# to determine the appropriate threshold of the z-score for the non-normally distributed data

I am interested in CPI. And I need to identify outliers in the series. For that, my instructor mentioned about the number of standard deviations from the mean that a data point is. This is Z-score.

I read the following information on Z-score:" if you are looking at data that is normally distributed, then a Z-score of 3 is a good threshold. However, if you are looking at data that is not normally distributed, then you may want to use a different threshold."

My data is not normally distributed. The descriptive statistics of the data is as follows:

How can I determine the appropriate threshold for the z-score?

I also calculated z-scores for all observations in the data as follows:

The Graph in the case that the threshold is 2

The Graph in the case that the threshold is 2.5

The Graph in the case that the threshold is 3

Or, there might be different threshold from these I posted.

Editted:

Firstly, fit a loess curve as follows:

Secondly fit a smooth spline curve as follows:

• If your distribution is normally distributed with known mean and standard deviation, then about $0.27\%$ of the distribution will be $3$ or more standard deviations from the mean; with $256$ observations, you might expect $0.69$ of them to be $3$ or more standard deviations from the mean. But in the worst case with a non-normal distribution, the one-sided Chebyshev inequality suggests up to $10\%$ of the distribution can be $3$ or more standard deviations from the mean. This is not helpful in your example. Commented Jun 17 at 10:57
• Thank you! Well, how can I determine outliers in my example? @Henry
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Commented Jun 17 at 12:13
• This needs a cross-reference to your other recent thread. stats.stackexchange.com/questions/649330/… Commented Jun 19 at 23:47

TL:DR I'm not sure this is needed, but, if it is, there will not be a simple method of doing it.