Statistical Prediction Intervals for the 

Evaluation of Ground-Water Quality 

by Robert D. Gibbons` 1 

ABSTRACT 

Factors for a norrnal distribution are given such that 

one may be 9996 conficlent that the two-sided prediction 

interval ac ± rs or the one-sided prediction interval ac + rs 

will contain all of the k future values, where x` and s are the 

sample mean and standard dcviation obtained from ii 

previous values. In the context of ground-water monitoring, 

the future samples may represent new monitoring values at 

cach of k downgradient wells, and the n previous values 

might be the historical monitoring results for one or morc 

upgradient wells. The 'I'ables provided in this paper allow 

the computation of one-sided and two-sided 9996 prediction 

intervals for previous sample sizes of n = 4 to 100 and 

future samples of k= 1 to 100. Modification of these 

intervals for log-normally distributed data is also presented. 

'I. lNTRODUCTION 

The principal objective of ground-water 

monitoring at hazardous waste disposal facilities is 

to empiricaIly determine if on-site chemical waste 

is migrating off-site in ground water. The principal 

method of ground water monitoring involves a 

conlparison of specific chemical constituents of 

ground water upgradient of the site with similarly 

collected and analyzed sarnples downgradierit of 

a University of Iltinois, Biometric l.aboratory, Illinois 

State Psychiatric Institute, 1601 W. 'I'aylor St., Chicago, 

Illinois 60612. 

Received June 1986, revised April 1987, accepted 

April 1987. 

Discussion open until January 1, 1988. 

Vol. 25, No. 4—GRaUND WATEIt---July-August 1987 

the site. '1'he null hypothesis of intzrest in such 

monitoring programs is that of no difference in 

chemical composition of ground water upgradient 

and downgradient of the facility. 'I'he logic in such 

a comparison is that presumably, the upgradient 

wells monitor ground water that is unaffected by 

the waste disposaI facility, whereas the downgradient wclls arc in the path of ground water that 

flows underneath the site, and therefore should 

provide an early warning system for any potential 

off-site migration of hazardous waste. 'I'he validity 

of this monitoring procedure is not the focus of 

this paper. I'he development of a statistically 

rigorous hypothesis testing strategy is the focus of 

this paper. 

If ground water were in the form of a wellmixed underground river or lake th.tt spanned the 

entire arca of the site, the most direct approach to 

testing for upgradient versus downgradient 

differences would be to obtain estimates of ineans 

and variances for individual compounds for all upgradient wells and similar estimates for alI downgradient wells. lf the variances of these two distributions were reasonabiy similar, the measurements independent (i.e., not replicate measurements obtained from the same sample), and the 

compounds in question liad approximately normal 

(i.e., Gaussian) distributions, one could readily 

compare the two distributions using Student's 

t-statistic for each indicator paramet:er separately, 

or if a sufficient number of monitoring wells were 

available, all of the indicator parameters could be 

455 
compared simultaneously using Hotelling's T 2 

statistic [i.e., the multivariate extension of the 

t-statistic (Anderson, 1958)], Alternatively, if the 

assumption of homogeneity of variance could not 

be supported, one could use an unequal variance 

t-statistic such as Cochran's approximation to the 

13ehrens-I-'ishcr t-statistic (5nedecor and Cochran, 

1980). 

I,'rom a statistical perspective it is unfortunate 

that the ground-water lake or river example falls 

considerably short of reality. Ground water oceurs 

in geologically discrete aquifers which generally 

move extremely slowly (i.e., inches per year) and 

in rare cases can exhibit a plume-like projection, 

such that only localized areas of the downgra.dient 

geology niay exhibit contarnination. In light of 

this, pooling information obtained from downgradient wells, soine of which may be contaminated and some of which may be perfectly normal, 

does not scem sensiblc. 

If wc cannot estimate a mean and variance for 

a collection of downgradicnt well measurcnients on 

a particular occasion, we cannot compute a 

t-statistic. The reason for this is that a t-statistic is 

appropriate for the Comparison of two mean 

values; however, if each dUwngradient well is 

examincd individually, its mcasurcment is not the 

mean of anything, it is sirnply a single new observation which rnust in turn hc cornparcd to a historical upgradient samplc of ineasurements in order to 

assess its significance. Stated statistically, we desire 

an interval or range of values that will have high 

probability of containing all new monitoring 

observations. if the monitoring program involves 

multiple wclls, wc must construct an interval that 

will include all of the k futurc obscrvations, where 

k is the number of wells that arc to bc evaluated on 

the next monitoring event. Intervals of this type 

arc called "prediction intervals" and necd to be distinguished both from a confidence interval on an 

unknown distribution paramcter (e.g., a confidence 

interval of a mean value), and from a tolerance 

interval which contains the values- of a specified 

proportion of the population with a certain level of 

confidence. I lahn (1.970) illustrates this distinction 

by providing an cxample of the typical astronaut's 

problem. "An astronaut who has bcen assigned to a 

limited number of spacc flights is not very 

interested in what will happen on the average in 

the population of all space flights, of which his 

happens to be a random sample, or even in what 

will happen in 99% of the population of such space 

flights. lIis main interest is in thc worst that Inay 

happen in the onc, or three, or fivc• flights in which 

he will be personally involved." In this example, 

what wi1l happen on the average in the population 

of all space flights represcnts a confidencc intcrval, 

what wiIl happcn in 99% of the population of 

space flights is a tolerance interval, and wllat will 

happen in the next one, or three, or five space 

flights rcprescnts a prediction interval. 

2. STATISTICAL DEVELOPME11tT 

()ur objective is to definc an interval 

ac ± r(k, n; 1-- a) s 

that will contain the values of all k ftiturc mcasurcments, where x and s arc the sample mean and 

standard deviation calculated froni the values 

x l,..., x n of n previous nieasurements; that is, 

n 

x = Z xi/n 

i=I 

n 

52= Z (x; - ac)2 /(n 

~ -1 

The original n observations and the additional 

k samples are assumed to be independent random 

samplcs (i.e., not analytical replicates) frorn thc.• 

same normal distribution. For the case of a single 

new observation, Wilks (1941) showed that when 

sampling from a N(p, (J 2 ) with ,u and 0 2unknown, 

the coverage c of the interval 

z+ rs ] 

c= f ~ exp {- 2 (x - μ)2 } ay 

z-rs 2 Q 

will on the average be 1- cx, if and only if 

r +1/n ttn_1;W21 

whcre tln-1;W21 is the 100(1 - a)% point of 

"Studcnt's" t-distribution with n-- 1 degrees of 

freedom. 

C:hcw (1968) extendcd this result to thc casc 

of k future observations using an approximate 

conservative method for obtaining prediction 

intcrvals bascd on thc Honferroni incquality 

(Miller, 1966). "I'his approximation leads to the 

100 (1 - a)% two-sided prediction limit : 

x±Nrl+ 1/n t(n—l;a/2k] s 

and thc 100(1 - a)% one-sided upper prediction 

limit: 

x+ 1+ 1/n t(n-1;a/kl s 

flahn (1969 and 1970) compared results using 

456 
this method and selected exact values obtained 

using numerical quadrature and found the approximation to be satisfactory for practical purposes. 

Coniparison of values in Tables 1-3, which provide 

values of r(k, n; .99) for n= 4 to 100 and k= 1 to 

100 for one-sided intervals, and '1'ables 4-6 for the 

corresponding two-sided intervals reveal that on 

the average, the bias is less than 1% with a maximum 

deviation of 7% for n= 4 and k= 1. In fact, for 

n> 4 the bias is never greater than 1% ► regardless of 

k. The small bias observed for n= 4 is always an 

underestilna.te; therefore, the approxilnation is 

conservative for very small n which should probably 

not be used in practice. For example, the exact 

one-tailed result for n= 15 and k= 4 is 

r(4, 15 ;.99) = 3.42, and the approxinlate value 

given in 'I'able 1 is 3.43. Usirng this example in a 

practical application, we wOuld conlpute the 99% 

upper prediction limit as x+ 3.43s, where in this 

exainple, Y and s would be computed on 15 background (upgradient) nleasurenlents, and each of 

the four new monitoring (downgradient) rneasuremcnts wotlld siniply be compared to this Iimit 

value. 

Table 1. Factors for Obtaining One-Sided 99% Prediction Limits for k 

Additiona! Samples Given a Background Sample of Size n 

Previvus 

n 1 2 3 4 6 

Number of new measurements (k) 

6 7 8 9 10 11 12 13 14 15 

4 4.70 6.81 6.51 7.06 7.49 7.86 8.17 8.45 8.71 8.94 9.15 9_36 9.53 9.70 9.86 

6 3.96 4.77 6.28 5.66 5.96 6.22 6.44 6.63 6.81 6.97 7.11 7.24 7.37 7.41+ 7.59 

6 3.67 4.23 4.63 4.93 5.17 6.37 6.54 5.69 6.83 5.96 6.06 6.16 6.26 6.3S 6.43 

7 3.32 3.89 4.24 4,49 469 4.86 5.00 5.13 6.24 6.34 6.43 5.62 5.60 6.67 5.74 

8 3.16 3.67 3.98 4,20 4.38 4.62 4.66 4.75 4.85 4.94 5.02 5.09 5.16 5.22 5.28 

9 3.04 3.51 3.79 3.99 4.15 4.28 4.39 4.49 4.68 4.68 4.73 4.79 4,85 4.91 4.96 

10 2.95 3.39 3,86 3.84 3.98 4.10 4.2 1 4.30 4.37 4.45 4.61 4.57 4.62 4.68 4.72 

11 2.88 3.30 3.54 3.72 385 3.97 4.06 4.14 4.22 4.28 4.34 4.40 4.45 4.5 ~. 4.54 

12 2.82 3.22 3.46 3.62 3.76 3.86 3.95 4.03 4.09 4.16 4.21 4.26 4.31 4.3E 4.40 

13 2.78 3.16 3.39 3_54 3.67 3.77 3.85 3.93 3.99 4.05 4_11 4.16 4.20 4.24 4.28 

14 2.74 3.11 3.33 3.48 3.60 3.70 3.78 3.85 3.91 3.97 4.02 4.07 4.11 4.15 4.19 

15 2.71 3.07 3.28 3.43 3.64 3.63 3.71 3.78 3.84 4.90 3.95 3.99 4.03 4.07 4.11 

16 2.68 3.03 3.24 3.38 3.49 3.68 3.66 3.72 3.78 3.84 3.88 3.93 3.97 4.00 4.04 

17 2.66 3.00 3.20 3.34 3.46 3.64 3.61 3.68 3.73 3.78 3.83 3.87 3.91 3.95 3.98 

18 2.64 2.97 3.17 3.31 3_41 3.60 3.67 3.63 3.69 3.74 3.78 3.82 3.236 3.88 3.93 

19 2.62 2.96 3.14 3.28 3.38 3.46 3.63 3.60 3,65 3.70 3.74 3.78 3.82 3.85 3.88 

20 2.60 2.93 3.12 3,25 3.36 3.43 3.50 3.56 3.61 3.66 3.70 3.74 3.78 3.81 3.84 

21 2.69 2.91 3.10 3.22 3.32 3.41 3.47 3.53 3.58 3.63 3.67 3.71 3.75 3_78 3.81 

22 2.57 2.89 3.08 3.20 3.30 3.38 3.45 3.61 3.66 3.60 3.64 3.68 3.71 3.75 3.7b 

23 2.66 2.88 3.06 3.18 3.28 3.36 3.42 3.48 3.63 3.58 3.62 3.65 3.69 3.72 3.75 

24 2.65 2.86 3.04 3.17 3.26 3.34 3.40 3.46 3,61 3.55 3.69 3.63 3.66 3.69 3.72 

25 2.54 2.86 3.03 3.16 3.24 3.32 3,39 3.44 3.49 3.53 3.57 3.61 3.64 3.67 3.70 

26 2.53 2.84 3.01 3.14 3.23 3.30 3.37 3.42 3,47 3.51 3.56 3.59 3.62 3.66 3.68 

27 2.52 2.83 3.00 3.12 3.21 3.29 3.36 3.41 3.46 3.50 3.53 3.67 3.60 3.63 3.66 

28 2.52 2.82 2.99 3 11 3.20 3.27 3.34 3.39 3.44 3 48 3.52 3.65 3.58 3.61 3.64 

29 2_51 2.81 2.98 3.10 3.19 3.26 3.32 3.38 3.42 3.46 3.50 3.54 3.57 3.60 3.62 

30 2.50 2.80 2.97 3.09 3.18 3.25 3.31 3.36 3.41 3.45 3.49 3.52 3.55 3.58 3.61 

31 2.60 2.79 2.96 3.08 3.17 314 3.30 3.35 3.40 3.44 3.47 3.61 3.54 3.57 3.59 

32 2.49 2.79 2.96 307 3.16 3.23 3.29 3.34 3.39 3 43 3.46 3.60 3.63 3.55 3.58 

33 2.49 2.78 2.94 3.06 3.16 3.22 3.28 3.33 3.37 3.41 346 3.48 3.51 3.54 3.57 

34 2.48 2.77 2.94 3.05 3.14 3.21 3.27 3.32 3.36 3.40 3.44 3.47 3.50 3.53 3.56 

35 2.48 2.77 2.93 304 3.13 3.20 3.26 3.31 3.36 3.39 3.43 3.46 3.49 3.52 3,54 

3g 2.47 2.76 2.92 3.04 3.12 3.19 3.26 3.30 3.35 3.39 3.42 3.46 3.48 3.51 3.53 

37 2.47 2.76 2.92 3.03 3.12 3.19 3.24 3.29 3.34 3.38 3.41 3.44 3.47 3.50 3.53 

38 2.46 2.76 2.91 3.02 3.11 3.18 3.24 3.29 3.33 3.37 3.40 3.44 3.46 3.49 3.52 

39 2.46 2.76 2.91 3.02 3.10 3.17 3.23 3.28 3.32 3.36 3.40 3.43 3.46 3.48 3.51 

40 2.46 2.74 2,90 3.01 310 3.17 3.22 3.27 3.31 3.36 3.39 3.42 3.45 3 47 3.50 

41 2.46 2.74 2.90 3.01 3.09 3.16 3.22 3.27 3.31 3.35 338 3 41 3.44 3.47 3.49 

42 2.45 2.73 2.89 3.00 3.09 3.15 3.21 3.26 3.30 3.34 3.37 3.41 3.43 3.46 3.48 

43 2.45 2.73 2.89 3.00 3.08 3.16 3.20 3.26 3.30 3.33 3.37 3.40 3.43 3.45 3.48 

44 2.44 2.73 2.88 299 3.08 3.14 3.20 5.25 3.29 3.33 3.36 3.39 3.42 3.45 3.47 

45 2,44 2.72 2.88 2.99 3.07 3.14 3.19 3.24 3.28 3 32 336 3.39 3.41 3.44 3.46 

46 2.44 2.72 2.88 2.98 3.07 3.13 3.19 3.24 3.28 3.32 3.35 3.38 3.41 3.43 3.46 

47 2.44 2.72 2,87 2.98 3.06 3.13 3,18 3.23 3.27 3.31 3.34 3.38 3.40 3.43 3.45 

48 2.43 2.71 2.87 2 98 3.06 3.12 3.18 3.23 3.27 3 31 3.34 3.37 3.40 3.42 3,45 

49 2.43 2.71 2.86 297 3.06 3.12 3.18 3.22 3.26 3 30 3 34 3.37 3.39 3.42 3.44 

50 2.43 211 2.86 2.97 3.06 3.12 3.17 3.22 3.26 3.30 3.33 3.36 3.39 3.41 3.44 

60 2.41 2.68 2,84 2.94 3.01 3.08 3.14 3.18 3.22 3.26 3.29 3.32 3.35 3.37 3.40 

70 2.40 2.67 2.82 2.92 3.00 3.06 3.12 3.16 3.20 3.24 3.27 3.30 3.32 3.3S 3.37 

80 2.39 2.66 2.80 2.91 2.98 3.05 3,10 3.14 3.18 3 22 3.25 3.28 3.30 3.33 3.35 

90 2.38 2.65 2.79 2.89 2.97 3.03 3.08 3.13 3.17 3.20 3.23 3.26 3.29 3.31 3.33 

100 2.38 2.64 2.79 2.89 2.96 3.02 3.07 3.12 316 3.19 3.22 3.26 3.27 3.30 3.32 

Factor = t t.l _ a~k~~ l + 1/n 

45i 
Tabie 2. Factors for Obtaining One-Sided 99% Prediction Limits for k 

Additional Samples Given a Background Sample of Size n 

Previous Nucnber of new nies surernerrts (k) 

n 16 _ 17 18 19 ZO 21 ZS 23 24 26 26 27 28 29 30 

4 10.01 10.16 10.30 10.43 10.55 10.6 10.79 10.90 11.00 11.11 11.31 11.30 11.40 11.49 11.67 

5 7.70 7.79 7.89 7.97 8.06 8.14 8.22 8.29 8.36 8.43 8.50 8.56 8.62 8.68 8.74 

6 6.51 6.58 6.65 6.74 6.79 6.85 6.90 6.96 7.02 7.07 7.12 7.17 7.41 7.26 7.30 

7 5.80 5.86 6.94 5.98 6.03 6.08 6.13 6.17 6.21 6.26 6.30 6.34 6.38 6.41 6.45 

8 5.34 6.39 6.44 5.49 5.63 6.67 5.61 6.66 5.69 6.73 5.76 6.80 6.83 6.86 5.89 

9 6.01 5.06 5.10 6.14 6.18 6.22 6.26 6.39 6.32 6.36 619 6.42 6.44 6.47 5.50 

10 4.77 4.81 4.86 4.89 4.92 4.96 4.99 6.02 5.06 6.08 6.11 5.14 6.16 5.19 5.21 

11 4.68 4.62 4.66 4.69 4.73 4.76 4.79 4.82 4.84 4.87 4.90 4.92 4.94 4.97 4.99 

12 4.44 4.47 4.61 4.64 4.57 4.60 4.63 4.66 4.68 4.71 4.73 4.76 4.77 4.80 4.82 

13 4.32 4.36 418 4.42 4.44 4.47 4.60 4.62 4.55 4.67 4.59 4.62 4.64 4.66 4.68 

14 4.22 4.26 4.28 4.31 4.34 4.37 4.39 4.42 4.44 4.46 4.48 4.50 4.62 4.54 4.56 

15 4.14 4.17 4.20 4.23 4.25 4.78 4.30 4.33 4.36 4.37 4.39 4.41 4.43 4.45 4.46 

16 4.07 4.10 4.13 4.15 4.18 4.40 4.23 4.25 4.27 4.29 4.31 4.33 4.35 4.37 4.38 

17 4.01 4.04 4.07 4.09 4.12 4.14 4.16 4.18 4.20 4.22 4.24 4.26 4.28 4.30 4.31 

18 3.96 3.99 4.01 4.04 4.06 4.08 4.11 4.13 4.16 4.17 4.18 4.20 4.22 4.24 4.25 

19 3.91 3.94 3.97 3.99 4.01 4.04 4.06 4.08 4.10 4.11 4.13 4.15 4.17 4.18 4.20 

20 3.87 3.90 3.92 3.96 3.97 3.99 4.01 4.03 4.06 4.07 4.09 4.10 4.12 4.14 4.15 

21 3.84 3.86 3.89 3.91 3.93 3.96 3.97 3.99 4.01 4.03 4.05 4.06 4.08 4.09 4.11 

ZZ 3.80 3.83 3.86 3.88 3.90 3.92 3.94 3.96 3.98 3.99 4.01 4.03 4.04 4.06 3 .07 

23 3.77 3.80 3.82 3.86 3.87 3.89 3.91 3.93 3.94 3.96 3.98 3.99 4.01 4.02 4.04 

24 3.75 3.77 3.80 3.82 3.84 3.86 3.88 3.90 3.95 3.93 3.95 3.96 3.98 3.99 4.01 

25 3.72 3.76 3.77 3.79 3.82 3.84 3.85 3.87 3.89 3.91 3.92 5.94 3.95 3.97 3.98 

26 370 3.73 3.75 3.77 3.79 3.81 3.83 3.86 3.87 3.88 3.90 3.91 3.93 5.94 3.95 

27 3.68 3471 3.73 3.75 3.77 3.79 3.81 3.83 3.84 3.86 3.87 3.89 3.90 3.92 3.93 

28 3.66 3.69 3.71 3.73 3.75 3.77 3.79 3.81 3.82 3.84 3.86 3.87 3.88 3.90 3.91 

29 3.66 3.67 3.69 3.71 3.73 3.76 3.77 3.79 3.80 3.82 3.83 3.85 3.86 3.88 3.89 

30 3.63 3.66 3.68 3.70 3.74 3.74 3.75 3.77 3.79 3.80 3.82 3.83 3.86 3.86 3.87 

31 3.62 3.64 3.66 3.68 3.70 3.72 3.74 3.76 3.77 3.79 3.80 3.81 3.83 3.84 3.86 

32 3.60 3.63 3.66 3.67 3.69 3.71 3.72 3.74 3.76 3.77 3.79 3.80 3.81 3.83 3.84 

33 3.59 3.61 3.64 3.66 3.67 3.69 3.71 3.73 3.74 3.76 3.77 3.79 3.80 3.81 3.82 

34 3.58 3.60 3.64 3.64 3.66 3.68 3.70 3.71 3.73 3.74 3.76 3.77 3.78 3.80 3.81 

35 3.57 3.69 3.61 3.63 3.66 3.67 3.89 3.70 3.75 3.73 3.75 3.76 3.77 3.78 3.80 

38 3.56 3.58 3.60 3.62 3.64 3.66 3.67 3.69 3.71 3.72 3.73 3.76 3.76 3.77 3.79 

37 3.66 3.67 3.69 3.81 3.63 3.66 3.66 3.68 3.69 3.71 3.72 3.74 3.75 3.76 3.77 

38 3.54 3.56 3.68 3.60 3.62 3.64 3.65 3.67 3.68 3.70 3.71 3.73 3.74 3.75 3.76 

39 3.63 3.65 3.57 3.59 3.61 3.83 3.64 3.66 3.67 3.69 3.70 3.72 3.73 3.74 3.75 

40 3.62 3.64 3.56 3.58 3.60 3.62 3.63 3.66 3.67 3.68 3.69 3.71 317 3.73 3.74 

41 3.61 3.64 3.66 3.68 3.69 3.61 3.63 3,64 3.66 3.67 3.68 3.70 3.71 3.72 3.73 

42 3.51 3.63 3.66 3.67 3.69 5.60 3.64 3.63 3.66 3.66 3.68 3.69 3.70 3.71 3.73 

43 3.60 3.62 3.64 3.66 3.68 3.69 3.61 3.63 3.64 3.65 3.67 3.68 3.69 3.71 3.72 

44 3.49 3.61 3.63 3.66 3.67 3.59 3.60 3.62 3.63 3.66 3.86 3.87 3.69 3.70 3.71 

46 3.49 3.61 3.63 3.66 3.66 3.68 3.60 3.61 3.63 3.64 3.66 3.67 3.68 3.69 3.70 

46 3.48 3.60 3.62 3.64 3.68 3.67 3.69 3.61 3.62 3.63 3.65 3.66 3.67 3.68 3.70 

47 3.48 3.50 3.62 3.63 3.65 3.67 3.68 3.60 3.61 3.63 3.64 3.66 3.67 3.68 3.69 

48 3.47 3.49 3.51 3.53 3.55 3.66 3.68 3.59 3.61 3.62 3.63 3.66 3.66 3.67 3.68 

49 3.46 3.48 3.60 3.62 3.64 3.66 3.67 3.69 3.60 3.62 3.63 3.64 3.6b 3.66 3.68 

b0 3.46 3.48 3.60 3.64 3.53 3.66 3.67 3.68 3.60 3.61 3.62 1.64 3.66 3.66 3.67 

60 3.42 3.44 3.46 3.47 3.49 3.61 3.52 3.64 3.66 3.66 3.68 3.69 3.60 3.61 3.62 

70 3.39 3.41 3.43 3.45 3.48 3.48 3.49 3.61 3.52 3.63 3.54 3.66 3.67 3.68 3.69 

80 3.37 3.39 3.41 3.42 3.44 3.46 3.47 3.48 3.60 3.51 3.52 3.53 3.54 3.56 3.67 

90 3.36 3.37 3.59 3.41 3.42 3.44 3.46 3.47 3.48 3,49 3.50 3.51 3.63 3.54 3.55 

100 3.34 3.36 3.38 3.39 3.41 3.42 3.44 3.45 3.46 3.48 3.49 3.60 3.51 3.62 3.53 

Factor = t(n-1,1-a/k) %/ 1 f- 1/ s 

3. PREDICTION LIMITS FOR THE 

LOGNORMAL DISTRIBUTION 

Whcn data exhibit a few elcvated values such 

that the frequency distrihution is "skewcd" with a. 

long right ta.il , estimates based oii the assuinption 

of norniality do not apply. In practicc, transfornlations of data are used to stabilize variance and 

bring about normality ( t3ox and (:ox, 1964). 

Perhaps the most commonly Lised transformation 

in rhis situation is the natural log transformation 

(Aroian, 1941), wlhere x is a lognorrnal randorn 

variable, such that y= logex- N(p, 0 2 ). Wlien 

jj and o 2 are unknown, we tnay substitut4 the 

saniple estiniates y and sz wherc 

n 

I loge(x; )In 

i =-1 

n 

ancl s= - Z[log, (x1) - y` J'I (n - 1). 

i=1 

Rased on a result obtained by Dahiya and 

Guttman (1982), thc 100(1 - a)96 prediction 

interval has thc familiar form 

exp {y ± 1 + ],/n tIn_l;a/21 s}. 

in light of this, the values in 'I'ables 1•3 may be 

used to obtain the one-sided 99% prediction limit: 

exp { y` + r(k, n; .99) s} 

458 
where y` and s are the mean and standard deviation 

computed on natural log transformed data. 

Similarly, Tables 4-6 may be used to computc: the 

corresponding two-sided 99% prediction limit: 

exp {y ± r(k, n; .99) s} 

A note of caution must be raised regarding the 

use of log transformation in calculating prediction 

limits. 1'his point is best illustrated by an example. 

Consider the following data for 'fOX, and for 

sirnplicity let us assume that these measurements 

were collected from a single upgradicnt well, 

quarterly for two years so that sl is an unbiased 

estimate of 02. 

Upgradient 

wcll Quarter Ycar TOX 10ge(TOX) 

Mw01 1 2985 48 3.87 

Mwol 2 1985 5 1.61 

MWoI 3 1985 44 3.78 

Mw01 4 1985 24 3.18 

MWoI 1 1986 17 2.83 

Mw01 2 1986 ti 1.79 

Mw01 3 1986 12 2.48 

Mw01 4 1986 60 4.09 

x=27.00 x'=2.96 

s = 20.97 s = 0.95 

n- 8 n=8 

Inspection of the original I'OX valucs reveals a distribution that is skewed to the right; thercfore, log 

Table 3. Factors for Obtaining One-Sided 9996 Prediction Limits for k 

Additional Samples Given a Background Sample of Size n 

Previoue Number of new measuremente (k) 

n 30 36 40 46 60 66 60 66 70 76 80 86 90 96 100 

4 11.67 11.98 12.34 12.66 1.96 13.22 13.47 13.70 13.91 14.11 14.31 14.49 14.66 14.82 14.98 

6 8.74 9.01 9.56 9.46 9.66 9.83 9.99 10.14 10.29 10.42 10.64 10.66 10.77 10.88 10.98 

6 7.30 7.61 7.68 7.84 7.99 8.12 8.24 8.36 8.46 8.66 8.66 8.73 8.82 8.90 8.97 

7 6.46 6.61 6.76 6.89 7.00 7.11 7.20 7.29 7.38 7.46 7.63 7.60 7.67 7.73 7.79 

8 6.89 6.03 6.16 6.26 6.36 6.46 6.63 6.61 6.68 6.74 6.81 6.87 6.92 6.97 7.02 

9 6.60 6.62 6.73 6.82 6.91 6.99 6.06 6.13 6.19 6.25 6.30 6.36 6.40 6.46 6.49 

10 5.21 5.32 6.42 6.60 6.68 6.66 6.72 5.77 6.83 6.88 6.93 6.98 6.02 6.06 6.10 

11 4.99 6.09 6.18 6.26 5.33 6.39 6.46 6.61 6.66 6.60 6.66 6.69 6.73 6.77 5.80 

12 4.82 4.91 4.99 6.07 6.13 6.19 6.26 6.30 6.34 6.39 6.43 6.47 6.60 6.64 5.57 

13 4.68 4.77 4.84 4.91 4.97 6.03 6.08 6.13 6.17 6.31 6.26 6.28 6.32 6.35 6.38 

14 4.66 4.66 4.72 4.78 4.84 4.89 4.94 4.99 6.03 6.07 6.10 6.14 6.17 6.20 5.23 

15 4.46 4.64 4.61 4.68 4.73 4.78 4.83 4.87 4.91 4.96 4.98 6.01 6.04 6.07 5.10 

16 4.38 4.46 4.63 4.59 4.64 4.69 4.73 4.77 4.81 4.84 4.88 4.91 4.94 4.96 4.99 

17 4.31 4.39 4.46 4.61 4.66 4.61 4.66 4.69 4.72 4.76 4.79 4.82 4.86 4.87 4.90 

18 4.26 4.32 4.39 4.44 4.49 4.64 4.68 4.61 4.66 4.68 4.71 4.74 4.77 4.79 4.82 

19 4.20 4.27 4.13 4.38 4.45 4.47 4.61 4.66 4.68 4.61 4.64 4.67 4.70 4.72 4.76 

20 4.16 4.22 4.28 4.33 4.38 4.42 4.46 4.49 4.63 4.66 4.69 4.61 4.64 4.66 4.68 

21 4.11 4.18 4.25 4.58 4.33 4.57 4.41 4.44 4.48 4.60 4.63 4.56 4.68 4,61 4.63 

22 4.07 4.14 4.19 4.24 4.29 4.33 4.36 4.40 4.43 4.46 4.49 4.61 4.64 4.66 4.58 

23 4.04 4.10 4.16 4.21 4.26 4.29 4.32 4.38 4.39 4.42 4.44 4.47 4.49 4.51 4.54 

24 4.01 4.07 4.12 4.17 4.21 4.26 4.29 4.51 4.36 4.38 4.41 4.43 4.46 4.48 4.50 

26 5.98 4.04 4.09 4.14 4.18 4.22 4.26 4.29 4.32 4.36 4.37 4.40 4.42 4.44 4.46 

26 3.96 4.01 4.07 4.11 4.16 4.19 4.23 4.26 4.29 4.31 4.34 4.36 4.39 4.41 4.43 

27 5.93 3.99 4.04 4.09 4.13 4.17 4.20 4.23 4.26 4.29 4.31 4.33 4.36 4.38 4.40 

28 3.91 3.97 4.02 4.06 4.11 4.14 4.17 4.21 4.23 4.26 4.28 4.31 4.33 4.36 4.37 

29 5.89 3.96 4.00 4.04 4.08 4.12 4.16 4.18 4.21 4.24 4.26 4.28 4.30 4.32 4.34 

SO 5.87 3.93 3.98 4.02 4.06 4.10 4.13 4.16 4.19 4.21 4.24 4.26 4.28 4.30 4.32 

31 5.86 3.91 3.96 4.00 4.04 4.08 4.11 4.14 4.17 4.19 4.22 4.24 4.26 4.28 4.30 

32 3.84 3.90 3.94 3.99 4.03 4.06 4.09 4.12 4.16 4.17 4.20 4.22 4.24 4.26 4.28 

33 3.82 3.88 3.93 3.97 4.01 4.04 4.08 4.10 4.13 4.16 4.18 4.30 4.22 4.24 4.26 

34 1.81 3.87 3.91 3.96 3.99 4.03 4.06 4.09 4.11 4.14 4.16 4.18 4.20 4.22 4.24 

36 3.80 3.86 3.90 3.94 3.98 4.01 4.04 4.07 4.10 4.12 4.16 4.17 4.19 4.21 4.22 

36 3.79 3.84 3.89 3.93 3.97 4.00 4.03 4.06 4.08 4.11 4.13 1.16 4.17 4.19 4.21 

37 3.77 3.83 3.88 3.92 3.96 3.99 4.02 4.04 4.07 4.09 4.12 4.14 4.16 4.18 4.19 

38 3.76 3.82 3.86 3.90 3.94 3.97 4.00 4.03 4.06 4.08 4.10 4.12 4.14 4.16 4.18 

39 3.75 3.81 3.85 3.89 5.93 3.96 5.99 4.02 4.06 4.07 4.09 4.11 4.13 4.15 4.17 

40 5.74 3.80 3.84 3.88 3.92 3.96 3.98 4.01 4.03 4.06 4.08 4.10 4.12 4.14 4.15 

41 3.73 3.79 3.83 3.87 3.91 3.94 3.97 4.00 4.02 4.05 4.07 4.09 4.11 4.12 4.14 

42 3.73 3.78 3.82 3.86 3.90 3.93 3.96 3.99 4.01 4.04 4.06 4.08 4.10 4.11 4.13 

43 3.72 3.77 3.82 3.86 3.89 3.92 3.96 3.98 4.00 4.03 4.06 4.07 4.09 4.10 4.12 

44 3.71 3.76 3.81 3.86 3.88 3.91 3.94 5.97 5.99 4.02 4.04 4.06 4.08 4.09 4.11 

46 5.70 3.76 3.80 3.84 3.87 5.91 3.95 3.96 3.98 4.01 4.05 4.06 4.07 4.08 4.10 

46 3.70 3.76 3.79 3.83 3.87 3.90 3.93 3.95 3.98 4.00 4.02 4.04 4.06 4.08 4.09 

47 3.09 3.74 3.78 3.82 3.86 3.89 3.92 3.94 3.97 3.99 4.01 4.03 4.05 4.07 4.08 

48 5.68 3.73 3.78 3.82 3.86 3.88 3.91 3.94 5.96 3.98 4.00 4.02 4.04 4.06 4.07 

49 3.88 3.73 3.77 3.81 3.84 3.88 3.90 3.93 3.96 3.98 4.00 4.02 4.03 4.05 4.07 

60 3.67 3.73 3.76 3.80 3.84 3.87 5.90 3.92 3.96 3.97 3.99 4.01 4.03 4.04 4.06 

60 3.62 3.67 3.71 3.76 3.78 3.81 3.84 3.87 3.89 3.91 3.93 3.96 3.97 3.98 4.00 

70 5.69 3.64 5.68 3.72 3.76 3.78 3.80 3.83 3.86 3.87 3.89 3.91 3.93 3.94 3.96 

80 3.67 3.61 3.66 3.69 3.72 3.76 3.78 1.80 3.82 3.84 3.86 3.88 3.89 3.91 3.93 

90 3.66 3.69 5.63 3.67 3.70 3.75 3.76 3.78 3.80 3.82 3.84 3.86 3.87 3.89 3.90 

100 3.63 5.58 5.62 3.66 3.68 3.71 3.74 3.76 3.78 5.60 3.82 3.84 5.86 3.87 3.88 

Fsctor = t ~w -l.l-a/k) ~1 ♦ I/R 

459 
Table 4. Factors for Obtaining Two-Sided 99% Prediction Limits for k 

Additional Samples Given a Background Sample of Size n 

Previous Number ot new measurements (k) 

n 1 2 3 4 6 6 7 8 9 10 11 12 13 14 15 

4 6.81 7.06 7.86 8.46 8.94 9.35 9.70 10.01 10.30 10.66 10.79 11.00 11.21 11.40 11.57 

5 4.77 5.66 6.22 6.63 6.97 7.24 7.48 7.70 7.89 8.06 8.22 8.36 8.50 8.62 8.74 

6 4.23 4.93 5.37 5.69 5.96 6.16 6.36 6.61 6.65 6.79 6.90 7.02 7.12 7.21 7.30 

7 3.89 4.49 4.86 5.13 6.34 5.62 6.67 6.80 6.92 6.03 6.13 6.21 6.30 6.38 6.45 

8 3.67 4.20 4.62 4.75 4.94 6.09 6.22 6.34 5.44 6.53 5.61 6.69 5.76 5.83 5.89 

9 3.51 3.99 4.28 4.49 4.66 4.79 4.91 6.01 5.10 6.18 5.25 6.32 6.39 6.44 6.60 

10 3.39 3.84 4.10 4.30 4.45 4.67 4.68 4.77 4.85 4.92 4.99 6.06 5.11 6.16 5.21 

11 3.30 3.72 3.97 4.14 4.28 4.40 4.60 4.68 4.66 4.73 4.79 4.84 4.90 4.94 4.99 

12 3.22 3.62 3.86 4.03 4.16 4.26 4.36 4.44 4.61 4.67 4.63 4.68 4.73 4.77 4.82 

13 3.16 5.64 3.77 3.93 4.05 4.16 4.24 4.32 4.38 4.44 4.50 4.56 4.69 4.64 4.68 

14 3.11 3.48 3.70 3.86 3.97 4.07 4.15 4.22 4.28 4.34 4.39 4.44 4.48 4.52 4.56 

15 3.07 3.43 3.63 3.78 3.90 3.99 4.07 4.14 4.20 4.26 4.30 416 4.39 4.43 4.46 

16 3.03 3.38 3.68 3.72 3.84 3.93 4.00 4.07 4.13 4.18 4.23 4.27 4.31 4.35 4.38 

17 3.00 3.34 3.54 3.68 3.78 3.87 3.95 4.01 4.07 4.15 4.16 4.40 4.24 4.28 4.31 

18 2.97 311 3.60 3.63 3.74 3.82 3.89 3.96 4.01 4.06 4.11 4.16 4.18 4.22 4.26 

19 2.96 3.28 3.46 3.60 3.70 3.78 3.85 3.91 3.97 4.01 4.06 4.10 4.13 4.17 4.20 

20 2.93 3.25 3.43 3.66 3.66 3.74 3.81 3.87 3.92 3.97 4.01 4.05 4.09 4.12 4,15 

21 2.91 3.22 3.41 3.65 3.63 3.71 3.78 3.84 3.89 3.93 3.97 4.01 4.06 4.08 4.11 

22 2.89 3.20 3.38 3.61 3.60 3.68 3.76 3.80 3.86 3.90 3.94 3.98 4.01 4.04 4.07 

23 2.88 5.18 3.36 3.48 3.68 3.66 3.72 3.77 3.82 3.87 3.91 3.94 3.98 4.01 4.04 

24 2.86 3.17 3.54 3.46 3.65 5.63 3.69 3.76 3.80 3.84 3.88 3.92 3.95 3.98 4.01 

25 2.85 3.15 3.32 3.44 3.53 3.61 3.67 3.72 3.77 3.82 3.85 3.89 3.92 3.95 3.98 

26 2.84 3.14 3.30 3.47 3.61 3.59 3.65 3.70 3.76 3.79 3.83 3.87 3.90 3_93 3.95 

27 2.83 3.12 3.59 3.41 3.50 3.67 3.63 3.68 3.73 3.77 3.81 3.84 3.87 3.90 3.93 

28 2.82 3.11 3.27 3.39 3.48 3.65 3.61 3.66 3.71 3.76 3.79 3.82 3.85 3.88 3.91 

29 2.81 3.10 3.26 3.38 3.46 3.54 3.60 3.66 3.69 3.73 3.77 3.80 3.83 3.86 3.89 

30 2.80 3.09 3.25 3.36 3.46 3.62 3.68 3.63 3.88 3.72 3.75 3.79 3.82 3_86 317 

31 2.79 3.08 3.24 3.36 3.44 3.61 3.67 1.65 3.66 3.70 3.74 3.77 3.80 3.83 346 

32 2.79 3.07 3.23 3.34 3.43 3.50 3.65 3.60 3.66 3.69 3.72 3.76 1.79 3.81 3.84 

33 2.78 3.06 3.22 3.33 3.41 3.48 3.54 3.69 3.64 3.67 3.71 3.74 3.77 3.80 3.82 

34 2.77 3.05 3.21 3.52 3.40 3.47 3.63 3.68 3.62 3.66 3.70 3.73 3.76 3.78 3.81 

35 2.77 3.04 3.20 3.31 3.39 3.46 3.65 317 3.e1 3.66 3.89 3.72 3.75 3.77 3.80 

36 2.76 3.04 3.19 3.30 3.39 3.45 3.51 3.66 3.60 3.64 3.67 3.71 3.73 3.76 3.79 

37 2.76 3.03 3.19 3.29 3.38 3.44 3.60 3.65 3.59 3.63 3.66 3.69 3.72 3.7L 3.77 

38 2.76 3.02 3.18 3.29 3.37 3.44 3.49 3.64 5.58 3.62 3.65 3.68 3.71 3.74 3.76 

39 2.75 3.02 3.17 3.28 3.36 3.43 3.48 3.63 3.57 3.61 3.64 3.67 3.70 3.73 3.75 

40 2.74 3.01 3.17 3.27 3.36 3.42 3.47 3.65 3.66 3.60 3.63 3.67 3.69 3.72 3.74 

41 2.74 3.01 3.16 3.27 3.35 3.41 3.47 3.51 3.66 3.59 3.63 3.66 3.68 3.71 3.73 

42 2.73 3.00 3.15 3.26 3.34 3.41 3.46 3.51 3.56 3.59 3.62 3.65 3.68 3.70 3.73 

43 2.73 3.00 3.15 3.26 3.33 3.40 3.45 3.60 3.64 3.58 3.61 3.64 3.67 3.69 3.72 

44 2.73 2.99 3.14 3.26 3.33 3.39 3.46 3.49 3.63 3.67 3.60 3.63 3.66 3.69 3.71 

45 2.72 2.99 3.14 3.54 3.32 3.39 3.44 3.49 313 3.66 3.60 3.63 3.66 3.68 3.70 

46 2.72 2.98 3.13 3.24 3.32 3.38 3.43 3.48 3.62 3.66 3.69 3.62 3.66 3.67 3.70 

47 2.72 2.98 3.13 3.23 3.51 3.38 3.43 3.48 3.62 3.65 3.68 3.61 S.64 3.67 3.69 

48 2.71 2.98 3.12 3.23 3.31 3.37 3.42 3.47 3.61 3.56 3.58 5.61 3.63 3.66 3.68 

49 2.71 2.97 3.12 3.22 3.30 3.37 3.42 3.46 3.60 3.64 3.57 3.60 3.63 3.65 3.68 

60 2.71 2.97 3.12 3.22 3.30 3.36 3.41 3.46 3.60 3.63 3.57 3.60 3.62 3.66 3.67 

60 2.68 2.94 3.08 3.18 3.28 3.32 3.37 3.42 3.46 3.49 3.62 3.55 3.68 5.60 3.62 

70 2.67 2.92 3.06 3.16 3.24 3.30 3.55 3.39 3.43 3.46 3.49 3.52 3.54 3.57 3.69 

80 2.66 2.91 3.05 3.14 3.21 3.28 3.33 3.37 3.41 3.44 3.17 3.60 3.52 3.54 3.57 

90 2.66 2.89 3.03 3.13 3.20 3.26 3.31 3.36 3.39 3.42 3.45 3.48 3.50 3.53 3.55 

100 2.64 2.89 3.02 3.12 3.19 3.55 3.30 3.34 3.38 3.41 3.44 3.46 3.49 3.51 3.53 

Factor = t(R_ 1 , 1 _0/2 k) 1+ 1 fn 

transformation appcars to be a natural choice for 

better approxima.ting the assumed normality of the 

statistical procedure. Indeed, the transforniation 

appears to have somewhat normali -r.ed the distribution of these nieasurements. 

Assuming that we are interested in evaluating 

a singlc new downgradient measuremcnt, thc 9996 

prediction limit based on the original data is: 

27.00 + 20.97(3.16) = 93.27 pph 

whereas, the 99% prediction limit for the log 

transformed data is: 

exp [2.96 + .95 (3.16)] = 388.39 ppb 

Alternatively, if we directly estinlate the niean and 

standard deviation of the lognormal distribution 

(Aitchison, 1955), we obtain for the 99% prediction limit 

28.16 + 27.71(3.1 6) = 115.73 ppb 

Inspection of these three prediction limits 

reveals that the estilnate based on log transformed 

data is remarkablv elevated relative to the normal 

. 

and lognornial interval estimatcs. The lognorrnal 

prediction limit is, of course, incorrect because the 

use of thesc liniits assumcs a normal distribution 

for xt, whereas these estimates are based on log460 
normal assumptions; however, they do seem to give 

quite reasonable estimates, at least, for practical 

purposes. Conversely, the interval estimatc 

obtained following log transformation of the data, 

a seemingly standard and benign practice, is 

extremely high and is likely to lcad to elevated 

false negative rates even for liberal effect magnitudes. In fact, the limit value for the log transformed data is almost an entire order of magnitude 

higher than the maximum value in the background 

sample. It is for this reason that log transformation 

must be used very cautiously when constructing 

prediction intervals. 

4. SAMPLING STRATEGIES 

When using prediction intervals, we assume 

that both the original observations (x l , ..., x n ) 

and the future observations (x„ + t, .. ., xn f k ) arise 

from the same probability distribution, namely, 

N(,u, o 2). Since, in practice, μ and o 2are unknown 

quantities, we must estimate their value from a 

random sample of previous measurements by 

obtaining the sufficient statistics z and s, that is, 

the sample mean and standard deviation. If our 

background measurcments consist of multiplc 

quarterly measurements of a single well or muttiple 

upgradient wells at a single time-point, 

Table 5. Factors for Obtaining Two-Sided 991/o Prediction Limits for k 

Additional Samples Given a Background Sample of Size n 

Previous IVumber of new messuraments (!t) 

n 16 17 18 19 20 21 22 23 24 26 26 27 28 29 30 

4 11. 74 11,90 12 . 05 12.20 12.34 12.47 12.60 12.72 12.84 12.95 13.06 13.17 13.27 13.37 13.47 

5 8.85 8.96 9.06 9.16 9.25 9.34 9.42 9.50 9.58 9.65 9.73 9.80 9.86 9.93 9.99 

6 7.39 7.47 7.64 7.61 7.68 7.76 7.81 7.87 7.93 7.99 8.04 8.09 8.14 8.19 8.24 

7 6.62 6.68 6.64 6.70 6.76 6.81 6.86 6.91 6.96 7.00 7.04 7.09 7.13 7.17 7.20 

8 5.95 6.00 6.06 6.10 6.16 6.20 6.24 6.28 6.32 6.36 6.40 6.43 6.47 6.50 6.53 

9 5.55 5.60 5.64 6.69 5.73 5.77 5.81 6.84 5.88 6.91 6.94 5.97 6.00 6.03 6.06 

10 6.26 6.30 614 6.38 6.42 5.45 5.49 5.52 5.55 5.58 5.61 5.64 5.66 5.69 5.72 

11 6.03 6.07 6.11 6.15 5.18 6.21 5.24 6.27 6.30 6,33 6.36 5.38 5.41 5.43 6.45 

12 4.86 4.89 4.93 4.96 4.99 5.02 5.05 6.08 6.11 5.13 5.16 5.18 5.20 5.22 5.25 

13 4.71 4.75 4.78 4.81 4.84 4.87 4.90 4.92 4.96 4.97 5.00 5.02 6.04 5.06 5.08 

14 4.60 4.63 4.66 4.69 4.72 4.76 4.77 4.80 4.82 4.84 4.86 4.88 4.90 4.92 4.94 

15 4.60 4.53 4.56 4.59 4.61 4.64 4.66 4.69 4.71 4.73 4_75 4.77 4.79 4.81 4.83 

16 4.41 4.45 4.47 4.50 4.63 4.56 4.67 4.60 4.62 4.64 4.66 4.68 4.70 4.71 4.73 

17 4.34 4.37 4.40 4.43 4.45 4.47 4.60 4.62 4.54 4.66 4.58 4.60 4.61 4.63 4.65 

l8 4.28 4.31 4.34 4.36 4.39 4.41 4.43 4.46 4.47 4.49 4.61 4.63 4.54 4.56 4.58 

19 4.23 4.26 4.28 4.31 4.33 4.35 4.37 4.39 4.41 4.43 4.46 4.47 4.48 4.50 4_51 

20 4.18 4.21 4.23 4.26 4.28 4.30 4.32 4.34 4.36 4.38 4.39 4.41 4.43 4.44 4.46 

21 4.14 4.16 4.19 4.21 4.23 4.25 4.27 4.29 4.31 4.33 4_35 4.36 4.38 4.39 4.41 

22 4.10 4.12 4.15 4.17 4.19 4.21 4.53 4.25 4.27 4.29 4.30 4.32 4.34 4.35 4.36 

23 4.06 4.09 4.11 4.14 4.16 4.18 4.20 4.21 4.23 4.26 4.27 4.28 4.30 4.31 4.32 

24 4.03 4.06 4.08 4.10 4.12 4.14 4.16 4.18 4.20 4.21 4.23 4.26 4.26 4.28 4.29 

26 4.00 4.03 4.05 4.07 4.09 4.11 4.13 4.15 4.17 4.18 4.20 4.21 4.23 4.24 4.26 

26 3.98 4.00 4.03 4.06 4.07 4.09 4.10 4.12 4.14 4.16 4.17 4.19 4.20 4.21 4.23 

27 3.96 3.98 4.00 4.02 4.04 4.06 4.08 4.10 4.11 4.13 4.14 4.16 4.17 4.19 4.20 

28 3.93 3.96 3.98 4.00 4.02 4.04 4.06 4.07 4.09 4.11 4.12 4.13 4.16 4.16 4.17 

29 3.91 3.94 3.96 3.98 4.00 4.02 4.03 4.05 4.07 4.08 4.10 4.11 4.13 4.14 4.15 

30 3.90 3.92 3.94 3.96 3.98 4.00 4.01 4.03 4.06 4.06 4.08 4.09 4.10 4.12 4.13 

31 3.88 3.90 3.92 3.94 3.96 3.98 4.00 4.01 4.03 4.04 4.06 4.07 4.09 4.10 4.11 

32 3.86 3.88 3.91 3.93 3.94 3.96 3.98 4.00 4.01 4.03 4.04 4.06 4.07 4.08 4.09 

33 3.85 3.87 3.89 3.91 3.93 3.95 3.96 3.98 3.99 4.01 4.02 4.04 4.05 4.06 4.08 

34 3.83 3.86 3.88 3.90 3.91 3.93 3.96 3.96 3.98 3.99 4.01 4.02 4.03 4.05 4.06 

36 3.82 3.84 3.86 3.88 3.90 3.92 3.93 3.96 3.97 3.98 3.99 4.01 4.02 4.03 4.04 

S6 3.81 3.83 3.86 3.87 3.89 3.90 3.92 3.94 3.96 3.97 3.98 3.99 4.01 4.02 4.03 

37 3.80 3.82 3.84 3.86 3.88 3.89 3.91 3.92 S.94 3.96 3.97 3.98 3.99 4.00 4.02 

38 3.79 3.81 3.83 3.85 3.86 3.88 3.90 3.91 3.93 3.94 3.95 3.97 3.98 3.99 4.00 

39 3.78 3.80 3.82 3.84 3.86 3.87 3.89 3.90 3.92 3.93 3.94 3.96 3.97 3.98 3.99 

40 3.77 3.79 3.81 3.83 3.84 3.86 3.88 3.89 3.91 3.92 3.93 3.96 3.96 3497 3.98 

41 3.76 3.78 3.80 3.82 3.83 3.86 3.87 3.88 3.90 3.91 3.92 3.94 3.95 3.96 3.97 

42 3.75 3.77 3.79 3.81 3.82 3.84 3.86 3.87 3.89 3.90 3.91 3.93 3.94 3.95 3.96 

43 3.74 3.76 3.78 3.80 3.82 3.83 3.85 3.86 3.88 3.89 3.90 3.92 3.93 3.94 3.95 

44 3.73 3.76 3.77 3.79 3.81 3.82 3.84 3.85 3.87 3.88 3.89 3.91 3.92 3.93 3.94 

46 3.72 3.74 3.76 3.78 3.80 3.82 3.83 3.86 3.86 3.87 3.89 3.90 3.91 3.92 3.93 

46 3.72 3.74 3.76 3.77 3.79 3.81 3.82 3.84 3.86 3.87 3.88 3.89 3.90 3.91 3.93 

47 3.71 3.73 3.75 3.77 3.78 3.80 3.82 3.83 3.84 3.86 3.87 3.88 3.90 3.91 3.92 

48 3.70 3.72 3.74 3.76 3.78 3.79 3.81 3.82 3.84 3.86 3.86 3.88 3.89 3.90 3.91 

49 3.70 3.72 3.74 3.76 3.77 3.79 3.80 3.82 3.83 3.84 3.86 3.87 3.88 3.89 3.90 

50 3.69 3.71 3.73 3.76 3.76 3.78 3.80 3.81 3.82 3.84 3.86 3.86 3.87 3.89 3.90 

60 3.64 3.66 3.68 3.70 3.71 3.73 3.74 3.76 3.77 3.78 3.80 3.81 3.82 3.83 3.84 

70 3.61 3.63 3.65 3.66 3.68 3.69 3.71 3.72 3.74 3.76 3.76 3.77 3.78 3.79 3.80 

80 3.59 3.60 3.62 3.64 3.65 3.67 3.68 3.70 3.71 3.72 3.73 3.74 3.76 3.76 3.78 

90 3.57 3.58 3.60 3.62 3.63 3.65 3.66 3.67 3.69 3.70 3.71 3.72 3.73 3.74 3.75 

100 3.56 3.57 3.59 3.60 3.62 3.63 3.65 3.66 3.67 3.68 3.69 3.71 3.72 3.73 3.74 

Factor = t ~„-~,~-~/~k)~ l + 1/n 

461 
Table 6. Factors for Obtaining Two•Sided 9996 Prediction Limits for k 

Additional Samples Given a Background Sample of Sixe n 

Previoue Number ot now measurements (k) 

n 30 35 40 46 60 55 60 65 70 76 80 85 90 95 100 

4 13.47 13.91 14.31 14 . 14.98 15.27 16.54 16.80 16.03 16.26 16.46 16.66 16.85 17.03 17.20 

5 9.99 10.29 10.64 10.77 10.98 11.17 11.36 11.61 11.67 11.81 11.94 12.07 12.19 12.31 12.42 

6 8.24 8.46 8.66 8.82 8.97 9.11 9.24 9.36 9.48 9.68 9.68 9.77 9.86 9.96 10.03 

7 7.20 7.38 7.53 7.67 7.79 7.90 8.01 8.10 8.19 8.28 8.35 8.43 8.50 8.57 8.63 

8 6.53 6.68 6.81 6.92 7.02 7.12 7.20 7.29 7.36 7.43 7.50 7.56 7.62 7.67 7.72 

9 6.06 6.19 6.30 6.40 6.49 6.57 6.66 6.72 6.78 6.84 6.90 6.95 7.00 7.05 7.10 

10 5.72 5.83 5.93 6.02 6.10 6.17 6.24 6.30 6.36 6.41 6.46 6.61 6.56 6.60 6.64 

11 5.45 5.56 5.65 5.73 5.80 5.87 5.93 6.99 6.04 6.09 6.13 6.18 6.22 6.25 6.29 

12 5.25 6.34 5.43 5.50 5.57 5.63 5.69 5.74 5.79 6.83 5.87 5.91 5.95 5.99 6.02 

13 5.08 6.17 6.26 5.32 6.38 6.44 6.49 5.64 5.68 6.63 5.66 5.70 6.74 5.77 5.80 

14 4.94 6.03 5.10 6.17 5.23 5.28 633 6.38 6.42 6.46 6.49 6.53 5.56 5.69 5.62 

15 4.83 4.91 4.98 5.04 5.10 5.16 6.20 6.24 6.28 5.32 6.36 6.38 5.42 5.45 5.47 

16 4.73 4.81 4.88 4.94 4.99 5.04 6.08 5.12 6.16 6.20 5.23 6.26 5.29 5.32 5.35 

17 4.65 4.72 4.79 4.85 4.90 4.94 4.99 6.03 6.06 5.10 6.13 6.16 6.19 6.21 5,24 

18 4.58 4.65 4.71 4.77 4.82 4.86 4.90 4.94 4.98 6.01 5.04 5.07 5.10 6.12 5.15 

19 4.51 4.58 4.64 4.70 4.75 4.79 4.83 4.87 4.90 4.93 4.96 4.99 6.02 5.04 5.06 

20 4.46 4.53 4.69 4.64 4.68 4.73 4.77 4.80 4.83 4.86 4.89 4.92 4.95 4.97 4.99 

21 4.41 4.48 4.53 4.58 4.63 4.67 4.71 4.74 4.77 4.80 4.83 4.86 4.88 4.01 4.93 

22 4.36 4.43 4.49 4.54 4.58 4.62 4.66 4.69 4.72 4.76 4.78 4.80 4.83 4.85 4.87 

23 4.32 4.39 4.44 4.49 4.64 4.68 4.61 4.64 4.67 4.70 4.73 4.75 4.78 4.80 4.82 

24 4.29 4.35 4.41 4.46 4.50 4.63 4.57 4.60 4.63 4.66 4.69 4.71 4.73 4.76 4.78 

26 4.26 4.32 4.37 4.42 4.48 4.50 4.53 4.66 4.59 4.62 4.65 4.67 4.69 4.71 4.73 

26 4.23 4.29 4.34 4.39 4.43 4.46 4.50 4.53 4.56 4.58 4.61 4.63 4.66 4.68 4.70 

27 4.20 4.26 4.31 4.36 4.40 4.43 4.47 4.60 4.63 4.65 4.58 4.60 4.62 4.64 4.66 

28 4.17 4.25 4.28 4.33 4.37 4.40 4.44 4.47 4.60 4.52 4.S5 4.57 4.59 4.61 4.63 

29 4.15 4.21 4.26 4.30 4.34 4.38 4.41 4.44 4.47 4.49 4.52 4.54 4.56 4.58 4.60 

30 4.13 4.19 4.24 4.28 4.32 4.35 4.39 4.42 4.44 4.47 4.49 4.61 4.63 4.65 4.57 

31 4.11 4.17 4.22 4.26 4.30 4.33 4.86 4.39 4.42 4.44 4.47 4.49 4.51 4.53 4.55 

32 4.09 4.15 4.20 4.24 4.28 4.31 4.34 4.37 4.40 4.42 4.45 4.47 4.49 4.51 4.52 

33 4.08 4.13 4.18 4.22 4.26 4.29 4.32 4.35 4.38 4.40 4.42 4.45 4.47 4.48 4.50 

34 4.06 4.11 4.16 4.20 4.24 4.27 4.30 4.33 4.36 4.38 4.40 4.43 4.45 4.46 4.48 

35 4.04 4.10 4.16 4.19 4.22 4.26 4.29 4.31 4.34 4.36 4.39 4.41 4.43 4.45 4.46 

36 4.03 4.08 4.13 4.17 4.21 4.24 4.27 4.30 4.32 4.35 4.37 4.39 4.41 4.43 4.45 

37 4.02 4.07 4.12 4.16 4.19 4.23 4.26 4.28 4.31 4.33 4.35 4.37 4.39 4.41 4.43 

38 4.00 4,06 4.10 4.14 4.18 4.21 4.24 4.27 4.29 4.32 4.34 4.36 4.38 4.40 4.41 

39 3.99 4.06 4.09 4.13 4.17 4.20 4.23 4.26 4.28 4.30 4.32 4.34 4.36 4.38 4.40 

40 3.98 4.03 4.08 4.12 4.15 4.19 4.41 4.24 4.27 4.29 4.31 4.33 4.36 4.37 4.38 

41 3.97 4.02 4.07 4.11 4.14 447 4.20 4.23 4.25 4.28 4.30 4.32 4.34 4.35 4.37 

42 3.96 4.01 4.06 4.10 4.15 4.16 4.19 4.22 4.24 4.26 4.29 4.31 4.32 4.34 4.36 

43 3.96 4.00 4.05 4.09 4.12 4.15 4.18 4.51 4.23 4.25 4.27 4.29 4.31 4.33 4.35 

44 3.94 3.99 4.04 4.08 4.11 4.14 4.17 4.20 4.22 4.24 4.26 4.28 4.30 4.32 4.33 

45 3.93 3.98 4.03 4.07 4.10 4.13 4.16 4.19 4.21 4.23 4.26 4.27 4.29 4.31 4.32 

46 3.93 3.98 4.02 4.06 4.09 4.12 4.16 4.18 4.20 4.22 4.24 4.28 4.28 4.30 4.31 

47 3.92 3.97 4.01 4.06 4.08 4.11 4.14 4.17 4.19 4.21 4.23 4.25 4-27 4.29 4.30 

48 3.91 3.96 4.00 4.04 4.07 4.11 4.13 4.16 4.18 4.20 4.22 4.24 4.26 4.28 4.29 

49 3.90 3.95 4.00 4.03 4.07 4.10 4.12 4.16 4.17 4.20 4.22 4.23 4.25 4.27 4.29 

5o 3.90 3.96 3.99 4.03 4.06 4.09 4.12 4.14 4.17 4.19 4.21 4.23 4.24 4.26 4.28 

60 3.84 3.89 3.93 3.97 4.00 4.03 4.05 4.08 4.10 4.12 4.14 4.16 4.18 4.19 4. 21 

70 3.80 3.86 3.89 3.93 3.96 3.98 4.01 4.03 4.06 4.08 4.09 4.11 4.13 4.14 4.16 

80 3.78 3.82 3.86 3.89 3.93 3.95 3.98 4.00 4.02 4.04 4.06 4.08 4.09 4.11 4.12 

90 3.75 3.80 3.84 3.87 3.90 3.93 3.95 3.98 4.00 4.02 4.03 4.05 4.07 4.08 4.10 

100 3.74 3.78 3.82 3.86 3.88 3.91 3.93 3.96 3.98 4.00 4.01 4.03 4.05 4.06 4.08 

F: ttor = f(n-1,1-a/1R)\/1 + 11/1 

s2= 

a z. Howcver, when multiplc upgradient wells are 

evaluated on several quarters, the previous 

equation yields biased results unless we make the 

demonstrably false assumption that temporal and 

spatial variability are identical. 

'I'he unbiased variance estimate for the case of 

multiple upgradient wells and multiple samplings 

of each well (selected at sufficicnt intervals to 

insure independence, for example, quarterly 

sampling) can be obtained froni the random effects 

xi1 -μ+wt +TI +Ei1 

where x;j is the measurement on background 

(upgradient) well i on sampling event j; p is the 

unknown mean level for background measurements 

(i. e., averaged across wells and time); w 1 is a random 

variable distributed N(0, a',,) that describes the 

n analysis of variance model [see Winer (1971), 

L (xr - ac)z/(n - 1) is an unbiased estiniatc of page 427, for a similar examplc] . l iere, we are 

i=1 . modeling thc background measurernents in terms 

of two random factors corresponding to wells and 

time-points. 'I'he general form of this model is 

given by : 

462 
deviation of well i from μ; rj is a random variable 

distributed N(0, o; ) that describes the deviation at 

time j from μ; and Frj is a random residual distributed 1V(0, vE). Assuming independence of the x=j, 

the expected value of a measurement is: 

E(xij)=μ 

with variance 

V(xij) = QW + O~ + QE 

1'his model corresporids to what is termed a 

randomized bdock design in the statistical literature, 

and the expected mean square for the wells, timepoints, and residuals are given, respectively, as: 

MS W = tow + oF 

w 

_ t £ (~'i - i)2 /(u,_ 1) 

i=f 

MST= wo T+ QE 

- z~ 2; (acj - ac)2 l(t - 1) 

j=1 

~~ 

w t _ 

_ £ £ (xij — xi — xj + x)2 l (W — ~ ) (t — .i* ) 

i-1 j=1 

wlhere xij is the measurement for well i on occasion 

j; w is the number of wells; t is the number of 

quarterly measurements per well; ac; is the niean for 

all measurernents from well i; acj is the mean for all 

nleasurements from tirne j; and x is the mean of all 

measurements across wells and tiines. 

Solving for the individual variance components 

Q 2,, , oT , and o ~ , we find that : 

2- MS w- MS E 

~ Wt 

2- MSr-1WSE 

W7 ~ 

and o f = MSC 

The unbiased estimate of the total variance is 

therefore: 

V(xtj) _' Q GJ + pT + Q F 

=MSw- MSE + Msr - MSE + mS 

t w E 

When the time-points corresponding to the different 

wells cannot be assumed to be equal, then the timepoint factor is said to be "nested" within the well 

factor [see Winer (1971), page 360, for a detailed 

description of nested factor designs) . This would 

occur, for instance, if the background observations 

for well 1 were made quarterly in 1.972, while the 

obscrvations for well 2 were made quarterly in 

1976, and those for well 3 were made in 1975, etc. 

'1'his nested model is given by: 

xzj = /,( + c.Wt + Trj + EiJ 

where xzj, A. wI, and E.ij are as before, while Tij is a 

random variable disrributed N'(0, QT) that describes 

the deviation of well i at time j fronl the overall 

mean value for well z(μ - • w i) averaged over time. 

If satnpling events are "nested" within wells, 

the expected mean squares must be nrtodified to 

reflect this design. Specifically, the expected mean 

square for the wells is: 

mSW = tnW + Qr + v~ 

zu 

~ t 

while the mean square for time is: 

MSr= QT+ oE 

w t 

_ £ Z (.'aCtj — ,XI )2/ [w{t A 

i=1 j ~ I 

With tirne ncstcct within wc:lls, the error term 

QE cannot be scparately cstimatcd since there is 

only one observatiotn per well x tiiile cell. "i'hus, 

solving for the individual variance components 02; 

and aT , we f ind r.hat : 

AISW ._ MS T 

t 

and v 7+ v£ =~YIST 

I'he tinbiased estimatc: of the total variance is now: 

Qw + oT + Qc 

MSw- MST 

= t + :ti'S7 

w(xp — x) 2 z~: t (x=j — x}z 

t ;~1 (w-- 1 ) i~I j~I [w (t 

t 

The above model assumes that all wells are 

rneasured at the same time-points, and thus, the 

two factors are referred to as "crossed" factors. 

w t (xij '" xi) , 

iz i j= t t zv (t - 01 

463 
It is of considerable interest to compare the 

biased variance estimate s 2with the unbiased 

variance cstimate V(xtj ) given above. The simple 

variance estimate that assumes equal spatial and 

temporal variability has expcctation (the author is 

grateful to an anonymous reviewer for raising this 

point): 

w t 

F. [ E E (xtj - xi) ll(n - 1)) 

i=1 j=1 

n- t 

= O ~ +O7 + Qf 

~ ' 1 

wherc n is the totai number of ineasurcments. '1'his 

cquation revcals that s 2will always underestimate 

V(xtj ). For cxample, with two wclls and four 

quarterly measurements, the contribution of the 

well variance Q2 , is underestimated by 43%. '['he 

addition of two upgradient wells decreases this bias 

to 20%, and the addition of a second years background increases the bias only slightly to 2 3%. 

When multiple upgradient wells comprise the 

background saniple, the unbiased estimate V(x=j ) 

should always be uscd in place of s 2 . "fhe importance of using multiple upgradieint wells is due 

both to the large spatial variability that is comnionly observed for naturally occurring constituents 

of ground watcr and bccause it eliminates the 

confound between spatial variability and contamination. In the casc of a single upgradicnt well, an 

upgradient versus downgradient differcncc may 

reflect either spatial variability or contamination 

(z.e., there is no estimate oi' Qt,, ). In c;ontrast, when 

multiplc upgradient wells are sampled, an estimate 

of sratial variability Qw is available, and the resulting prediction limit will provide for spatial cf'fects. 

A critical assumption, however, is that both spatial 

and temporal variability are the same for t.ipgradient 

and downgradient locations. 'Chis is a cornplex 

problcm, and the fcasibility of tllis assumption 

should always be cvaluatcd on a site-specific basis 

to hring into consideration hydrogeology as well as 

problems with previous conditions and sources of 

off-site contamination. 

5. ILLUSTRATIaN 

Considcr thc following data for pli obtained 

from two upgradient wells eac:h with four 

quarterly,measurements. 

Quarter Upgradient well 1 Upgradient well 2 

1 7.1 7.7 

2 7.4 7.8 

3 7.7 7.9 

4 8.0 8.0 

The summary statistics for the nested design 

are: 

ac= (7.1+7.4+7.7+8.0+7.7+7.8+7.9+8.0)/8=7.70 

xwelll = (7.1 + 7.4 + 7.7 + 8.0)/4 = 7.55 

xwc112 =(7.7 + 7.8 + 7.9 + 8.0)/4 = 7.85 

M.SW = 4 [(7.5 5-7.70) 2+(7.85-7.70) Z ] /(2-1)=0.180 

1lgST= [(7.1--7.55) 2 +(7.4-7.55) 1 +(7.7-7.55) 2 + 

(8.0--7.55) 1 +(7.7-7.85) 2 +(7.8 - 7.85) 2 + 

(7.9- 7.85)2 +(8.0 - 7.85) a ] /(2(4- 1)] 

~ 0.08 3 

The resulting variance estimates are thereforc:: 

ZMsw - MS T 

Qw = 

t 

= (.180 -- .08 3 )/4 = 0.024 

o

2 =MST =0.083 

V (x,j ) = 0.108 

As expectcd, the biased variancc estintate 

s 2- .097 underestima.tes V(xtj) since, in this 

example, s 2= 4/7u ~2,, + vT + aE . The variance 

estimate for the randomired blocks design 

V(xij )= .120 was somewhat larger than the result 

obtained for the nested model, because a unique 

estimate of the re5idual error is available 

(ow = .037, QT = .050, and QE = .033). 

In ternls of establishing prediction intervals 

for future observations, we obtain thc. ~ following 

estiniates from Tables 4-6 assuming the more 

general nested sampling design (Z.e., the time-points 

are not the same for all wells). 

Number ef 

downgradient 

wells Prediction interval 

1 7.70 f 3.67( .108) = 6.49 - 8.91 

2 7.70 i 4.20( .108) = 6.32 - 9.08 

3 7.70 t 4.52( .108) = 6.22 - 9.19 

4 7.70 t 4.75 ( 108) = 6.13 - 9.26 

5 7.70 f 4.94(V.108) = 6.08 - 9.32 

6 7.70 f 5.09(V_.108) = 6.03 - 9.37 

7 7.70 -.t- 5.22(V_.108) = 5.98 - 9.42 

8 7.70 ± 5.34(v'108) - 5.94 - 9.46 

9 7.70 f 5.44 ( 108) = 5.91 - 9.49 

10 7.70 f 5.53 (.108) = 5.88 - 9.52 

20 7.70 f 6.15( .108) = 5.68 - 9.72 

30 7.70 1 6.53 ( 108) = 5.55 - 9.85 

40 7.70 ± 6. 81( .108 )-- 5. 46 - 9. 94 

50 7.70 ± 7.02(V_._1O9) = 5.39 - 10.01 

100 7.70 f 7.72( .108) = 5.16 - 10.24 

464 
With a doubling of sample size from 8 to 16 (e.g., 

four upgradient wells with quarterly sampling for 

one year), the prediction intervals would range 

from 6.70 to 8.70 for a single downgradient well 

to 5.94 to 9.46 for a site with 100 downgradient 

wells, assuming that x and V(x;j ) remained 

constant. 

6. DISCUSSlON 

Although the statistical monitoring strategy 

developed here may appear comprehensive, it is 

only a small first step. For example, none of the 

methodology presented here is directly appropriate 

for the analysis of indicator compounds that 

exhibit values below method detection iimits, or go 

undetected in a proportion of the sample. Perhaps, 

the simplest solution to this problem is to use the 

prediction limits developed here with estimates of 

the mean and variance of a censored normal distribution (Cohen, 1961) or censored lognormal distribution (Aitchison, 1955). Using these procedures, however, it is not at all clear how unbiased 

variance estimates would be obtained, as are 

described here for the case of complcte information. As an even more extreme omission, the 

strategy presented here is completely inappropriate 

for the analysis of volatile organic compounds that 

occur in less than 5% of all measurements obtained 

from clean upgradient wells, trip blanks, and field 

blanks (Plumb and Parolini, 1986; Hurd, 1987). 

Inspections of these potentially useful indicator 

compounds suggest that they may arise from a 

Poisson process (Gibbons, 1987) or possibly even 

a censored Poisson distribution. Statistical methods 

for monitoring infrequcntly detected compounds, 

such as the volatile organic priority pollutant compounds are sorely needed in ground-water 

monitoring, and statistical research in this area is 

strongly encouraged. 

ACKNODULEDGMENTS 

Z'he work reported herein was funded by a 

grant from Waste Management lncorporated. Waste 

Management also provided access to an extensive 

national data base that proved to be critical in the 

development and illustration of these statistical 

methods. I am indebted to Donald I iecdekcr of the 

University of Chicago for his excellent cornputational assistance and for reviewing the manuscript. 

I also wish to thank an anonyrnous reviewer for 

clarifying a number of points in an earlier draft. 

REFERENCES 

Aitchison, ,l . 1955. On the distribution of a positive 

random variable having a discrete probability mass at 

the origin. Journal of the American Statistical 

Association, v. 50, pp. 901-908. 

Anderson, T. W. 1958. An Introduction to Multivariate 

Statistical Analysis. Wiley, New York. 

Aroian, L. A. 1941. A study of Fisher's z distribution and 

the related F distribution. Ann. Math. Statist. v. 12, 

pp. 429-448. 

I3ox, G.E.P. and D. R. Cox. 1964. An analysis of transforrnations (with discussion.) J. Roy. Statist. Soc. 

Ser. B. v. 26, pp. 211-252. 

Chcw, V. 1968. Simultaneous prediction intervals. 1'echnornetrics. v. 10, pp. 323-331. 

Cohen, A. C. 1961. Tables for maximum likelihood 

estimates from singly truncated and singly censored 

samples. Technometrics. v. 3, pp. 5 35-541. 

Dahiya, R. C. and I. Guttman. 1982. Shortest confidence 

and prediction intervals for the log-normal. The 

Canadian Journal of Statistics. v. 10, pp. 277-291. 

Gibbons, R. D. 1987. Statistical models for the analysis of 

rare volatile organic compounds in waste disposal 

sites. Ground Water. ln press. 

Hahn, G. J. 1969. Facturs for calculating two-sided prediction intervals for samples from a normal distribution. 

Journal of the American Statistical Association. v. 64, 

pp. 878-888. 

tiahn, G. J. 1970. Additional factors for calculating prediction intervals for samplcs from a normal distribution. 

}ournal of the Amcrican Statistical Association. 

v. 65, pp. 1668-1676. 

Hurd, M. 1987. Determining the impact of land disposal: 

The review of organic analytical data. Unpublished 

manuscript, Environmental 1'esting and Certification 

Corp. 

,tililler, R. G. 1966. Simultaneous Statistical Inferenee. 

Springer-Verlag, New York. 

Plumb , R. H. and J. R. Parolini. 1986. Urgailic contamination of groundwater near hazardous waste disposal 

sites. 'I'echnical Report: EPA Las Vegas Laboratory. 

Snedecor, G, W. and W. G. Cochran. 1980. Statistical 

Methods: Seventh Edition. iowa State University 

Press, Ames, Iowa. 

Wilks, S. S. 1941. Determination of sample sizes for setting 

tolerance limits. Ann. Math. Statist. v. 12, pp 91-96. 

Winer, B. J. 1971. Statistieal Principles in Experimental 

Design. Second E;dition. McGraw ~ -Hill, New York. 

Robert D. Gibbons received bis Pb,U. in Statistics 

from the University of Chicago in 1981, and is currently 

an Associate Professor of Biostatistics at the University of 

lllinois. !n 1985, he received a Young Scientist Award from 

the C)f'fice of rVaval Researcb for studies in Theoretical 

Statistics. A mcijor area of his applied statistical work over 

the past f our years bas been related to problems irr the 

statistical monitoring of ground water at hazardous waste 

disposal facilities. 

465