Rosner’s test for multiple outliers

This vignette explains Rosner’s generalized extreme Studentized deviate test for multiple potential outliers. It includes the sequential testing procedure, critical values, and the simulation tables used to assess Type I error and determine when warnings are issued.

Related vignettes:

• Distribution diagnostics, transformations, and probability plots
• Probability distributions and parameter estimation

rosnerTest

Help topic: rosnerTest.

Let $x_1, x_2, \ldots, x_n$ denote the $n$ observations. We assume that $n-k$ of these observations come from the same normal (Gaussian) distribution, and that the $k$ most “extreme” observations may or may not represent observations from a different distribution. Let $x^{*}_1, x^{*}_2, \ldots, x^{*}_{n-i}$ denote the $n-i$ observations left after omiting the $i$ most extreme observations, where $i = 0, 1, \ldots, k-1$. Let $\bar{x}^{(i)}$ and $s^{(i)}$ denote the mean and standard deviation, respectively, of the $n-i$ observations in the data that remain after removing the $i$ most extreme observations. Thus, $\bar{x}^{(0)}$ and $s^{(0)}$ denote the mean and standard deviation for the full sample, and in general

$$\bar{x}^{(i)} = \frac{1}{n-i}\sum_{j=1}^{n-i} x^{*}_j \;\;\;\;\;\; (1)$$

$$s^{(i)} = \sqrt{\frac{1}{n-i-1} \sum_{j=1}^{n-i} (x^{*}_j - \bar{x}^{(i)})^2} \;\;\;\;\;\; (2)$$

For a specified value of $i$, the most extreme observation $x^{(i)}$ is the one that is the greatest distance from the mean for that data set, i.e.,

$$x^{(i)} = \max_{j=1,2,\ldots,n-i} |x^{*}_j - \bar{x}^{(i)}| \;\;\;\;\;\; (3)$$

Thus, an extreme observation may be the smallest or the largest one in that data set.

Rosner’s test is based on the $k$ statistics $R_1, R_2, \ldots, R_k$, which represent the extreme Studentized deviates computed from successively reduced samples of size $n, n-1, \ldots, n-k+1$:

$$R_{i+1} = \frac{|x^{(i)} - \bar{x}^{(i)}|}{s^{(i)}} \;\;\;\;\;\; (4)$$

Critical values for $R_{i+1}$ are denoted $\lambda_{i+1}$ and are computed as:

$$\lambda_{i+1} = \frac{t_{p, n-i-2} (n-i-1)}{\sqrt{(n-i-2 + t^{2}_{p, n-i-2}) (n-i)}} \;\;\;\;\;\; (5)$$

where $t_{p, \nu}$ denotes the $p$’th quantile of Student’s t-distribution with $\nu$ degrees of freedom, and in this case

$$p = 1 - \frac{\alpha/2}{n - i} \;\;\;\;\;\; (6)$$

where $\alpha$ denotes the Type I error level.

The algorithm for determining the number of outliers is as follows:

1. Compare $R_k$ with $\lambda_k$. If $R_k > \lambda_k$ then conclude the $k$ most extreme values are outliers.
2. If $R_k \le \lambda_k$ then compare $R_{k-1}$ with $\lambda_{k-1}$. If $R_{k-1} > \lambda_{k-1}$ then conclude the $k-1$ most extreme values are outliers.
3. Continue in this fashion until a certain number of outliers have been identified or Rosner’s test finds no outliers at all.

Based on a study using N=1,000 simulations, Rosner’s (1983) Table 1 shows the estimated true Type I error of declaring at least one outlier when none exists for various sample sizes $n$ ranging from 10 to 100, and the declared maximum number of outliers $k$ ranging from 1 to 10. Based on that table, Roser (1983) declared that for an assumed Type I error level of 0.05, as long as $n \ge 25$, the estimated $\alpha$ levels are quite close to 0.05, and that similar results were obtained assuming a Type I error level of 0.01. However, the table below is an expanded version of Rosner’s (1983) Table 1 and shows results based on N=10,000 simulations. You can see that for an assumed Type I error of 0.05, the test maintains the Type I error fairly well for sample sizes as small as $n = 3$ as long as $k = 1$, and for $n \ge 15$, as long as $k \le 2$. Also, for an assumed Type I error of 0.01, the test maintains the Type I error fairly well for sample sizes as small as $n = 15$ as long as $k \le 7$.

Based on these results, when warn=TRUE, a warning is issued for the following cases indicating that the assumed Type I error may not be correct:

• alpha is greater than 0.01, the sample size is less than 15, and k is greater than 1.
• alpha is greater than 0.01, the sample size is at least 15 and less than 25, and k is greater than 2.
• alpha is less than or equal to 0.01, the sample size is less than 15, and k is greater than 1.
• k is greater than 10, or greater than the floor of half of the sample size (i.e., greater than the greatest integer less than or equal to half of the sample size). A warning is given for this case because simulations have not been done for this case.

Table 1a. Observed Type I Error Levels based on 10,000 Simulations, $n =$ 3 to 5.

		Assumed	$\alpha=0.05$		Assumed	$\alpha=0.01$
$n$	$k$	$\hat{\alpha}$	95% LCL	95% UCL	$\hat{\alpha}$	95% LCL	95% UCL
3	1	0.047	0.043	0.051	0.009	0.007	0.01
4	1	0.049	0.045	0.053	0.010	0.008	0.012
	2	0.107	0.101	0.113	0.021	0.018	0.024
5	1	0.048	0.044	0.053	0.008	0.006	0.009
	2	0.095	0.090	0.101	0.020	0.018	0.023

Table 1b. Observed Type I Error Levels based on 10,000 Simulations, $n =$ 6 to 10.

		Assumed	$\alpha=0.05$		Assumed	$\alpha=0.01$
$n$	$k$	$\hat{\alpha}$	95% LCL	95% UCL	$\hat{\alpha}$	95% LCL	95% UCL
6	1	0.048	0.044	0.053	0.010	0.009	0.012
	2	0.085	0.080	0.091	0.017	0.015	0.020
	3	0.141	0.134	0.148	0.028	0.025	0.031
7	1	0.048	0.044	0.053	0.013	0.011	0.015
	2	0.080	0.075	0.086	0.017	0.015	0.020
	3	0.112	0.106	0.118	0.022	0.019	0.025
8	1	0.048	0.044	0.053	0.011	0.009	0.013
	2	0.080	0.074	0.085	0.017	0.014	0.019
	3	0.102	0.096	0.108	0.020	0.017	0.023
	4	0.143	0.136	0.150	0.028	0.025	0.031
9	1	0.052	0.048	0.057	0.010	0.008	0.012
	2	0.069	0.064	0.074	0.014	0.012	0.016
	3	0.097	0.091	0.103	0.018	0.015	0.021
	4	0.120	0.114	0.126	0.024	0.021	0.027
10	1	0.051	0.047	0.056	0.010	0.008	0.012
	2	0.068	0.063	0.073	0.012	0.010	0.014
	3	0.085	0.080	0.091	0.015	0.013	0.017
	4	0.106	0.100	0.112	0.021	0.018	0.024
	5	0.135	0.128	0.142	0.025	0.022	0.028

Table 1c. Observed Type I Error Levels based on 10,000 Simulations, $n =$ 11 to 15.

		Assumed	$\alpha=0.05$		Assumed	$\alpha=0.01$
$n$	$k$	$\hat{\alpha}$	95% LCL	95% UCL	$\hat{\alpha}$	95% LCL	95% UCL
11	1	0.052	0.048	0.056	0.012	0.010	0.014
	2	0.070	0.065	0.075	0.014	0.012	0.017
	3	0.082	0.077	0.088	0.014	0.011	0.016
	4	0.101	0.095	0.107	0.019	0.016	0.021
	5	0.116	0.110	0.123	0.022	0.019	0.024
12	1	0.052	0.047	0.056	0.011	0.009	0.013
	2	0.067	0.062	0.072	0.011	0.009	0.013
	3	0.074	0.069	0.080	0.016	0.013	0.018
	4	0.088	0.082	0.093	0.016	0.014	0.019
	5	0.099	0.093	0.105	0.016	0.013	0.018
	6	0.117	0.111	0.123	0.021	0.018	0.023
13	1	0.048	0.044	0.052	0.010	0.008	0.012
	2	0.064	0.059	0.069	0.014	0.012	0.016
	3	0.070	0.065	0.075	0.013	0.011	0.015
	4	0.079	0.074	0.084	0.014	0.012	0.017
	5	0.088	0.083	0.094	0.015	0.013	0.018
	6	0.109	0.103	0.115	0.020	0.017	0.022
14	1	0.046	0.042	0.051	0.009	0.007	0.011
	2	0.062	0.057	0.066	0.012	0.010	0.014
	3	0.069	0.064	0.074	0.012	0.010	0.014
	4	0.077	0.072	0.082	0.015	0.013	0.018
	5	0.084	0.079	0.090	0.016	0.013	0.018
	6	0.091	0.085	0.097	0.017	0.014	0.019
	7	0.107	0.101	0.113	0.018	0.016	0.021
15	1	0.054	0.050	0.059	0.010	0.008	0.012
	2	0.057	0.053	0.062	0.010	0.008	0.012
	3	0.065	0.060	0.069	0.013	0.011	0.016
	4	0.073	0.068	0.078	0.014	0.011	0.016
	5	0.074	0.069	0.079	0.012	0.010	0.014
	6	0.086	0.081	0.092	0.015	0.013	0.017
	7	0.099	0.094	0.105	0.018	0.015	0.020

Table 1d. Observed Type I Error Levels based on 10,000 Simulations, $n =$ 16 to 20.

		Assumed	$\alpha=0.05$		Assumed	$\alpha=0.01$
$n$	$k$	$\hat{\alpha}$	95% LCL	95% UCL	$\hat{\alpha}$	95% LCL	95% UCL
16	1	0.052	0.048	0.057	0.010	0.008	0.012
	2	0.055	0.051	0.059	0.011	0.009	0.013
	3	0.068	0.063	0.073	0.011	0.009	0.013
	4	0.074	0.069	0.079	0.015	0.013	0.017
	5	0.077	0.072	0.082	0.015	0.013	0.018
	6	0.075	0.070	0.080	0.013	0.011	0.016
	7	0.087	0.082	0.093	0.017	0.014	0.020
	8	0.096	0.090	0.101	0.016	0.014	0.019
17	1	0.047	0.043	0.051	0.008	0.007	0.010
	2	0.059	0.054	0.063	0.011	0.009	0.013
	3	0.062	0.057	0.067	0.012	0.010	0.014
	4	0.070	0.065	0.075	0.012	0.009	0.014
	5	0.069	0.064	0.074	0.012	0.010	0.015
	6	0.071	0.066	0.076	0.015	0.012	0.017
	7	0.081	0.076	0.087	0.014	0.012	0.016
	8	0.083	0.078	0.088	0.015	0.013	0.017
18	1	0.051	0.047	0.055	0.010	0.008	0.012
	2	0.056	0.052	0.061	0.012	0.010	0.014
	3	0.065	0.060	0.070	0.012	0.010	0.015
	4	0.065	0.060	0.070	0.013	0.011	0.015
	5	0.069	0.064	0.074	0.012	0.010	0.014
	6	0.068	0.063	0.073	0.014	0.011	0.016
	7	0.072	0.067	0.077	0.014	0.011	0.016
	8	0.076	0.071	0.081	0.012	0.010	0.014
	9	0.081	0.076	0.086	0.012	0.010	0.014
19	1	0.051	0.046	0.055	0.008	0.006	0.010
	2	0.059	0.055	0.064	0.012	0.010	0.014
	3	0.059	0.054	0.064	0.011	0.009	0.013
	4	0.061	0.057	0.066	0.012	0.010	0.014
	5	0.067	0.062	0.072	0.013	0.010	0.015
	6	0.066	0.061	0.071	0.011	0.009	0.013
	7	0.069	0.064	0.074	0.013	0.011	0.015
	8	0.074	0.069	0.079	0.012	0.010	0.014
	9	0.082	0.077	0.087	0.015	0.013	0.018
20	1	0.053	0.048	0.057	0.011	0.009	0.013
	2	0.056	0.052	0.061	0.010	0.008	0.012
	3	0.060	0.056	0.065	0.009	0.007	0.011
	4	0.063	0.058	0.068	0.012	0.010	0.014
	5	0.063	0.059	0.068	0.014	0.011	0.016
	6	0.063	0.058	0.067	0.011	0.009	0.013
	7	0.065	0.061	0.070	0.011	0.009	0.013
	8	0.070	0.065	0.076	0.012	0.010	0.014
	9	0.076	0.070	0.081	0.013	0.011	0.015
	10	0.081	0.076	0.087	0.012	0.010	0.014

Table 1e. Observed Type I Error Levels based on 10,000 Simulations, $n =$ 21 to 25.

		Assumed	$\alpha=0.05$		Assumed	$\alpha=0.01$
$n$	$k$	$\hat{\alpha}$	95% LCL	95% UCL	$\hat{\alpha}$	95% LCL	95% UCL
21	1	0.054	0.049	0.058	0.013	0.011	0.015
	2	0.054	0.049	0.058	0.012	0.010	0.014
	3	0.058	0.054	0.063	0.012	0.010	0.014
	4	0.058	0.054	0.063	0.011	0.009	0.013
	5	0.064	0.059	0.069	0.013	0.011	0.016
	6	0.066	0.061	0.071	0.012	0.010	0.015
	7	0.063	0.058	0.068	0.013	0.011	0.015
	8	0.066	0.061	0.071	0.010	0.008	0.012
	9	0.073	0.068	0.078	0.013	0.011	0.015
	10	0.071	0.066	0.076	0.012	0.010	0.014
22	1	0.047	0.042	0.051	0.010	0.008	0.012
	2	0.058	0.053	0.062	0.012	0.010	0.015
	3	0.056	0.052	0.061	0.010	0.008	0.012
	4	0.059	0.055	0.064	0.012	0.010	0.014
	5	0.061	0.057	0.066	0.009	0.008	0.011
	6	0.063	0.058	0.068	0.013	0.010	0.015
	7	0.065	0.060	0.070	0.013	0.010	0.015
	8	0.065	0.060	0.070	0.014	0.012	0.016
	9	0.065	0.060	0.070	0.012	0.010	0.014
	10	0.067	0.062	0.072	0.012	0.009	0.014
23	1	0.051	0.047	0.056	0.008	0.007	0.010
	2	0.056	0.052	0.061	0.010	0.009	0.012
	3	0.056	0.052	0.061	0.011	0.009	0.013
	4	0.062	0.057	0.066	0.011	0.009	0.013
	5	0.061	0.056	0.065	0.010	0.009	0.012
	6	0.060	0.055	0.064	0.012	0.010	0.014
	7	0.062	0.057	0.066	0.011	0.009	0.013
	8	0.063	0.058	0.068	0.012	0.010	0.014
	9	0.066	0.061	0.071	0.012	0.010	0.014
	10	0.068	0.063	0.073	0.014	0.012	0.017
24	1	0.051	0.046	0.055	0.010	0.008	0.012
	2	0.056	0.051	0.060	0.011	0.009	0.013
	3	0.058	0.053	0.062	0.010	0.008	0.012
	4	0.060	0.056	0.065	0.013	0.011	0.015
	5	0.057	0.053	0.062	0.012	0.010	0.014
	6	0.065	0.060	0.069	0.011	0.009	0.013
	7	0.062	0.057	0.066	0.012	0.010	0.014
	8	0.060	0.055	0.065	0.012	0.010	0.014
	9	0.066	0.061	0.071	0.012	0.010	0.014
	10	0.064	0.059	0.068	0.012	0.010	0.015
25	1	0.054	0.050	0.059	0.012	0.009	0.014
	2	0.055	0.051	0.060	0.010	0.008	0.012
	3	0.057	0.052	0.062	0.011	0.009	0.013
	4	0.055	0.051	0.060	0.011	0.009	0.013
	5	0.060	0.055	0.065	0.012	0.010	0.014
	6	0.060	0.055	0.064	0.011	0.009	0.013
	7	0.057	0.052	0.061	0.011	0.009	0.013
	8	0.062	0.058	0.067	0.011	0.009	0.013
	9	0.058	0.053	0.062	0.012	0.010	0.014
	10	0.061	0.057	0.066	0.010	0.008	0.012

Table 1f. Observed Type I Error Levels based on 10,000 Simulations, $n =$ 26 to 30.

		Assumed	$\alpha=0.05$		Assumed	$\alpha=0.01$
$n$	$k$	$\hat{\alpha}$	95% LCL	95% UCL	$\hat{\alpha}$	95% LCL	95% UCL
26	1	0.051	0.047	0.055	0.012	0.010	0.014
	2	0.057	0.053	0.062	0.013	0.011	0.015
	3	0.055	0.050	0.059	0.012	0.010	0.014
	4	0.055	0.051	0.060	0.010	0.008	0.012
	5	0.058	0.054	0.063	0.011	0.009	0.013
	6	0.061	0.056	0.066	0.012	0.010	0.014
	7	0.059	0.054	0.064	0.011	0.009	0.013
	8	0.060	0.056	0.065	0.010	0.008	0.012
	9	0.060	0.056	0.065	0.011	0.009	0.013
	10	0.061	0.056	0.065	0.011	0.009	0.013
27	1	0.050	0.046	0.054	0.009	0.007	0.011
	2	0.054	0.050	0.059	0.011	0.009	0.013
	3	0.062	0.057	0.066	0.012	0.010	0.014
	4	0.063	0.058	0.068	0.011	0.009	0.013
	5	0.051	0.047	0.055	0.010	0.008	0.012
	6	0.058	0.053	0.062	0.011	0.009	0.013
	7	0.060	0.056	0.065	0.010	0.008	0.012
	8	0.056	0.052	0.061	0.010	0.008	0.012
	9	0.061	0.056	0.066	0.012	0.010	0.014
	10	0.055	0.051	0.060	0.008	0.006	0.010
28	1	0.049	0.045	0.053	0.010	0.008	0.011
	2	0.057	0.052	0.061	0.011	0.009	0.013
	3	0.056	0.052	0.061	0.012	0.009	0.014
	4	0.057	0.053	0.062	0.011	0.009	0.013
	5	0.057	0.053	0.062	0.010	0.008	0.012
	6	0.056	0.051	0.060	0.010	0.008	0.012
	7	0.057	0.052	0.061	0.010	0.008	0.012
	8	0.058	0.054	0.063	0.011	0.009	0.013
	9	0.054	0.050	0.058	0.011	0.009	0.013
	10	0.062	0.057	0.067	0.011	0.009	0.013
29	1	0.049	0.045	0.053	0.011	0.009	0.013
	2	0.053	0.048	0.057	0.010	0.008	0.012
	3	0.056	0.051	0.060	0.010	0.009	0.012
	4	0.055	0.050	0.059	0.010	0.008	0.012
	5	0.056	0.051	0.060	0.010	0.008	0.012
	6	0.057	0.053	0.062	0.012	0.010	0.014
	7	0.055	0.050	0.059	0.010	0.008	0.012
	8	0.057	0.052	0.061	0.011	0.009	0.013
	9	0.056	0.051	0.061	0.011	0.009	0.013
	10	0.057	0.052	0.061	0.011	0.009	0.013
30	1	0.050	0.046	0.054	0.009	0.007	0.011
	2	0.054	0.049	0.058	0.011	0.009	0.013
	3	0.056	0.052	0.061	0.012	0.010	0.015
	4	0.054	0.049	0.058	0.010	0.008	0.012
	5	0.058	0.053	0.063	0.012	0.010	0.014
	6	0.062	0.058	0.067	0.012	0.010	0.014
	7	0.056	0.052	0.061	0.012	0.010	0.014
	8	0.059	0.054	0.064	0.011	0.009	0.013
	9	0.056	0.052	0.061	0.010	0.009	0.012
	10	0.058	0.053	0.062	0.012	0.010	0.015

Table 1g. Observed Type I Error Levels based on 10,000 Simulations, n = 31 to 35.

		Assumed	$\alpha=0.05$		Assumed	$\alpha=0.01$
$n$	$k$	$\hat{\alpha}$	95% LCL	95% UCL	$\hat{\alpha}$	95% LCL	95% UCL
31	1	0.051	0.047	0.056	0.009	0.007	0.011
	2	0.054	0.050	0.059	0.010	0.009	0.012
	3	0.053	0.049	0.058	0.010	0.008	0.012
	4	0.055	0.050	0.059	0.010	0.008	0.012
	5	0.053	0.049	0.057	0.011	0.009	0.013
	6	0.055	0.050	0.059	0.010	0.008	0.012
	7	0.055	0.050	0.059	0.012	0.010	0.014
	8	0.056	0.051	0.060	0.010	0.008	0.012
	9	0.057	0.053	0.062	0.011	0.009	0.013
	10	0.058	0.053	0.062	0.011	0.009	0.013
32	1	0.054	0.049	0.058	0.010	0.008	0.012
	2	0.054	0.050	0.059	0.010	0.008	0.012
	3	0.052	0.047	0.056	0.009	0.007	0.011
	4	0.056	0.051	0.060	0.011	0.009	0.013
	5	0.056	0.052	0.061	0.011	0.009	0.013
	6	0.055	0.051	0.060	0.011	0.009	0.013
	7	0.055	0.051	0.060	0.010	0.008	0.012
	8	0.055	0.051	0.060	0.010	0.008	0.012
	9	0.057	0.053	0.062	0.012	0.010	0.014
	10	0.054	0.050	0.059	0.010	0.008	0.012
33	1	0.051	0.046	0.055	0.011	0.009	0.013
	2	0.055	0.051	0.060	0.011	0.009	0.013
	3	0.056	0.052	0.061	0.010	0.008	0.012
	4	0.052	0.048	0.057	0.010	0.008	0.012
	5	0.055	0.050	0.059	0.010	0.008	0.012
	6	0.058	0.053	0.062	0.011	0.009	0.013
	7	0.057	0.052	0.061	0.010	0.008	0.012
	8	0.058	0.054	0.063	0.011	0.009	0.013
	9	0.057	0.053	0.062	0.012	0.010	0.014
	10	0.055	0.051	0.060	0.011	0.009	0.013
34	1	0.052	0.048	0.056	0.009	0.007	0.011
	2	0.053	0.049	0.058	0.011	0.009	0.013
	3	0.055	0.050	0.059	0.012	0.010	0.014
	4	0.056	0.052	0.061	0.010	0.008	0.012
	5	0.053	0.048	0.057	0.009	0.007	0.011
	6	0.055	0.050	0.059	0.010	0.008	0.012
	7	0.052	0.048	0.057	0.012	0.010	0.014
	8	0.055	0.050	0.059	0.009	0.008	0.011
	9	0.055	0.051	0.060	0.011	0.009	0.013
	10	0.054	0.049	0.058	0.010	0.008	0.012
35	1	0.051	0.046	0.055	0.010	0.009	0.012
	2	0.054	0.049	0.058	0.010	0.009	0.012
	3	0.055	0.050	0.059	0.010	0.009	0.012
	4	0.053	0.048	0.057	0.011	0.009	0.013
	5	0.056	0.051	0.061	0.011	0.009	0.013
	6	0.055	0.051	0.059	0.012	0.010	0.014
	7	0.054	0.050	0.059	0.011	0.009	0.013
	8	0.054	0.049	0.058	0.011	0.009	0.013
	9	0.061	0.056	0.066	0.012	0.010	0.014
	10	0.053	0.048	0.057	0.011	0.009	0.013

Table 1h. Observed Type I Error Levels based on 10,000 Simulations, n = 36 to 40.

		Assumed	$\alpha=0.05$		Assumed	$\alpha=0.01$
$n$	$k$	$\hat{\alpha}$	95% LCL	95% UCL	$\hat{\alpha}$	95% LCL	95% UCL
36	1	0.047	0.043	0.051	0.010	0.008	0.012
	2	0.058	0.053	0.062	0.012	0.010	0.015
	3	0.052	0.047	0.056	0.009	0.007	0.011
	4	0.052	0.048	0.056	0.012	0.010	0.014
	5	0.052	0.048	0.057	0.010	0.008	0.012
	6	0.055	0.051	0.059	0.012	0.010	0.014
	7	0.053	0.048	0.057	0.011	0.009	0.013
	8	0.056	0.051	0.060	0.012	0.010	0.014
	9	0.056	0.051	0.060	0.011	0.009	0.013
	10	0.056	0.051	0.060	0.011	0.009	0.013
37	1	0.050	0.046	0.055	0.010	0.008	0.012
	2	0.054	0.049	0.058	0.011	0.009	0.013
	3	0.054	0.049	0.058	0.011	0.009	0.013
	4	0.054	0.050	0.058	0.010	0.008	0.012
	5	0.054	0.049	0.058	0.010	0.008	0.012
	6	0.054	0.050	0.058	0.011	0.009	0.013
	7	0.055	0.051	0.060	0.010	0.008	0.012
	8	0.055	0.050	0.059	0.011	0.009	0.013
	9	0.053	0.049	0.058	0.011	0.009	0.013
	10	0.049	0.045	0.054	0.009	0.007	0.011
38	1	0.049	0.045	0.053	0.009	0.007	0.011
	2	0.052	0.047	0.056	0.008	0.007	0.010
	3	0.054	0.050	0.059	0.011	0.009	0.013
	4	0.055	0.050	0.059	0.011	0.009	0.013
	5	0.056	0.052	0.061	0.012	0.010	0.014
	6	0.055	0.050	0.059	0.011	0.009	0.013
	7	0.049	0.045	0.053	0.009	0.007	0.011
	8	0.052	0.048	0.057	0.010	0.008	0.012
	9	0.054	0.050	0.059	0.010	0.009	0.012
	10	0.055	0.050	0.059	0.011	0.009	0.013
39	1	0.047	0.043	0.051	0.010	0.008	0.012
	2	0.055	0.051	0.059	0.010	0.008	0.012
	3	0.053	0.049	0.057	0.010	0.008	0.012
	4	0.053	0.049	0.058	0.010	0.009	0.012
	5	0.052	0.048	0.057	0.010	0.008	0.012
	6	0.053	0.049	0.058	0.010	0.008	0.012
	7	0.057	0.052	0.061	0.011	0.009	0.013
	8	0.057	0.053	0.062	0.012	0.010	0.014
	9	0.050	0.046	0.055	0.010	0.008	0.012
	10	0.056	0.051	0.060	0.011	0.009	0.013
40	1	0.049	0.045	0.054	0.010	0.008	0.012
	2	0.052	0.048	0.057	0.010	0.009	0.012
	3	0.055	0.050	0.059	0.011	0.009	0.013
	4	0.054	0.050	0.059	0.011	0.009	0.013
	5	0.054	0.050	0.059	0.010	0.008	0.012
	6	0.049	0.045	0.053	0.010	0.008	0.012
	7	0.056	0.051	0.060	0.011	0.009	0.013
	8	0.054	0.050	0.059	0.011	0.009	0.013
	9	0.047	0.043	0.052	0.010	0.008	0.011
	10	0.058	0.054	0.063	0.010	0.008	0.012