Background Threshold validation

Free upper background limit calculations for environmental datasets: UCL, UPL, UTL and USL with distribution selection and non-detect handling.

Scenarios: 9
Values agreeing: 40
Values differing: 2
Validated against: USEPA ProUCL, R EnvStats / nortest

Open Background Threshold

Prediction, tolerance and simultaneous limits are compared against USEPA ProUCL at three sample sizes. Prediction and tolerance limits are also compared against R EnvStats. Tolerance factors follow ProUCL's published table up to thirty results and the Natrella approximation above it; EnvStats computes the exact non-central t factor, which differs by about 0.1 per cent above thirty results. The censored-data estimators are shared with Descriptive Statistics.

Scenarios

Upper tolerance limit covering 95% of the population at 95% confidence (N10)Boundary case
Upper tolerance limit covering 95% of the population at 95% confidence (N50)Boundary case
Upper prediction limit for a single future result (N10)Typical use
Upper prediction limit for a single future result (N20)Typical use
Upper prediction limit for a single future result (N50)Typical use
Upper simultaneous limit bounding every observation at once (N10)Typical use
Upper simultaneous limit bounding every observation at once (N20)Typical use
Upper simultaneous limit bounding every observation at once (N50)Typical use
Upper tolerance limit covering 95% of the population at 95% confidence (N20)Boundary case

Upper tolerance limit covering 95% of the population at 95% confidence (N10)

Boundary case

Input

10 values

value
4.2
5.1
6.3
7.8
9.1
11.4
3.3
8.8
12.1
6.6

Settings: limitType = utl, distribution = normal, confidence = 0.95, coverage = 0.95

Against USEPA ProUCL (ProUCL 5.2.0)

Value	USEPA ProUCL	EnviroAutomate	Difference	Agreement band	Verdict
Sample size (n) n	10	10	0	exact match	Agree
Limit limitValue	15.997954893310197	15.997954893310197	0	within 0.000001 relative	Agree

Against R EnvStats / nortest (EnvStats 3.1.0, nortest 1.0-4, R 4.6.1)

Value	R EnvStats / nortest	EnviroAutomate	Difference	Agreement band	Verdict
Sample size (n) n	10	10	0	exact match	Agree
Mean mean	7.47	7.469999999999999	8.88e-16	within 0.000001 relative	Agree
Standard deviation sd	2.929561969532874	2.9295619695328745	4.44e-16	within 0.000001 relative	Agree
Limit limitValue	15.997847709663812	15.997954893310197	0.000107	within 0.000001 relative	Differs
Why these differ: EnvStats computes the tolerance factor exactly, from the non-central t distribution. This toolkit uses the factor ProUCL uses - a published table up to thirty results, and the Natrella approximation above it - because we assume a reviewer checking this limit will be checking it against ProUCL, and the two agree here to the last digit. Below thirty results the table and the exact value are within a rounding of each other. Above it the approximation is about a tenth of a percent low against the exact figure, which is the difference visible on this row. Adopting the table was itself a correction: the approximation alone gave 2.8748 at ten results where the table gives 2.911, and that 1.2% error understated a background threshold.

Upper tolerance limit covering 95% of the population at 95% confidence (N50)

Boundary case

Input

50 values

value
6.64
10.98
8.53
12.87
10.42
7.97
12.31
9.86
7.41
11.75
9.3
6.85
11.19
8.74
13.08
10.63
8.18
12.52
10.07
7.62
11.96
9.51
7.06
11.4
8.95
13.29
10.84
8.39
12.73
10.28
7.83
12.17
9.72
7.27
11.61
9.16
6.71
11.05
8.6
12.94
10.49
8.04
12.38
9.93
7.48
11.82
9.37
6.92
11.26
8.81

Settings: limitType = utl, distribution = normal, confidence = 0.95, coverage = 0.95

Against USEPA ProUCL (ProUCL 5.2.0)

Value	USEPA ProUCL	EnviroAutomate	Difference	Agreement band	Verdict
Sample size (n) n	50	50	0	exact match	Agree
Limit limitValue	13.979968008806239	13.979968005971887	2.83e-9	within 0.000001 relative	Agree

Against R EnvStats / nortest (EnvStats 3.1.0, nortest 1.0-4, R 4.6.1)

Value	R EnvStats / nortest	EnviroAutomate	Difference	Agreement band	Verdict
Sample size (n) n	50	50	0	exact match	Agree
Mean mean	9.8978	9.897800000000002	1.78e-15	within 0.000001 relative	Agree
Standard deviation sd	1.9830890045582932	1.9830890045582927	4.44e-16	within 0.000001 relative	Agree
Limit limitValue	13.992865741217095	13.979968005971887	0.0129	within 0.000001 relative	Differs
Why these differ: EnvStats computes the tolerance factor exactly, from the non-central t distribution. This toolkit uses the factor ProUCL uses - a published table up to thirty results, and the Natrella approximation above it - because we assume a reviewer checking this limit will be checking it against ProUCL, and the two agree here to the last digit. Below thirty results the table and the exact value are within a rounding of each other. Above it the approximation is about a tenth of a percent low against the exact figure, which is the difference visible on this row. Adopting the table was itself a correction: the approximation alone gave 2.8748 at ten results where the table gives 2.911, and that 1.2% error understated a background threshold.

Upper prediction limit for a single future result (N10)

Typical use

Input

10 values

value
4.2
5.1
6.3
7.8
9.1
11.4
3.3
8.8
12.1
6.6

Settings: limitType = upl, distribution = normal, confidence = 0.95, coverage = 0.95

Against USEPA ProUCL (ProUCL 5.2.0)

Value	USEPA ProUCL	EnviroAutomate	Difference	Agreement band	Verdict
Sample size (n) n	10	10	0	exact match	Agree
Limit limitValue	13.102332124158544	13.102332085118997	3.90e-8	within 0.000001 relative	Agree

Against R EnvStats / nortest (EnvStats 3.1.0, nortest 1.0-4, R 4.6.1)

Value	R EnvStats / nortest	EnviroAutomate	Difference	Agreement band	Verdict
Sample size (n) n	10	10	0	exact match	Agree
Mean mean	7.47	7.469999999999999	8.88e-16	within 0.000001 relative	Agree
Standard deviation sd	2.929561969532874	2.9295619695328745	4.44e-16	within 0.000001 relative	Agree
Limit limitValue	13.102332085119	13.102332085118997	3.55e-15	within 0.000001 relative	Agree

Upper prediction limit for a single future result (N20)

Typical use

Input

20 values

value
9.8
10.4
11.2
9.1
10.8
12
8.7
10.1
11.6
9.5
10.9
10.3
10
9.9
10.6
11
9.4
10.2
10.7
9.7

Settings: limitType = upl, distribution = normal, confidence = 0.95, coverage = 0.95

Against USEPA ProUCL (ProUCL 5.2.0)

Value	USEPA ProUCL	EnviroAutomate	Difference	Agreement band	Verdict
Sample size (n) n	20	20	0	exact match	Agree
Limit limitValue	11.766767699534803	11.766767691283933	8.25e-9	within 0.000001 relative	Agree

Against R EnvStats / nortest (EnvStats 3.1.0, nortest 1.0-4, R 4.6.1)

Value	R EnvStats / nortest	EnviroAutomate	Difference	Agreement band	Verdict
Sample size (n) n	20	20	0	exact match	Agree
Mean mean	10.295	10.294999999999998	1.78e-15	within 0.000001 relative	Agree
Standard deviation sd	0.8306465458969454	0.8306465458969454	0	within 0.000001 relative	Agree
Limit limitValue	11.766767691283949	11.766767691283933	1.60e-14	within 0.000001 relative	Agree

Upper prediction limit for a single future result (N50)

Typical use

Input

50 values

value
6.64
10.98
8.53
12.87
10.42
7.97
12.31
9.86
7.41
11.75
9.3
6.85
11.19
8.74
13.08
10.63
8.18
12.52
10.07
7.62
11.96
9.51
7.06
11.4
8.95
13.29
10.84
8.39
12.73
10.28
7.83
12.17
9.72
7.27
11.61
9.16
6.71
11.05
8.6
12.94
10.49
8.04
12.38
9.93
7.48
11.82
9.37
6.92
11.26
8.81

Settings: limitType = upl, distribution = normal, confidence = 0.95, coverage = 0.95

Against USEPA ProUCL (ProUCL 5.2.0)

Value	USEPA ProUCL	EnviroAutomate	Difference	Agreement band	Verdict
Sample size (n) n	50	50	0	exact match	Agree
Limit limitValue	13.255632324245093	13.255632541539594	2.17e-7	within 0.000001 relative	Agree

Against R EnvStats / nortest (EnvStats 3.1.0, nortest 1.0-4, R 4.6.1)

Value	R EnvStats / nortest	EnviroAutomate	Difference	Agreement band	Verdict
Sample size (n) n	50	50	0	exact match	Agree
Mean mean	9.8978	9.897800000000002	1.78e-15	within 0.000001 relative	Agree
Standard deviation sd	1.9830890045582932	1.9830890045582927	4.44e-16	within 0.000001 relative	Agree
Limit limitValue	13.255632541537901	13.255632541539594	1.69e-12	within 0.000001 relative	Agree

Upper simultaneous limit bounding every observation at once (N10)

Typical use

Input

10 values

value
4.2
5.1
6.3
7.8
9.1
11.4
3.3
8.8
12.1
6.6

Settings: limitType = usl, distribution = normal, confidence = 0.95, coverage = 0.95

Against USEPA ProUCL (ProUCL 5.2.0)

Value	USEPA ProUCL	EnviroAutomate	Difference	Agreement band	Verdict
Sample size (n) n	10	10	0	exact match	Agree
Limit limitValue	13.844927248373635	13.844927210733864	3.76e-8	within 0.000001 relative	Agree

Not run for this scenario

R EnvStats / nortest - This limit is a Bonferroni bound over the sample, which EnvStats does not offer; it is checked against ProUCL.

Upper simultaneous limit bounding every observation at once (N20)

Typical use

Input

20 values

value
9.8
10.4
11.2
9.1
10.8
12
8.7
10.1
11.6
9.5
10.9
10.3
10
9.9
10.6
11
9.4
10.2
10.7
9.7

Settings: limitType = usl, distribution = normal, confidence = 0.95, coverage = 0.95

Against USEPA ProUCL (ProUCL 5.2.0)

Value	USEPA ProUCL	EnviroAutomate	Difference	Agreement band	Verdict
Sample size (n) n	20	20	0	exact match	Agree
Limit limitValue	12.418615439838208	12.418615454801007	1.50e-8	within 0.000001 relative	Agree

Not run for this scenario

R EnvStats / nortest - This limit is a Bonferroni bound over the sample, which EnvStats does not offer; it is checked against ProUCL.

Upper simultaneous limit bounding every observation at once (N50)

Typical use

Input

50 values

value
6.64
10.98
8.53
12.87
10.42
7.97
12.31
9.86
7.41
11.75
9.3
6.85
11.19
8.74
13.08
10.63
8.18
12.52
10.07
7.62
11.96
9.51
7.06
11.4
8.95
13.29
10.84
8.39
12.73
10.28
7.83
12.17
9.72
7.27
11.61
9.16
6.71
11.05
8.6
12.94
10.49
8.04
12.38
9.93
7.48
11.82
9.37
6.92
11.26
8.81

Settings: limitType = usl, distribution = normal, confidence = 0.95, coverage = 0.95

Against USEPA ProUCL (ProUCL 5.2.0)

Value	USEPA ProUCL	EnviroAutomate	Difference	Agreement band	Verdict
Sample size (n) n	50	50	0	exact match	Agree
Limit limitValue	15.761744301336439	15.761744306137661	4.80e-9	within 0.000001 relative	Agree

Not run for this scenario

R EnvStats / nortest - This limit is a Bonferroni bound over the sample, which EnvStats does not offer; it is checked against ProUCL.

Upper tolerance limit covering 95% of the population at 95% confidence (N20)

Boundary case

Input

20 values

value
9.8
10.4
11.2
9.1
10.8
12
8.7
10.1
11.6
9.5
10.9
10.3
10
9.9
10.6
11
9.4
10.2
10.7
9.7

Settings: limitType = utl, distribution = normal, confidence = 0.95, coverage = 0.95

Against USEPA ProUCL (ProUCL 5.2.0)

Value	USEPA ProUCL	EnviroAutomate	Difference	Agreement band	Verdict
Sample size (n) n	20	20	0	exact match	Agree
Limit limitValue	12.28522912396908	12.28522912396908	0	within 0.000001 relative	Agree

Against R EnvStats / nortest (EnvStats 3.1.0, nortest 1.0-4, R 4.6.1)

Value	R EnvStats / nortest	EnviroAutomate	Difference	Agreement band	Verdict
Sample size (n) n	20	20	0	exact match	Agree
Mean mean	10.295	10.294999999999998	1.78e-15	within 0.000001 relative	Agree
Standard deviation sd	0.8306465458969454	0.8306465458969454	0	within 0.000001 relative	Agree
Limit limitValue	12.285230522573283	12.28522912396908	1.40e-6	within 0.000001 relative	Agree

Evidence generated 2026-08-26 from a full comparison run against the versions named above.