# Standard error of mean

I am learning statistical analysis, and am now confused about calculating the standard error of the mean.

My dataset looks like..

Condition A:

animal 1 - 3,4,4,3,6
animal 2 - 5,5,5,8,7


Condition B:

animal 3 - 3,4,5,1,6
animal 4 - 3,1,1,4,8


How can I calculate the standard error of the mean for condition A? I am confused by two methods

1) Simply taking the mean for two animals and use the means of them, like

mean for animal 1: 4

mean for animal 2: 6

overall mean : 5

SD: $\sqrt{\frac{\left[\left(6-5\right)^{2} + \left(4-5\right)^{2}\right]}{2-1}} = \sqrt{2}$

SE: $\frac{\sqrt{2}}{\sqrt{2}} = 1$

or maybe using all the measurements like

SD: $\sqrt{\frac{\left[\left(3-5\right)^{2} + \left(4-5\right)^{2} + \left(4-5\right)^2 + ...... + \left(8-5\right)^2 + \left(7-5\right)^{2}\right]}{\left(10-1\right)}} = \sqrt{\frac{24}{9}}$

SE: $\frac{\sqrt{\frac{24}{9}}}{\sqrt{10}} = 0.51$

Whichi is right? Thanks in advance.

• Novice, I have taken the liberty of encoding your notation using MathJax (essentially an embedded LaTeX renderer). If you click the "edit" link for your question (lower left) you can see how I did this, and perhaps use as a model for your future questions and answers. Jun 10, 2014 at 17:38

There is no obvious right answer, here. Different means that you calculate answer different questions.

1) The first analysis would be the correct answer for calculating the standard error of the mean (of means across animals).

2) The second answer is problematic in its present form. While the formula is applied correctly, this SE would not be usable for inference purposes as the observations are not independent (5 of them belong to one animal, the other 5 to the second).

In the spirit of this approach, you could calculate SEs for each of the animals individually (now you would average across measurements).

So more important than the question: what is the right way to calculate the measure is the question: for what purpose do you intend to use the measure. If you specify this purpose then you can restrict the ways of analyzing the data.

• It helps me a lot. Thank you for the clear answer. I will look over the research questions to clarify what is appropriate for me. Jun 10, 2014 at 20:50
• @jank Even though I agree with the limitation discussed by, I have to highlight one which did not receive enough attention, at least not in a sense that highlighted the problem of the approach. While the SE can be used to calculate the error of the mean inside a population, I am not sure you can use it at all to calculate the error among means which come in themselves from different populations. So I wonder if the error of the mean does not have to be calculated simply by error propagation in this last step. Jun 10, 2014 at 22:54
• If you for example operationally defined the mean of values for each animal as a measure of interest, nothing prevents you from analysing the distribution of this measure. It would be my intuition that sometimes the imprecision in the measurement of individual variation (and both animals and measurements in this case could be sampled randomly) could favor this agnostic method, as long as the distribution of the resulting measure is analysed carefully.
– jank
Jun 11, 2014 at 5:02