# Rosner's Outlier Test

Rosner's test for multiple outliers is used by VSP to detect up to 10 outliers among the selected data values. This test will detect outliers that are either much smaller or much larger than the rest of the data. Rosner's approach is designed to avoid the problem of masking, where an outlier that is close in value to another outlier can go undetected.

Rosner's test is appropriate only when the data, excluding the suspected outliers, are approximately normally distributed, and when the sample size is greater than or equal to 25.

Data should not be excluded from analysis solely on the basis of the results of this or any other statistical test. If any values are flagged as possible outliers, further investigation is recommended to determine whether there is a plausible explanation that justifies removing or replacing them.

## Performing Rosner's Test

The $$n$$ observed values are ordered from smallest to largest. We specify the maximum number of suspected outliers $$k$$ , where $$k$$ is between 1 and 10. Then we calculate a series of test statistics by removing the datum (large or small) that is farthest from the mean and recomputing the test statistic according to the following equation:

$$\large R_{i+1} = \frac{|x^{(i)} - \bar x^{(i)}|}{s^{(i)}}$$

where $$\bar x^{(i)}$$ is the sample mean and $$s^{(i)}$$ is the standard deviation of the data after the $$i$$ most extreme observations have been removed, and $$x^{(i)}$$ is the observation in that subset of the data that is furthest from $$\bar x^{(i)}$$ .

Once all of the test statistics $$R_1 \dotsc R_k$$ are computed, a series of hypothesis tests are performed. We first test the hypothesis that there are $$k$$ outliers by comparing $$R_k$$ to the critical value $$\lambda_k$$ , obtained from a table (Table A-4, EPA) for the specified significance level $$\alpha$$.

If $$R_k > \lambda_k$$ , then the test is significant and we can reject the null hypothesis that there are no outliers in the data and conclude that the $$k$$ most extreme values are outliers.

If $$R_k \leq \lambda_k$$ , we move on to test the hypothesis that there are $$k-1$$ outliers by comparing $$R_{k-1}$$ to the critical value $$\lambda_{k-1}$$ . This process is continued until one of the tests is significant and we can conclude that there are a certain number of outliers, or until all the tests have been performed and none were found to be significant. If none of the tests are significant, then we conclude that there are no outliers in the data.

## References:

EPA. 2006. Data Quality Assessment: Statistical Methods for Practitioners EPA QA/G-9S, EPA/240/B-06/003, U.S. Environmental Protection Agency, Office of Environmental Information, Washington DC.

Gilbert, R.O. 1987. Statistical Methods for Environmental Pollution Monitoring. Wiley & Sons, Inc., New York, NY.

Rosner, B. 1983. Percentage Points for a Generalized ESD Many-Outlier Procedure. Technometrics . Vol. 25, Pages 165-172.