probability. The old mean and standard deviation (σ) method supposes data match normal distribution, and then use 95% (2σ) or 99.7% (3σ) boundary to identify "outliers". Boxplot devides ordered data into four quarters, let the lower hinge (defined as the 25th percentile) be q1 and upper hinge (the 75th percentile) be q3, then call the difference of them IQR (q3-q1), data out of fence q1-1.5*IQR and q3+1.5*IQR are identified as outliers. Median and MAD method calculates the median and MADn of the data, MADn = b*medi|xi-medjxj|, medixi is the median of data {x1,,…,xn}, and b=1.4826, then uses median±kMADn to detect outliers. While mean and standard deviation method uses fixed coefficient (2 or 3) multiplied by standard deviation (σ), Chauvenet"s criterion uses variable coefficient related to the number of data. In recent work, Ross [13] suggested to return to the criterion of Peirce, who is a forerunner of the probability approach. Besides, Rousseeuw presented some robust algorithms like LMS and LTS [14], which were developed from the well-known least squares (LS) method. Differing from LS by using idea of the least sum of squares as a regression estimator, LMS uses the least median of squares and LTS uses the least trimmed squares. After that, outliers are identified as those points that lie far away from the robust fit (a same reasonable ratio like 1.5 of Boxplot or k of Median and MAD is predetermined). Since we have so many algorithms, we will face how to choose, especially when contradiction takes place. Turkey advised, "It is perfectly proper to use both classical and robust methods routinely, and only worry when they differ enough to matter. But when they differ, you should think hard" [9]. We are still in difficulty, which can be seen by some examples here. In Table 1, to median and MAD method (abbreviation "MAD"), we use k=3; mean and standard deviation is abbreviated to "Mean"; robust method LTS and LMS are included in PROCESS [14].

The following data are all taken from web [16]. ROSNER contains 10 monthly diastolic blood pressure measurements, GRUBBS1 data are strengths of hard-drawn copper wire, GRUBBS3 data are percent elongations of plastic material and CUSHNY measures the difference in hours of sleep due to two different drugs on ten patients. Obvious and common outliers are marked by italic and bold font. 1. ROSNER: 90, 93, 86, 92, 95, 83, 75, 40, 88, 80 2. BARNETT: 3, 4, 7, 8, 10, *949, 951* 3. GRUBBS1: 568, 570, 570, 570, 572, 572, 572, 578, 584, 596 4. GRUBBS3: 2.02, 2.22, 3.04, 3.23, 3.59, 3.73, 3.94, 4.05, 4.11, 4.13 5. CUSHNY: 0, 0.8, 1, 1.2, 1.3. 1.3, 1.4, 1.8, 2.4, 4.6

Table 1: Outliers detected by various methods.

Mean Boxplot MAD Piece"s LTS LMS IIR

ROSNER none 40 40 40 40 40 40 BARNETT none none 949,951 none 949,951 949,951 949,951 GRUBBS1 none 596 584,596 596 578,584,596 578,584,596 596 GRUBBS3 none none none none 2.02,2.22 2.02,2.22 2.02,2.22 CUSHNY none 4.6 4.6 4.6 0,2.4,4.6 2.4,4.6 4.6

What should be mentioned is Piece"s criterion to BARNETT data, if firstly assuming two doubtful observation, 949 and 951 can be detected; and MAD to GRUBBS3, if using k=2, 2.02 and 2.22 can be detected. Different methods lead to different results. Is 2