If the distribution is skewed, you should expect outliers.

Outliers may result from experimental problems. You would first check if the values are theoretically implausible (like a negative mass or time, or a person's height of just a few cm or of several meters...) andexclude those with good reasons. If the values are not that implausible of if you can't check that, it is likely that these values are absolutely ok and that they give you relevant information. Removing them is a very bad idea.

Paired t test for two independent sample means with different sample sizes.

https://www.google.com/url?sa=t&source=web&rct=j&url=https://www.researchgate.net/post/Paired_t_test_unequal_sample_sizes/amp&ved=2ahUKEwighoSu69HkAhUlAhAIHaidAfsQFjABegQIDhAH&usg=AOvVaw1I3T6HyJ6-b24fOR3R4qwq&cf=1

https://www.google.com/url?sa=t&source=web&rct=j&url=https://www.khanacademy.org/math/ap-statistics/two-sample-inference/two-sample-t-test-means/v/two-sample-t-test-for-difference-of-means&ved=2ahUKEwighoSu69HkAhUlAhAIHaidAfsQwqsBMA56BAgFEA8&usg=AOvVaw1qawBgeDApM9xp7DpKnjpI

If the distribution is skewed, you should expect outliers.

Outliers may result from experimental problems. You would first check if the values are theoretically implausible (like a negative mass or time, or a person's height of just a few cm or of several meters...) andexclude those with good reasons. If the values are not that implausible of if you can't check that, it is likely that these values are absolutely ok and that they give you relevant information. Removing them is a very bad idea.

I definitely agree with you, Jochen Wilhelm . A little more background might elucidate what I am trying to accomplish with this exercise. The extreme outliers are associated with a physical process distinct from other processes impacting observations in the time series. In my case, I am looking the effects of storms on the spatial derivatives in the surface wind. When a storm occurs, you get spikes in divergence and vorticity 1-2 orders of magnitude greater than the background. When I remove just a couple of observations associated with storms (maybe about 2% of observations), the sign of the mean changes, which has rather important consequences. My conclusion is that storm-induced variability is an important process for shaping the time mean wind field because the mean changes so drastically. The prevailing view has been that a long enough time series mitigates the effects of "noise" from storms on low order statistics of the wind field, which is evidently not the case.

What I am seeking is some insight into whether I could rigorously test the statistical significance of the change in mean with and without these extreme values.

Instead of removing these events, why don't you model them (set up a model that includes "storm" event)?

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maybe you could try a more robust estimator of the mean than the empirical mean (which is obviously very sensitive to outliers) ; i would try the "median of means" (MoM) estimator : it is fairly easy to compute and has strong robustness properties

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MoM groups the data in N mutually exclusive subsets of cardinality M, computes the empirical mean of each subset and returns the median of these N values

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**very** heuristically (the only purpose here is to explain why we can expect this estimator to be robust, do not take this as a proper dimensionning !) : if less than N/2 subsets are outlier-contaminated, our median estimator is correct ; therefore, assuming one outlier per subset, if the proportion of outliers is less than 1/(2M), the estimator is correct (the proper treatment gives a much tighter bound, see below)

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a summary of MoM properties can be found in the following :

http://www.cmat.edu.uy/~mordecki/seminario/seminario2.pdf (chapter 5)

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The main condition is that researcher believes his/her data is valid and representative. This is the same as to accept our datae as a matter of faith, because increasing sample size is expensive in size, cost and time. Therefore "median of means" would also be an acceptable estimator if you develop a new algorythm that fits extreme values and "median of means" for your proposal. (I ignore how to get it in your case). OK, good luck. Emilio Chaves

Fabrice Clerot , I don't understand why the MoM should be better than the mean.

I simulated some data and compared the methods and I found that both estimators have the same variance but MoM is usually biased. The direction and amount of the bias depends on the source distribution but also on the block size (for calculating the medians).

I attached the R script and a plot (generated with that script). The red line is the theoretical (true) mean, the blue dotted line is the mean of the estimates. The variance of the estimates is given in the titles above the plots.

There are lots of methods for handling variability due to sources that effect your study but are not interesting in their own right. I agree with Jochen Wilhelm ... Try using a model that accounts for the condition. One relatively straight forward method is to develop a mean model that depends on not only your observed divergence but also on a condition variable (condition_i is an element of {stormy, not stormy}).

Then, using either the lme4 or nmle packages (in R - there are other mixed effects PROC in SAS, SPSS, and Jump), you can easily construct a model that includes condition as a random effect. You'll capture the variability due to storms separately from the mean divergence.

Such a model would assume a normal distribution for the storm effects. It also assumes additivity -- storm driven divergence can be simply added to overall mean divergence. These may or may not be a justifiable assumptions... Only you know. If you really, really need to nail down the distribution of the variance in divergence due to storms, a commonly employed tool is the comprehensive Bayesian model. And that's probably a bit beyond the scope of these forums.