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I'm trying to understand when to report the standard error.

When reporting results in a scientific paper or essay, should any mean values also have the standard error reported alongside?

For example, let's say you measure the amount of toys a child buys each week over a year. You decide to calculate monthly averages. Should you also calculate the standard error and report this alongside?

Such as "In January the monthly mean of toys bought were 4 (+/- 0.7)"

Example Dataset

Toys Bought each week
Week 1: 4
Week 2: 3
Week 3: 4
Week 4: 1
Monthly average: 3
Standard Deviation: 1.41421
Standard Error: 0.707106781187

Summary Question

Should you always report the standard error when reporting a mean value, or is the standard error only applicable to certain things?

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    $\begingroup$ I think you meant standard deviation $\endgroup$ – rep_ho Oct 8 '16 at 16:18
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It is good practice to report some measure of variability with any result. It does not really matter whether you report the standard deviation, the standard error, a confidence interval as the reader can convert between them as long as s/he knows the sample size. You do not say exactly which field of science you work in but there is probably a convention which it is best to follow so as not to annoy the reader.

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    $\begingroup$ Just to add to this answer - make sure the thing quoted is meaningful. In the example the data is clearly not going to be normally distributed, so quoting a +/- based on an interval that assumes normality can be erroneous; it might suggest children can buy a negative number of toys! $\endgroup$ – Mooks Oct 8 '16 at 17:31
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I would recommend including a confidence interval when reporting means instead of the standard error. The confidence interval of a mean is its standard error multiplied by a critical $t$ value. This lends itself to more useful interpretation in most cases. Read more about interpreting confidence intervals in the following article:

Cumming, G., & Finch, S. (2005). Inference by eye: confidence intervals and how to read pictures of data. American Psychologist, 60(2), 170.

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    $\begingroup$ Confidence interval has an implicit assumption in it, e.g. t-distribution in your example. I'd rather see the descriptive such as standard deviation, which doesn't have any assumptions in it $\endgroup$ – Aksakal Jul 30 '18 at 21:37
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For means based on a sample, if you say something like, "Mean size declined from (mean1) to (mean2)" without a measure of variability, you really have provided no evidence that there was any change at all (i.e., you don't know there was a "decline") unless some measure of variability and the sample size are included - all you can say is that the means were different. If you had measured every individual in two populations (which would not be samples), it would be fine. It's when the population is sampled that you need the statistical statement. Probably the best definition of statistics is that it's about dealing with samples.

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