Standard deviation

Covering all sciences: Economics, biology, engineering, etc... there is no value that is "high." In one application I might expect a standard deviation that is close to zero no matter what the mean is. I am looking at retention time of a chemical on a chromatography column within the limits of that columns capacity. I expect that the deviation in retention time will have almost no variability. This is in contrast to I am looking at the reproductive rate of a small fish in the lower Mississippi delta over 5 years. Here, I might be lucky if my standard deviation is less than five times my mean. What I would be more concerned about is whether a sample size of 4 is capable of accurately estimating a standard deviation.

Trigger warning to stats pedants-- I am doing some hand wavy things below to help answer this person's question in an easy to understand way. Please don't get triggered. ☺ If he wants I can "do the exactly correct math" and go through the epistemological underpinnings of inference... but that doesn't seem to be what is asked. (Not that what I do is wrong, but it would get points taken off a university exam). My view is that we cannot trust the 0.10m value, because the standard deviation is 0.30m. The standard deviation measures the variability of one bar to the next-- I think about it as "a common difference an individual bar is from the average length*". So, if the real average is indeed 5.1, many bars will be in the range 4.8 to 5.4. Using something called "the empirical rule", we might guess that somewhere in the neighborhood of 2/3 of the bars might be in this range-- check this for fun! (Note, this makes some assumptions about the "shape" of the distribution of lengths of course, but it wouldn't surprise me if this was approximately true-- say between 55% and 80% of bars in the range I mentioned.) The standard error seems to be a good fit if your goal is to see how accurate your .10 measurement difference might be. You say that you measured hundreds... let's assume that means 200 for the moment. Take your sd=0.3 and divide by sqrt(200) and we get 0.021. This is a measurement of the "precision" to which we can estimate the average length of the entire population of bars. In a similar manner to what I said above, we can be reasonably confident that the average of all bars generated by the same process (assuming the process isn't varying over time, like having different people cut them) will be in the range 5.1 +/- 0.021. It is common to multiply this 0.021 by a number around 2 to make a "95% confidence interval for the mean". If we do this, we can say something kind of like: Given this data we can be around 95% confident that the range 5.1 +/- 0.042 or 5.058 to 5.142 might contain the true average length (not strictly the correct interpretation, but close enough as an introduction to these concepts on Reddit). So, this helps you understand how much you can trust the 0.1 difference. While it probably isn't exactly 0.1, the average is probably between 0.058 and 0.142 longer than 5 meters. Please feel free to ask followup questions if you like.

As other users have mentioned in the comments, "small" and "large" are arbitrary and depend on the context. However, one very simple way to think about whether a standard deviation is small or large is as follows. If you assume that your data is normally distributed, then approximately 68% of your data points fall between one standard deviation below the mean, and one standard deviation above the mean. In the case of your data, this would mean 68% of students scored between roughly 63 and 95, and conversely 32% scored either above 95 or below 63. This gives a practical way to understand what your standard deviation is telling you (again, under the assumption that your data is normal). If you would have expected a greater percentage to fall between 63 and 95, then your standard deviation may be considered large, and if you would have expected a smaller percentage, then your standard deviation may be considered small.