do you comply, or respectfully suggest that the request is inappropriate?

Here's a paper (see Table 1) where I used 2*SE along w/ water-quality means, to emphasize the differences between the 2 ponds studied:

Vadas, R.L. Jr. 1992. The springtime phytoplankton of two calcareous ponds in Ohio. Journal of Freshwater Ecology 7: 407-418 (cf. http://www.tandfonline.com/doi/abs/10.1080/02705060.1992.9664710).

And here's another paper (see Table 4) where I just showed spatiotemporal ranges for mean-microhabitat values, to show differences among mesohabitat types & streams (as I didn’t always have N info for lit. data):

Vadas, R.L. Jr., and D.J. Orth. 1998. Use of physical variables to discriminate visually determined mesohabitat types in North American streams. Rivers 6: 143-159 (cf. http://vtshabman.blogspot.com/2008/09/beyond-pools-and-

riffles.html).

I haven’t had much trouble from sci. journals over what to present for stats, although Env. Biol. Fish. told me (many yrs. ago) that they didn't want pie charts of fish diets that didn't have error bars depicted. That paper has been on the backburner...

-Bob

There was 1 other stats issue w/ a fish-sci. journal, although it's tangential to this discussion. They didn't like my use of standard cluster anal., which can be a bit pathological in what it puts tog. into grps. That's why I started using varimax-factor anal. to better cluster variables, e.g., for phytoplankton assemblages (Vadas 1992).

BTW, have you seen this controversial paper yet?

Amrhein, V., S. Greenland, and B. McShane. 2019. Scientists rise up against statistical significance. Nature 567: 305-307 (https://www.nature.com/articles/d41586-019-00857-9).

The “red herring” there is that we should be looking a multiple explanatory variables, rather than std. single-factor analyses, something that I’ve regularly done for my aquatic/riparian research.

The use of SD or SE depends on the specific issue that might be addressed.

The p-value issue is old, but may be gaining more traction.

https://amstat.tandfonline.com/doi/abs/10.1080/00031305.2016.1154108#.XLi1We-WxPY

and another excellent recent article (below). The other articles in this special issue are also interesting.

https://amstat.tandfonline.com/doi/full/10.1080/00031305.2019.1583913#.XLi17u-WxPY