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In many plots there are the group means and +/- 1 SD. Does it make sense to use the +/- 2*SD for making a simple, rough and quick visual comparison of the means?

For example, in this plot are 10 independent groups. We can see the mean and +/- 1 SD (we don't have standard error). Then, while I am presenting this plot in a meeting someone asks if the last group could be different compared to 1st group (comparison of these two groups only, we don't care for the rest). Does it make sense to say, look the 2SD of 1st group and the 2SD of the last group are not overlapping so yes, the chances to be different is high. Is this thought somehow valid? or the response should be something like.. we can not make this conclusion from this plot?

enter image description here

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    $\begingroup$ Generally this is not a good idea. The situation is more complicated than that, but it is possible to develop relatively simple rules of thumb to compare means based on their individual standard errors (or, almost equivalently, their individual confidence intervals). Comparing SDs of data is almost meaningless because you need to factor in the sizes of the datasets. $\endgroup$
    – whuber
    Feb 26, 2020 at 14:28
  • $\begingroup$ But the 95% of the data should be within 2 SD from the mean. Hence, somewhere inside this interval should be the "real mean" which the 95%CI try to find. But still, when a pair of 2SD are not overlapped we cannot expect to be different? $\endgroup$
    – Lefty
    Feb 26, 2020 at 15:49
  • $\begingroup$ The 95% CI will typically be much smaller than the $\pm$2 SD interval, depending on how much data there are in the group. Moreover, the standard error of the difference of means will be about 30% smaller than the sum of the standard errors of each mean (assuming the two SEs are about equal). Thus, when the $\pm$2 SD intervals do not overlap, it's usually fair to conclude the means differ. But this is a truly severe test! You also need to account, in your application, for the fact you are doing many comparisons, not just one. $\endgroup$
    – whuber
    Feb 26, 2020 at 15:55

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Are the groups independently and identically distributed? The standard deviation measures the dispersion of the distribution, while if you are only interested in comparing the means it may be better to plot the standard error of the mean (i.e. confidence intervals around the point-estimate of the sample mean - not the distribution of the data itself).

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$$ \hat\mu \pm 2 \times \frac{\hat\sigma}{ \sqrt{n}} $$

is roughly the normal 95% confidence interval for the mean, but notice that it uses standard error, not standard deviation.

You can check also the What is the difference between confidence intervals and hypothesis testing? thread.

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