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The equation for a standard error of the mean can be written like this:

$\sqrt{\frac{s^2}{n}}$

or like this

$\sqrt{\frac{\left(\frac{\sum (X - \bar{X}) ^2}{n-1}\right)}{n}}$

They both say the same thing, but the second is more explicit. How should I refer to the first equation and how should I refer to the second equation? Eg, should I refer to the first equation as a 'simple standard error equation' and the second equation as a 'complex standard error equation'?

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    $\begingroup$ It might be better to avoid words such as "complex" which also have other meanings. Why not something like "The standard error is defined as $\sqrt{\frac{s^2}{n}}$ where $s^2$ is the sample variance $\frac{1}{n-1}\sum (X - \bar{X})^2$"? No muss, no fuss, no unnecessary name-calling, and the meaning of $s^2$ is there, front and center, for everyone to see and make note of. $\endgroup$ – Dilip Sarwate Mar 5 '15 at 16:20
  • $\begingroup$ I just wondered if there was some widely accepted terminology for the two different types of equations? $\endgroup$ – luciano Mar 5 '15 at 16:55
  • $\begingroup$ Also note that "equation" is not a good word, as these are expressions or formulas (if written in se = ... style). I second Dilip, in proposing not to use the second one at all. $\endgroup$ – psarka Mar 5 '15 at 16:59
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I don't believe you're going to find widely accepted terminology to distinguish between the two formulas, as they are both used to calculate the standard error of the mean. In my experience, you'll only find the expanded formula

$\sqrt{\frac{\left(\frac{\sum (X - \bar{X}) ^2}{n-1}\right)}{n}}$

an a setting where some of the audience might not know what variance is. I think the most important thing is that you define the terms you are going to use and that you stay consistent. I would also avoid using terms such as 'complex' and if you're going to use 'simple' it might be better as simplified, but since they are both for calculating the same quantity, I would hesitate to differentiate between them so strongly. Personally, I would use terminology like 'standard error in variance form' when referring to

$\sqrt{\frac{s^2}{n}}$

and 'sum of squares form of standard error' when referring to

$\sqrt{\frac{\left(\frac{\sum (X - \bar{X}) ^2}{n-1}\right)}{n}}$

or something of that nature.

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