Search ASTM
A01 STEEL, STAINLESS STEEL AND RELATED ALLOYS A04 IRON CASTINGS A05 METALLIC-COATED IRON AND STEEL PRODUCTS B01 ELECTRICAL CONDUCTORS B05 COPPER AND COPPER ALLOYS B07 LIGHT METALS AND ALLOYS C01 CEMENT C04 VITRIFIED CLAY PIPE C07 LIME AND LIMESTONE C09 CONCRETE AND CONCRETE AGGREGATES C11 GYPSUM AND RELATED BUILDING MATERIALS AND SYSTEMS C12 MORTARS AND GROUTS FOR UNIT MASONRY C13 CONCRETE PIPE C14 GLASS AND GLASS PRODUCTS C15 MANUFACTURED MASONRY UNITS C16 THERMAL INSULATION C17 FIBER-REINFORCED CEMENT PRODUCTS C18 DIMENSION STONE C21 CERAMIC WHITEWARES AND RELATED PRODUCTS C24 BUILDING SEALS AND SEALANTS C27 PRECAST CONCRETE PRODUCTS D01 PAINT AND RELATED COATINGS, MATERIALS, AND APPLICATIONS D04 ROAD AND PAVING MATERIALS D07 WOOD D08 ROOFING AND WATERPROOFING D09 ELECTRICAL AND ELECTRONIC INSULATING MATERIALS D11 RUBBER D14 ADHESIVES D18 SOIL AND ROCK D20 PLASTICS D35 GEOSYNTHETICS E05 FIRE STANDARDS E06 PERFORMANCE OF BUILDINGS E33 BUILDING AND ENVIRONMENTAL ACOUSTICS E36 ACCREDITATION & CERTIFICATION E57 3D IMAGING SYSTEMS E60 SUSTAINABILITY F01 ELECTRONICS F06 RESILIENT FLOOR COVERINGS F13 PEDESTRIAN/WALKWAY SAFETY AND FOOTWEAR F16 FASTENERS F17 PLASTIC PIPING SYSTEMS F33 DETENTION AND CORRECTIONAL FACILITIES F36 TECHNOLOGY AND UNDERGROUND UTILITIES G03 WEATHERING AND DURABILITY C14 GLASS AND GLASS PRODUCTS C21 CERAMIC WHITEWARES AND RELATED PRODUCTS D01 PAINT AND RELATED COATINGS, MATERIALS, AND APPLICATIONS D06 PAPER AND PAPER PRODUCTS D09 ELECTRICAL AND ELECTRONIC INSULATING MATERIALS D10 PACKAGING D11 RUBBER D12 SOAPS AND OTHER DETERGENTS D13 TEXTILES D14 ADHESIVES D15 ENGINE COOLANTS AND RELATED FLUIDS D20 PLASTICS D21 POLISHES D31 LEATHER E12 COLOR AND APPEARANCE E18 SENSORY EVALUATION E20 TEMPERATURE MEASUREMENT E35 PESTICIDES, ANTIMICROBIALS, AND ALTERNATIVE CONTROL AGENTS E41 LABORATORY APPARATUS E53 ASSET MANAGEMENT E57 3D IMAGING SYSTEMS F02 FLEXIBLE BARRIER PACKAGING F05 BUSINESS IMAGING PRODUCTS F06 RESILIENT FLOOR COVERINGS F08 SPORTS EQUIPMENT, PLAYING SURFACES, AND FACILITIES F09 TIRES F10 LIVESTOCK, MEAT, AND POULTRY EVALUATION SYSTEMS F11 VACUUM CLEANERS F13 PEDESTRIAN/WALKWAY SAFETY AND FOOTWEAR F14 FENCES F15 CONSUMER PRODUCTS F16 FASTENERS F24 AMUSEMENT RIDES AND DEVICES F26 FOOD SERVICE EQUIPMENT F27 SNOW SKIING F37 LIGHT SPORT AIRCRAFT F43 LANGUAGE SERVICES AND PRODUCTS F44 GENERAL AVIATION AIRCRAFT A01 STEEL, STAINLESS STEEL AND RELATED ALLOYS A04 IRON CASTINGS A05 METALLIC-COATED IRON AND STEEL PRODUCTS A06 MAGNETIC PROPERTIES B01 ELECTRICAL CONDUCTORS B02 NONFERROUS METALS AND ALLOYS B05 COPPER AND COPPER ALLOYS B07 LIGHT METALS AND ALLOYS B08 METALLIC AND INORGANIC COATINGS B09 METAL POWDERS AND METAL POWDER PRODUCTS B10 REACTIVE AND REFRACTORY METALS AND ALLOYS C03 CHEMICAL-RESISTANT NONMETALLIC MATERIALS C08 REFRACTORIES C28 ADVANCED CERAMICS D01 PAINT AND RELATED COATINGS, MATERIALS, AND APPLICATIONS D20 PLASTICS D30 COMPOSITE MATERIALS E01 ANALYTICAL CHEMISTRY FOR METALS, ORES, AND RELATED MATERIALS E04 METALLOGRAPHY E07 NONDESTRUCTIVE TESTING E08 FATIGUE AND FRACTURE E12 COLOR AND APPEARANCE E13 MOLECULAR SPECTROSCOPY AND SEPARATION SCIENCE E28 MECHANICAL TESTING E29 PARTICLE AND SPRAY CHARACTERIZATION E37 THERMAL MEASUREMENTS E42 SURFACE ANALYSIS F01 ELECTRONICS F34 ROLLING ELEMENT BEARINGS F40 DECLARABLE SUBSTANCES IN MATERIALS F42 ADDITIVE MANUFACTURING TECHNOLOGIES G01 CORROSION OF METALS G03 WEATHERING AND DURABILITY D21 POLISHES D26 HALOGENATED ORGANIC SOLVENTS AND FIRE EXTINGUISHING AGENTS D33 PROTECTIVE COATING AND LINING WORK FOR POWER GENERATION FACILITIES E05 FIRE STANDARDS E27 HAZARD POTENTIAL OF CHEMICALS E30 FORENSIC SCIENCES E34 OCCUPATIONAL HEALTH AND SAFETY E35 PESTICIDES, ANTIMICROBIALS, AND ALTERNATIVE CONTROL AGENTS E52 FORENSIC PSYCHOPHYSIOLOGY E54 HOMELAND SECURITY APPLICATIONS E58 FORENSIC ENGINEERING F06 RESILIENT FLOOR COVERINGS F08 SPORTS EQUIPMENT, PLAYING SURFACES, AND FACILITIES F10 LIVESTOCK, MEAT, AND POULTRY EVALUATION SYSTEMS F12 SECURITY SYSTEMS AND EQUIPMENT F13 PEDESTRIAN/WALKWAY SAFETY AND FOOTWEAR F15 CONSUMER PRODUCTS F18 ELECTRICAL PROTECTIVE EQUIPMENT FOR WORKERS F23 PERSONAL PROTECTIVE CLOTHING AND EQUIPMENT F26 FOOD SERVICE EQUIPMENT F32 SEARCH AND RESCUE F33 DETENTION AND CORRECTIONAL FACILITIES G04 COMPATIBILITY AND SENSITIVITY OF MATERIALS IN OXYGEN ENRICHED ATMOSPHERES D08 ROOFING AND WATERPROOFING D18 SOIL AND ROCK D19 WATER D20 PLASTICS D22 AIR QUALITY D34 WASTE MANAGEMENT D35 GEOSYNTHETICS E06 PERFORMANCE OF BUILDINGS E44 SOLAR, GEOTHERMAL AND OTHER ALTERNATIVE ENERGY SOURCES E47 E48 BIOENERGY AND INDUSTRIAL CHEMICALS FROM BIOMASS E50 ENVIRONMENTAL ASSESSMENT, RISK MANAGEMENT AND CORRECTIVE ACTION E60 SUSTAINABILITY F20 HAZARDOUS SUBSTANCES AND OIL SPILL RESPONSE F40 DECLARABLE SUBSTANCES IN MATERIALS G02 WEAR AND EROSION B01 ELECTRICAL CONDUCTORS C26 NUCLEAR FUEL CYCLE D02 PETROLEUM PRODUCTS, LIQUID FUELS, AND LUBRICANTS D03 GASEOUS FUELS D05 COAL AND COKE D19 WATER D27 ELECTRICAL INSULATING LIQUIDS AND GASES D33 PROTECTIVE COATING AND LINING WORK FOR POWER GENERATION FACILITIES E10 NUCLEAR TECHNOLOGY AND APPLICATIONS E44 SOLAR, GEOTHERMAL AND OTHER ALTERNATIVE ENERGY SOURCES E48 BIOENERGY AND INDUSTRIAL CHEMICALS FROM BIOMASS A01 STEEL, STAINLESS STEEL AND RELATED ALLOYS C01 CEMENT C09 CONCRETE AND CONCRETE AGGREGATES D02 PETROLEUM PRODUCTS, LIQUID FUELS, AND LUBRICANTS D03 GASEOUS FUELS D04 ROAD AND PAVING MATERIALS D15 ENGINE COOLANTS AND RELATED FLUIDS D18 SOIL AND ROCK D24 CARBON BLACK D35 GEOSYNTHETICS E12 COLOR AND APPEARANCE E17 VEHICLE - PAVEMENT SYSTEMS E21 SPACE SIMULATION AND APPLICATIONS OF SPACE TECHNOLOGY E36 ACCREDITATION & CERTIFICATION E57 3D IMAGING SYSTEMS F03 GASKETS F07 AEROSPACE AND AIRCRAFT F09 TIRES F16 FASTENERS F25 SHIPS AND MARINE TECHNOLOGY F37 LIGHT SPORT AIRCRAFT F38 UNMANNED AIRCRAFT SYSTEMS F39 AIRCRAFT SYSTEMS F41 UNMANNED MARITIME VEHICLE SYSTEMS (UMVS) F44 GENERAL AVIATION AIRCRAFT F45 DRIVERLESS AUTOMATIC GUIDED INDUSTRIAL VEHICLES D10 PACKAGING D11 RUBBER E31 HEALTHCARE INFORMATICS E35 PESTICIDES, ANTIMICROBIALS, AND ALTERNATIVE CONTROL AGENTS E54 HOMELAND SECURITY APPLICATIONS E55 MANUFACTURE OF PHARMACEUTICAL PRODUCTS E56 NANOTECHNOLOGY F02 FLEXIBLE BARRIER PACKAGING F04 MEDICAL AND SURGICAL MATERIALS AND DEVICES F29 ANESTHETIC AND RESPIRATORY EQUIPMENT F30 EMERGENCY MEDICAL SERVICES G04 COMPATIBILITY AND SENSITIVITY OF MATERIALS IN OXYGEN ENRICHED ATMOSPHERES C07 LIME AND LIMESTONE D14 ADHESIVES D16 AROMATIC HYDROCARBONS AND RELATED CHEMICALS D20 PLASTICS D26 HALOGENATED ORGANIC SOLVENTS AND FIRE EXTINGUISHING AGENTS D28 ACTIVATED CARBON D32 CATALYSTS E13 MOLECULAR SPECTROSCOPY AND SEPARATION SCIENCE E15 INDUSTRIAL AND SPECIALTY CHEMICALS E27 HAZARD POTENTIAL OF CHEMICALS E35 PESTICIDES, ANTIMICROBIALS, AND ALTERNATIVE CONTROL AGENTS F40 DECLARABLE SUBSTANCES IN MATERIALS E11 QUALITY AND STATISTICS E36 ACCREDITATION & CERTIFICATION E43 SI PRACTICE E55 MANUFACTURE OF PHARMACEUTICAL PRODUCTS E56 NANOTECHNOLOGY F42 ADDITIVE MANUFACTURING TECHNOLOGIES
Bookmark and Share

DataPoints

DataPoints

Statistical Intervals: Nonparametric

Part 1

Q: How are nonparametric intervals applied and used?

A. In several previous articles of this column we have discussed confidence, prediction and tolerance intervals where the underlying distribution was of the normal type. In this article we develop these concepts further using nonparametric methods that do not assume an underlying normal distribution. We continue to assume that the sample is a random representation of a population or from a process in a state of statistical control.

We turn first to prediction-type intervals when we do not know the underlying distribution of the variable. In this scenario, the practitioner has a random sample of n observations taken from some population or process under study and would like to create an interval using the sample maximum and/or minimum that predicts one or more future values with some confidence C. Any such set of n observations will partition the support of the distribution into n + 1 compartments or blocks. For example, if we have four observations we get 4 + 1 = 5 compartments. This is depicted below in Figure 1, using an arbitrary distribution, for the case of n = 4. The subscripts in the diagram denote the ordered values in the sample: the so-called order statistics. Thus x(1) is the smallest value and x(4) the largest in this example, and so on.

FIGURE 1 – Any distribution is partitioned into n + 1 compartments with a sample of size n. In this example n = 4.

Associated with each compartment is its probability (area under the curve) and, surprisingly, this may be shown to be 1/(n + 1) on average for all such compartments and for any distribution so partitioned. From these simple facts, we can estimate the probability that the next value will fall between any two order statistics. For example, if n = 9, the estimated probability that the next observation will fall below the largest observed value is approximated as 9/(9 + 1) or 0.9. It is common practice to call this estimated probability the “confidence” in the interval. This same confidence could also be applied to a future observation being greater than the smallest observed value in n. Both of these cases are examples of one-sided intervals. That is, the first interval is of the form (-∞, x(n)] and the second of the form [x(1), ∞), where x(1) and x(n) are the sample minimum and maximum values. One-sided intervals are very important in practice since many types of properties are bounded either as a maximum or a minimum.

If we wanted to use the same interval as a basis for predicting several, say k, future observations, then the formula for the associated confidence is adjusted as C = n/(n + k) for either interval. For example, suppose n = 22, and we wanted to use the sample maximum as an upper bound for the next three observations and wanted to know the associated confidence in this interval. The answer is C = 22/(22 + 3) = 0.88, or 88 percent. It is not difficult to determine the sample size for a single-sided interval that would predict where the next k observations would fall. This is n = kC/(1 - C). Thus, for 95 percent confidence and k = 3, we should use a sample size of 57. It is also a simple matter to create intervals using arbitrary order statistics, although intervals based on the sample minimum and maximum are more common.

Next, consider the case where the interval is constructed from the largest and smallest values in the sample. This is an interval of the form [x(1), x(n)]. Using this interval as a basis for predicting the next (k = 1) observation carries a confidence of C = (n - 1)/(n + 1). Thus, if n = 22, then C = 21/23 = 0.913, or 91.3 percent. We summarize these basic cases below.

Case 1: The next k observations are less than the maximum (or the next k values are greater than the minimum) with confidence C.

C= n n+k MathType@MTEF@5@5@+=faaagCart1ev2aqaKnaaaaWenf2ys9wBH5garuqsUrxzYnhDHrxzGWuANHgDaeXatLxBI9gBaerbd9wDYLwzYbqefqvATv2CaebbnrfifHhDYfgasaacPqpw0le9v8qqaqFD0xXdHaFibbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dc9Gqpi0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaacaWGdbGaeyypa0ZaaSaaaeaacaWGUbaabaGaamOBaiabgUcaRiaadUgaaaaaaa@3C18@

(1)

Case 2: The next single value falls between the sample minimum and maximum with confidence C.

C= n1 n+1 MathType@MTEF@5@5@+=faaagCart1ev2aqaKnaaaaWenf2ys9wBH5garuqsUrxzYnhDHrxzGWuANHgDaeXatLxBI9gBaerbd9wDYLwzYbqefqvATv2CaebbnrfifHhDYfgasaacPqpw0le9v8qqaqFD0xXdHaFibbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dc9Gqpi0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaacaWGdbGaeyypa0ZaaSaaaeaacaWGUbGaeyOeI0IaaGymaaqaaiaad6gacqGHRaWkcaaIXaaaaaaa@3D8B@

(2)

We can also calculate the confidence that the next k observations will fall between the maximum and the minimum of the original sample. The formula is not as intuitive as the single observation case, but by a subtle conditional probability argument it may be shown that the formula is Equation 3 below.

Case 3: The next k observations fall between the sample minimum and maximum.

C= n(n1) (n+m)(n+m1) MathType@MTEF@5@5@+=faaagCart1ev2aqaKnaaaaWenf2ys9wBH5garuqsUrxzYnhDHrxzGWuANHgDaeXatLxBI9gBaerbd9wDYLwzYbqefqvATv2CaebbnrfifHhDYfgasaacPqpw0le9v8qqaqFD0xXdHaFibbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dc9Gqpi0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaacaWGdbGaeyypa0ZaaSaaaeaacaWGUbGaaiikaiaad6gacqGHsislcaaIXaGaaiykaaqaaiaacIcacaWGUbGaey4kaSIaamyBaiaacMcacaGGOaGaamOBaiabgUcaRiaad2gacqGHsislcaaIXaGaaiykaaaaaaa@472F@

(3)

Example

Suppose one has 39 observations and wants to use the sample minimum and maximum as a prediction interval for the next two observations. What is the confidence in using this interval? Using Equation 3, we find that C is 0.903, or approximately 90 percent. Note that in Equation 3, when m = 1, the formula reduces to Equation 2 for the single value prediction.

Example

If we have 19 observations where the smallest value is 23.48 and the largest is 39.57, the confidence that the next observation lies between these two values is (19 - 1)/(19 + 1) = 0.9, or 90 percent confidence. The confidence that the next observation falls below the maximum (or above the minimum) is 19/(20 + 1) = 0.905, or 90.5 percent confidence.

Example

Assuming we have n = 29 observations, what is the confidence that the next m = 3 observations fall between the minimum and the maximum of the original sample.

C= 29(291) (29+3)(29+31) =0.819 MathType@MTEF@5@5@+=faaagCart1ev2aqaKnaaaaWenf2ys9wBH5garuqsUrxzYnhDHrxzGWuANHgDaeXatLxBI9gBaerbd9wDYLwzYbqefqvATv2CaebbnrfifHhDYfgasaacPqpw0le9v8qqaqFD0xXdHaFibbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dc9Gqpi0dmeaabaqaciGacaGaaeqabaWaaqaafaaakeaacaWGdbGaeyypa0ZaaSaaaeaacaaIYaGaaGyoaiaacIcacaaIYaGaaGyoaiabgkHiTiaaigdacaGGPaaabaGaaiikaiaaikdacaaI5aGaey4kaSIaaG4maiaacMcacaGGOaGaaGOmaiaaiMdacqGHRaWkcaaIZaGaeyOeI0IaaGymaiaacMcaaaGaeyypa0JaaGimaiaac6cacaaI4aGaaGymaiaaiMdaaaa@4DA7@

or approximately 82 percent confidence.

We can also develop the confidence level for cases of prediction for k out of the next m observations, but these cases are less common and more complicated to use. Interested readers should consult Mathematical Statistics by S. S. Wilks for details.1 It is important to note that the prediction interval is similar to a confidence interval in that the capture probability (confidence) is a long run result. That is, confidence is the long run proportion of cases, under the same conditions and with differing data that would predict correctly what we say it would. For this and many other cases, including a comprehensive literature reference readers are encouraged to see Statistical Intervals: A Guide for Practitioners, by G. J. Hahn and W. Q. Meeker.2

References

1. Wilks, S. S., Mathematical Statistics, John Wiley & Sons, New York, N.Y., 1963.

2. Hahn, G. J., and Meeker, W. Q., Statistical Intervals: A Guide for Practitioners, Wiley InterScience, John Wiley and Sons Inc., New York, N.Y., 1991.

Stephen N. Luko, United Technologies Aerospace Systems, Windsor Locks, Conn., is an ASTM fellow; a past chairman of Committee E11 on Quality and Statistics, he is current chairman of Subcommittee E11.30 on Statistical Quality Control.

Dean V. Neubauer, Corning Inc., Corning, N.Y., is an ASTM fellow; he serves as chairman of Committee E11 on Quality and Statistics, chairman of E11.90.03 on Publications and coordinator of the DataPoints column.

Go to other DataPoints articles.

This article appears in the issue of Standardization News.