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I need to determine if a single number "fits" with a group of numbers. All the numbers will be percentages, so we'll use decimals.

For example:

The group is: 0.1, 0.25, 0.3, 0.4, 0.9 and we'll test against two single numbers: $y_1$ = 0.2 and $y_2$ = 0.8

The group is all low numbers, except for the one outlier, 0.9.

So I need to know that $y_1$ fits in with the group, but $y_2$ doesn't.

I'm thinking I need to do confidence interval. So I take the median of the group, in this case 0.3, then check to see if the single number is within a tolerance of the median. So in this example, if the single number is "good" it should be within the range 0.3 +/- 0.2. But how do you find that tolerance?

Note: this is not a math homework problem, so I don't get mean or variance or anything just handed to me. All I have to work with are the group of numbers and the single number.

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  • $\begingroup$ May I ask what the purpose of the outlier detection is here? Is determining that $.9$ doesn't fit of scientific interest here, or are you looking for a way of weeding out "bad" observations, such as measurement errors? $\endgroup$
    – Macro
    Commented May 8, 2012 at 3:26

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On another post I recommended the use of Dixon's ratio test to detect for a single outlier. For your original data Dixon's ratio is [X(n-1)-X(1)]/[X(n)-X(1)] in you case this ratio statistic is [0.4-0.1]/[0.9-0.1]=3/8=0.375. Compare this to the critical value for Dixon's test. I am pretty sure that values below 0.5 would be below the critical value and hence you would take 0.9 to be an outlier. Assume then that you have reason to exclude 0.9. Your data set becomes 0.1 0.25 0.3 0.4 then apply Dixon's test to see if 0.8 is an outlier. In this case the ratio is 3/7=0.428. Still likely below the critical value. Now add 0.2 to the adjusted data set. Dixon's ratio will be [0.3-0.1]/[0.4-0.1]=2/3=0.667. This can be shown to be above the critical value. I don't know the critical values for Dixon's test when n=5 but I bet it would go the way I conjectured. You need the tables for Dixon's ratio which is probably in his 1950 paper. You can find my 1982 paper in the American Statistician "A Note on the Robustness of Dixon's Ratio in Small Samples" on line and the references are in my paper.

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  • $\begingroup$ The only thing to keep in mind is that the critical values for the test statistic are designed for doing one hypothesis test. In the scenario I suggested you use Dixon's test three times. When multiple testing is done adjustments for multiplicity are made. It could be adjusted to control the Familywise error rate (FWER). One way would be to adjust the critical value so that an overall type I error is controlled to be below say 0.05. The Bonferroni bound is one way. $\endgroup$
    – Michael R. Chernick
    Commented May 8, 2012 at 4:36
  • $\begingroup$ For three tests you can test each at the 0.05/3 = 0.0164, then the FWER is guaranteed to be less than 0.05 based on the Bonferroni bound. $\endgroup$
    – Michael R. Chernick
    Commented May 8, 2012 at 4:36

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