For example, if the suspected outlier is the smallest value in the sample, but the sample also includes two unusually large values, then r12 is the appropriate test statistic. The test statistic r10 , (also called Dixon's Q), is appropriate when the sample includes only one extreme value.
Critical values for Dixon's test statistics are tabulated in Rorabacher (1991).
Term | Description |
---|---|
rij | Dixon's test statistic (i = 1, 2; j = 0, 1, 2) |
yi | the ith smallest value in the sample |
n | the number of observations in the sample |
Term | Description |
---|---|
the sample mean | |
yi | the ith smallest value in the sample |
s | the standard deviation of the sample |
n | the number of observations in the sample |
Minitab evaluates the inner integral using a 30-point Gauss-Laguerre quadrature. Minitab evaluates the outer integral using a 30-point Gauss-Hermite quadrature.
Similar to McBane (2006), Minitab calculates Fij(r) using a 16-point Gauss-Legendre quadrature method.
Also, King observes that the above approximation becomes an equality for .
Term | Description |
---|---|
rij | the Dixon test statistic where i = 1, 2; j = 0, 1, 2 |
yi | the ith smallest value in the sample |
n | the number of observations in the sample |
W.J. Dixon (1951). "Ratios Involving Extreme Values," Annals of Mathematical Statistics, 22(1), 68-78.
E.P. King (1953). "On Some Procedures for the Rejection of Suspected Data," Journal of the American Statistical Association, Vol. 48, No. 263, pages 531-533.
G.C. McBane (2006). "Programs to Compute Distribution Functions and Critical Values for Extreme Value Ratios for Outlier Detection," Journal of Statistical Software, Vol. 16, No. 3, pages 1-9.
If not, the calculated p-value represents an upper bound for the exact p-value. However, the upper bound is a very good approximation of the exact p-value.
Term | Description |
---|---|
G | Grubbs' test statistic |
n | the number of observations in the sample |
T | a random variable distributed as a t-distribution with n – 2 degrees of freedom |