# Finding outlier values for non-normally distributed data

I have univariate data (38 is the sample size).The distribution is certainly not normal. How can I find the outliers? I used z-score but am not getting a desired result.

• One of many definitions of outliers is that they are values surprising on the current model of the data. As you are clear that a normal distribution is an implausible model, you should assess your data in the context of a better model (lognormal? gamma? we can't tell). There is no canonical, universal definition of outliers that makes them unambiguously identifiable. Conversely, $z$ scores tell you little or nothing here as they are based on mean and SD which may well be unhelpful summaries any way. For better advice, post your data. For more advice, see several threads here on outliers. – Nick Cox May 20 '16 at 14:12
• Compare stats.stackexchange.com/questions/78063/… for the origin of a similar statement. – Nick Cox May 20 '16 at 14:13
• Thanks for your reply Nick Cox. Well here are my data: 2668, 159, 1167, 765, 491, 979, 1216, 403, 1459, 980, 271, 591, 215, 296, 871, 523, 1105, 698, 852, 409, 493, 252, 818, 743, 731, 439, 488, 306, 546, 1170, 201, 350, 1963, 653, 597, 377, 345, 758. Do I need non parametric tests? If so,what tests should i use? – user3798510 May 20 '16 at 14:45
• How could there be a non-parametric test for an outlier? – Nick Cox May 20 '16 at 14:50
• well..by that i meant if I should consider detecting outliers without assuming that the data follows any sort of distribution.What does the data suggest? – user3798510 May 20 '16 at 14:55

## 1 Answer

Given some data, my first line of attack is always a plot. A quantile plot shows (ordered) values plotted against cumulative probabilities, or if you like an implicit rank. Here the original data are clearly all positive and collectively positively skewed (left-hand panel), but a logarithmic scale is thereby suggested. When that is tried (right-hand panel), the data look like a very respectable sample from a lognormal distribution, i.e. the logarithms look like a very respectable sample from a normal distribution. Here "very respectable" means very close to the straight line fit which a perfect sample would show. I see no reason to shout "outlier" here, but every reason to work with a logarithmic transformation or a logarithmic link function.

• so,you are saying that the data doesn't contain outliers? – user3798510 May 20 '16 at 15:40
• The whole point is that calling outliers depends on a model for the data, which in turn depends on subject-matter knowledge. Clearly I know nothing about the scientific or practical context for the data and how far a logarithmic scale makes sense for analyses. But if you press me for an emphatic Yes or No, I give an emphatic No, with as already said a rider that you must think on logarithmic scale. – Nick Cox May 20 '16 at 15:47