Outlier Detection using Chauvenet's Criterion

What is an outlier and why we care about them

Outliers are values in a dataset that seem to be too extreme compared to other values in a dataset. Naturally, such definition is too loose, but in reality, there are lots of different views about what outliers are how to deal with them.

For now, we will stick with the idea that outliers are often a symptom of certain behavior of the underlying population, and the presence from outliers could be an indication that the underlying population is not normally distributed.

How is Chauvenet's criterion calculated?

Informally, Chauvenet's criterion is based on the idea that if the underlying population is normally distributed then it would be reasonable to find all or most of the values of a sample within a certain "band" around the mean of the distribution.

Now, this deviation is measured in relative terms, counting how many standard deviations away from the mean sample data are. In other words, we are dealing with z-scores

Mathematically, using Chauvenet's criterion, the band around the mean where the "reasonable" data values live is \(P = 1- \frac{1}{2n}\). So then, the total area where outliers live is \(\frac{1}{4n}\), allocated on the two tails, where \(n\) is the sample size

So in other words, we find a threshold value \(D_{max}\) that satisfies the following condition

\[ \Pr(Z > D_{max} = \displaystyle \frac{1}{4n}\]

and a value \(X\) will be an outlier if its association Z-score has an absolute value that exceeds \(D_{max}\), this is \(|Z| > D_{max}\).

Why are Outliers are so relevant

As we mentioned before, outliers could be a symptom indicating lack of normality, which would indicate that different statistical procedures like the z-tests and t-tests would yield unreliable conclusions.

Using Chauvenet's criterion is not the only way of finding outliers, as you can also find outliers using the IQR rule. Now, detecting outliers is just a part of larger scheme, as whenever you want to run a statistical analysis, you probably need to previously run a descriptive statistics analysis to assess the distributional properties of the sample used.