# Interpretations of time-series data with standard errors

I have plotted time-series data on estimates of pollutant emissions superimposed with standard errors of the means. See the figure below. In reality, quantities of emissions should be non-negative. However, the data I presented here were calculated from different data sources. So in the figure, the means are all non-negative, however, when superimposed with standard errors, the lower bounds of SE before 1950 might become negative, which is unreal and indicates that there is a possibility that the emissions did not happen. So I just want to understand the possibility of no emissions, can I say that

1. before 1915, the possibility of no emissions is nearly 50%.
2. from 1915 to 1950, the possibility of no emissions is at least 15.8% and at most 50%.
3. from 1950 onwards, the possibility of having pollutant emissions is at least 84.2%, and conclude that this period is significantly different from the period pre-1950 because the lower bounds are above zero which is unreal. All these are incorrect because of misunderstandings about what probabilities, error bars and confidence bands mean. I recommend reading a good textbook about them, and this paper specifically about interpreting error bars: Error bars in experimental biology.

1. No, this is wrong for multiple reasons. (i) The line does not appear to go exactly to zero, just some low positive value. (ii) A standard error is not the same thing as a probability. (iii) There are issues relating to detection limits and data quality which get worse the further back in time we go. (iv) It's unclear what pollutant this figure refers to, but if you dig far enough, you will almost certainly find examples of emissions before the date you have chosen. For example, the internal combustion engine was invented in the 19th century. Even if we could not detect its effects on the atmosphere till much later, we know for a fact that pollution was not precisely zero, which refutes your claim ("possibility of no emissions is 50%").
2. No. Again, this conflates estimates of uncertainty with probability of a quantity being zero.
3. No. Aside from the same conflation issue, there are different types of error bars and they mean different things.

Finally, there is a fundamental problem here. Your estimates of uncertainty are assuming that pollutant emissions can go negative, which is not logically possible unless you assume that we can and did pull pollutants out of the environment (which may be true, but is unlikely and cannot be judged based on your question).

[Edited in response to question edit]

A better approach to displaying and assessing uncertainty would consider what the distribution should look like. It would certainly be asymmetric, but it probably would not look like just a truncated version of the above plot. Zero might well be a very unlikely value when the appropriate distribution is used.

• thank you for your reply, I have amended my question according to the confusions you had. It is all because I simplified the explanation. The emissions are always non-negative. But before a certain time, the emissions indeed non-exist, thus equal to zero. I would like to know when emissions become above zero. My biggest puzzle is the interpretation of the standard errors. You have pointed out SE does not equal to probability. Is that true I have to convert SE to confidential intervals to have a possibility-wise understanding. – Elizabeth Sep 25 '19 at 16:26
• @Elizabeth 1) Possibility is not the same thing as probability. Probability has a technical, mathematical meaning while possibility does not. 2) Switching to confidence intervals does not fix the problem. The confidence interval also expresses uncertainty in the mean value. A credible interval is closer to what you need. But I really recommend reading a good text about these first. – mkt - Reinstate Monica Sep 26 '19 at 7:36