What is an outlier? Explain how you might screen for outliers and what would you do if you found them in your dataset. Also, explain what an inlier is and how you might screen for them and what would you do if you found them in your dataset

*Outliers:*

– An observation point that is distant from other observations

– Can occur by chance in any distribution

– Often, they indicate measurement error or a heavy-tailed distribution

– Measurement error: discard them or use robust statistics

– Heavy-tailed distribution: high skewness, can’t use tools assuming a normal distribution

– Three-sigma rules (normally distributed data): 1 in 22 observations will differ by twice the standard deviation from the mean

– Three-sigma rules: 1 in 370 observations will differ by three times the standard deviation from the mean

Three-sigma rules example: in a sample of 1000 observations, the presence of up to 5 observations deviating from the mean by more than three times the standard deviation is within the range of what can be expected, being less than twice the expected number and hence within 1 standard deviation of the expected number (Poisson distribution).

If the nature of the distribution is known a priori, it is possible to see if the number of outliers deviate significantly from what can be expected. For a given cutoff (samples fall beyond the cutoff with probability p), the number of outliers can be approximated with a Poisson distribution with lambda=pn. Example: if one takes a normal distribution with a cutoff 3 standard deviations from the mean, p=0.3% and thus we can approximate the number of samples whose deviation exceed 3 sigmas by a Poisson with lambda=3

*Identifying outliers:*

– No rigid mathematical method

– Subjective exercise: be careful

– Boxplots

– QQ plots (sample quantiles Vs theoretical quantiles)

*Handling outliers:*

– Depends on the cause

– Retention: when the underlying model is confidently known

– Regression problems: only exclude points which exhibit a large degree of influence on the estimated coefficients (Cook’s distance)

*Inlier:*

– Observation lying within the general distribution of other observed values

– Doesn’t perturb the results but are non-conforming and unusual

– Simple example: observation recorded in the wrong unit (°F instead of °C)

*Identifying inliers:*

– Mahalanobi’s distance

– Used to calculate the distance between two random vectors

– Difference with Euclidean distance: accounts for correlations

– Discard them

Source

One way of detecting outliers is if the mean/average of a data set is significantly different from the median of the data set. For example, if you have a data set,

10,20,34, 45, 50,60,93 the median of the data set is 45 while the mean/average = 312/7 = 44.5 ~ 45

If now, we add -200 to this set, the median is 45+34/2 = 39.5~40 whereas the mean = 112/8 = 14

You can try the opposite by adding, say 400 to our original data set. In that case, median will still be 40, while the mean = 712/8=89.

In datasets that are normally distributed, mean and median will be very close as in our first example.