# How can I identify outliers?

I have a collection of data that is best described using a weighted mean, that is there are a variable number of repeated measure of a single variable for each study subject.

My goal is to identify "outliers" - I do not mean that in the Tukey sense of the word, but rather justify that some individuals are outside the "typical" range of data.

Can I do that by calculating Z-scores from the weighted means and weighted variance? Am I best using the the method of IQR to find "outliers"? What is the most robust and accepted method to identify subjects with significant higher values?

Is there a treatise or paper I can reference to justify the analysis?

• What exactly do you mean by the "Tukey sense"??
– whuber
Commented Mar 25, 2023 at 13:18

Welcome to CV.SE!

There is no such thing as a general method to identify outliers. Before you think about identifying outliers, you want to think about what is the generator for your dataset.

The generator is the process that outputs data points. For instance, if the generator is a standard normal distribution ($$\mu = 0$$ and $$\sigma = 1$$), you might see data that looks like $$X = (-1.48039459, 1.53427447, -0.76417673, 1.41476498, -0.96253268, -0.10245806, -0.19721027, 0.6472976 , 1.60706016, -1.97137177, \cdots)$$.

An outlier is a data point that was generated by a process that is different from whatever process you think generates your dataset. Following the example, if you observe next $$X_i = 100$$, that was probably generated by a different distribution.

After you understand the generator for your process, you need to think about what might cause an outlier, and how to deal with it. Is it measurement error (i.e. the true generator is the data generator plus the instrument error)? Is your hypothesis about the data generator wrong (i.e. the true data generator is not the standard normal distribution, as you previously thought)? Perhaps the true generator is a combination of multiple processes? Should you exclude data based on the answer to these questions?

Then, and only then, should you think about how to identify outliers. The method for identifying outliers should be based on what you think is the data generator for your dataset, as well as the answers to the questions above.

Your description of your dataset is not enough to devise a strategy for dealing with outliers. What is the distribution of each variable, for instance? How certain are you of it? How have you measured it?

• +1 This is well-measured, well articulated advice.
– whuber
Commented Mar 25, 2023 at 13:17
• "What is the distribution of each variable, for instance? How certain are you of it?" No real data will follow any standard distribution. I'm very certain of that. What is correct about this answer is that it makes sense to define outlier identification relative to a model assumption for the non-outliers, see jstor.org/stable/2290763. But this is a reference choice; it's not that in any real situation we could say with any certainty that such an assumption is true. An "outlier" is an observation that lies out. There is never a guarantee that it's from a "different process". Commented Mar 25, 2023 at 15:41