# On the Mathematical Interpretation of Standard Error

:)

My colleagues are using Standard Error in a way that I do not find intuitive. Here is the situation we are dealing with:

1. We take the average for the last 12 months worth of sales data.

2. We take the standard deviation for those same 12 months.

3. This is the part that confuses me: the standard error (SE) is then calculated using the last 6 months instead of 12 to create an "average" for the last 6 months. In effect, the mathematical formula being used is SD / sqrt(6), not SD / sqrt(12).

4. The SE is then used with the average to create the Confidence Interval.

Here's where I need guidance: by using 6 months instead of 12, that implies that we care about the dispersion of the mean for 12 months while we only care about the standard error of the mean for 6 of those months. So, isn't this introducing a sort of randomness in that we are not using the full sample size available? We can't "choose" which 6 months to use out of the available 12. My colleagues insist this doesn't matter since it's an average, but that doesn't make sense to me at all. The Standard Error isn't an average, and yet it's being used as such.

Phrased another way: do we have to use the full sample size available for calculating SE? My colleagues believe we don't. I believe we have to.

Finally, my colleagues refer to this as the Confidence Interval. I thought when we use the Standard Error to create intervals that was called the Margin of Error, not Confidence interval.

Any guidance is greatly appreciated!

-valentinexoxo

• Your colleague is doing weird statistics, which may come down to the fact that most people don't know the difference between standard deviation and standard error. By using the method your colleague gives, you end up with a much wider confidence interval than you want for 95%. That saps the power from your testing methodology. I'd recommend checking out the Workplace stack for advice on how to handle a colleague making a technical mistake or how to challenge a superior on an issue like this. – Dave Dec 20 '19 at 16:04
• What you describe sounds strange, but perhaps your colleagues want to achieve an objective that is something other than a simple summary of the average sales. What is the analysis going to be used for? – Michael Lew Dec 20 '19 at 19:51
• Thank you both for the feedback; I agree that the two terms (SE and standard deviation) have a lot of confusion between the two! I'll be sure to checkout the Workplace Stack Dave. Michael: my colleagues are using this to try to get an average SE for the last 6 months. – valentinexoxo Dec 21 '19 at 17:20
• @valentinexoxo: note that the average SE is not the same as the SE of the average. – ocram Dec 22 '19 at 5:56

The standard error of the mean is $$\text{se}_{\bar{x}} = \frac{s}{\sqrt{n}}$$ when the sample is of size $$n$$.
All three quantities -- $$\bar{x}$$, $$s$$, and $$\text{se}_{\bar{x}}$$ -- must use the same $$n$$.
That is, if you want to use the past 6 months only, then the mean and the standard deviation must be estimated using the past 6 months only, and $$n=6$$ will be used to compute $$\text{se}_{\bar{x}}$$.
Then a confidence interval for the mean can be obtained by $$\bar{x} \pm k \times \text{se}_{\bar{x}}$$, where $$k$$ controls the confidence level ($$k \approx 2$$ for 95% confidence).