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£DJ X \s> 

532 

UNITED STATES ENVIRONMENTAL PROTECTION AGENCY 

Office of Air Quality Planning and Standards 

Research Triangle Park, North Carolina 27711 

\ PRQlt0 

•30 JAN 1986 

MEMORANDUM 

SUBJECT: A Comparison of the Precision Associated with Alternativ e 

Sampling Plans fo r One Versus 3 Years of Information 

and Monthly Versus Quarterly Averaging Times 

FROM: William F. Hunt, Jr. C%^£*-~* ~ J*: 

Chief, Data Analysis Section (MD-14) 

TO: John Haines, Chief 

Standards Development Section B (MD-12) 

This memorandum supersedes my earlier memo of December 24, 1985. The 

computer program, which calculated the precision estimates in the earlier 

memo, was found to contain an error. The precision estimates have been 

recalculated for both monthly and quarterly averaging times, for one versus 

3 years of information and for four categories of monitoring sites: 

(1) al l sites, (2) source-oriented sites with maximum annual quarterly 

averages less than the current standard of 1.5 ug/m3, (3) source-oriented 

sites with maximum annual quarterly averages greater than the current standard, and (4) urban-maximum concentration sites. 

ASSUMPTIONS 

The analysis presented in this memorandum will serve as interim 

guidance, while the statistical approach used to calculating precision is 

explored more fully. A number of assumptions have been made in applying 

the formulas which follow. The data are assumed to be approximately normally 

distributed, autocorrelation is assumed not to be a factor, and the confidence 

interval is determined without examining the effect that the monthly 

(quarterly) average is an order statistic. With these simplified assumptions, 

the statistical approach used 1n this memo provides an indication of how 

sampling frequency, 1 versus 3 years of information, averaging times and 

site-types relate to one another. 

CALCULATING PRECISION 

One way of focussing on the tables which follow is to ask the question: 

How many days in a given time period should the air be sampled, so that the 

true monthly (quarterly) arithmetic mean lies within +10 percent of the 

sample arithmetic mean? 
P.3 

The precision is calculated using the following formula: 

P I *-.*,_ % %^-^y^ </+ <$ + /,_ ^ 1^ {7^% "^ / - «*

 c O 

where 

tf = the sample arithmetic monthly (quarterly) mean. 

£t-\ s the t statistic associated with n-1 degrees of freedom. 

s = the sample monthly (quarterly) standard deviation. 

n = the number of samples taken from a finite population. 

N = the total number of observations for the finite population, 

i.e. N = 30 for 1 month, N = 90 for 1 quarter, etc. 

V''" 4i = finit e population correction coefficient (FPC). 

1- «.= the level of confidence, i.e. for * = 0.05, then 1 -* = 0.95 

or i t may be expected that 95 percent of the confidence intervals 

wil l contain the true arithmetic mean. 

The formula which calculates precision can be reformulated to express 

precision as a percentage of the mean. This is accomplished by dividing 

the Increment, +_ t ;-*/,. V>n; JV-""V%. by y and multiplying by 100. 

In doing so, a new variable is created, s/^ , or the coefficient of variation, 

(c). Precision can now be expressed as: 

Both of the above formulas, (1) and (2), can be expressed without the 

FPC, yTTT^j . whether or not the FPC is used, depends upon what you are 

trying to estimate. 

FINITE POPULATION CORRECTION COEFFICIENT 

For each of the comparisons which follow, precision is computed in two 

ways: with and without the finite population correction coefficient (FPC). 

The FPC is used when you want to know the confidence interval about the 

true mean for a specific time period with a finite number of possible 

observations. If the time period, on the other hand, is a sample taken 

from a much larger population, then the FPC is not used. For example, say 

you want to know the confidence interval about the true monthly mean for 

November 1985 at a site. There are only 30 possible sampling days in the 

month of November. To take this factor into account, the FPC is used. If 

every day is sampled, the true mean for this particular November 1985 would 
P.4 

maximum quarterly averages below the standard are upwind sites of lead 

sources, while the coefficients of variation for the source sites with 

maximum quarterly averages above the standard are downwind sites of lead 

sources and are being Impacted to a greater degree by the source. In both 

cases, it 1s assumed that these coefficients of variation will be indicative 

of future coefficients of variation as the source comes into compliance. 

This assumption may be overly conservative. The coefficients of variation 

for source-oriented sites, with higher concentrations, may decrease and 

resemble those with lower concentrations, as the source comes into compliance. 

In any event, these two types of source-oriented sites can provide a bound 

on where the precision associated with future source-oriented sites may lie 

as the source comes into compliance. 

RESULTS 

Tables 4, 5, 6, and 7 are structured in the same way. The firs t column 

indicates how much information is being used - 1 versus 3 years. The 

second column indicates the sampling plan: E6 (75 percent) represents 

every sixth day sampling with 75 percent data capture, E6 represents every 

sixth day sampling with 100 percent data capture and so on. The sampling 

increases from every sixth day (E6) to every third day (E3) to every other 

day (E2) to every day (ED). The third and fourth columns summarize precision 

according to the amount of information used - 1 year versus 3 years, 

and the sampling plan for the monthly averaging time. The third column 

labeled, FPC PERCENT OF MEAN, list s the precision calculations using 

formula 2. The fourth column labelled, WO - FPC PERCENT OF MEAN, 

recomputes the precision calculation in formula 2 without the FPC, 

'V i - ^/^ . The calculation using the FPC (columns 3 and 5) assumes 

that sampling is taking place from a finite population, while the precision 

estimates in columns 4 and 6 assume that sampling is taking place from an 

infinite population. The fift h and sixth columns summarize precision 

according to the amount of information used - 1 year versus 3 years, and 

the sampling plan for the quarterly averaging time. The fift h column 

computes the precision estimates with the FPC, while the sixth column 

computes the precision estimates without the FPC. 

As one would expect, Tables 4, 5, 6, and 7 all suggest that the precision 

associated with the quarterly mean is tighter than the precision associated 

with the monthly mean. Further, the precision associated with 3 years of 

information is tighter than the precision associated with 1 year of information. 

Finally, the precision calculated using the FPC is tighter than the precision 

calculated without the FPC, as would be expected. 

CONCLUSIONS 

In selecting criteria for an air quality standard, 3 years of 

information wil l provide tighter estimates of precision than 1 year of 

information, as would be expected. Further, the quarterly averaging time 

is preferable to the monthly averaging time in order to achieve tighter 

precision. An examination of Tables 4 through 7 indicates that the required 

sampling could vary according to the site type. Using the general rule, 
P.5 

that 1t would be desirable to have the true arithmetic quarterly mean be 

within _+ 10 percent of the sample arithmetic quarterly mean, 95 percent of 

the time, the following observations can be made: 

With the FPC: 

° For source oriented sites with maximum quarterly averages less than the 

current standard, the sampling frequency would have to be increased to 

every other day sampling (Table 5). 

0 For source-oriented with maximum quarterly averages greater than the 

current standard, the sampling frequency would have to be Increased to 

every day sampling (Table 6) . 

° For the NAMS maximum urban concentration sites, the sampling frequency 

would have to be increased to every third day sampling (Table 7) . 

Without the FPC: 

° For source-oriented sites with maximum quarterly averages less than the 

current standard, the sampling frequency would have to be Increased to 

every day sampling (Table 5). 

o For source-oriented sites with maximum quarterly averages greater than 

the current standard, the "sampling frequency would have to be increased 

to every day sampling (Table 6). Even with every day sampling for the 3 

quarters in the specific 3 year time period, precision of + 10 percent 

could not be achieved. Keep in mind that the 3 quarters o7 data over the 

specific 3 year period is treated as a complete sample of a "typical" 

3-year period, drawn from the population of al l possible 3-year periods, 

and, as such, the FPC is not used. 

0

 For the NAMS maximum urban concentration sites, the sampling frequency 

would have to be increased to every other day (Table 7). 

In conclusion, the use of the FPC is appropriate in making inferences 

about past events, but i t is probably not appropriate for expected events. 

The sampling frequency, discussed above, is intended to apply to monitoring 

sites near the standard level. As ambient lead levels approach the standard, 

the variability in the ambient lead data may decrease. Therefore, the 

sampling requirements could probably be relaxed. 

REFERENCES 

1. A. D. Thrall, J . L. Baptlsta and C. S. Burton, Final Report, An 

Examination of Air Quality Data Completeness Requirements, Systems 

Applications, Inc. Prepared for Monitoring and Data Analysis Division, 

Office of Air Quality Planning and Standards, U. S. Environmental 

Protection Agency, Research Triangle Park, N.C, May 1984. 
P.6 

2. G. Yarn, U. S. Environmental Protection Agency, "Source Oriented 

A1r Quality Monitors for Lead," letter to Eric Goldstein, Natural 

Resources Defense Council, March 5, 1985. 

Attachments 

cc: R. Rhoads (MD-14) 

J. O'Connor (MD-12) 

B. Jordan (MD-12) 

J. Cohen (MD-12) 

D. Sledge (MD-12) 
Table 1. List of Source-Oriented Sites with Maximum Quarterly Lead 

Averages Less than the Standard 

Site Location 

030240004F01 Douglas, AZ 

101800082G02 Hillsboro Co., FL 

130840006F02 Kellog, ID 

151180001H02 E. Chicago, IN 

151180004H02 E. Chicago, IN 

243300046F02 St. Paul, MN 

390720717F02 Berks Co., PA 

491840057A02 Seattle, WA 

Composite Avg. C.V.'s 

Source 

Type* 

PCS 

SLS 

PLS 

IS 

IS 

LOM 

SLS 

SLS 

n 

8 

15 

15 

9 

10 

14 

12 

8 

QUARTERLY 

Avg S.D. 

0.31 

0.23 

0.39 

0.71 

0.79 

0.25 

0.40 

0.73 

0.19 

0.15 

0.57 

0.74 

0.80 

0.18 

0.33 

0.38 

CV. 

0.61 

0.65 

1.46 

1.04 

1.01 

0.72 

0.83 

0.52 

n 

3 

4 

5 

3 

5 

5 

5 

3 

MONTHLY 

Avg. S.D. 

0.44 

0.33 

0.85 

1.58 

1.06 

0.42 

0.60 

0.72 

0.26 

0.26 

0.79 

0.69 

1.06 

0.17 

0.38 

0.54 

C V. 

0.59 

0.79 

0.93 

0.44 

1.00 

0.40 

0.63 

0.75 

0.86 0.69 

* LOM = Lead Oxide Manufacturer 

PCS = Primary Copper Smelter 

PLS = Primary Lead Smelter 

IS = Iron and Steel Plant 

SLS = Secondary Lead Smelter 
Table 2. List of Source-Oriented Sites with Maximum Quarterly Lead 

Averages Greater than the Standard (1.5 ug/m3) 

Site Location 

012140004G02 Leeds, AL 

013240003F02 Troy, AL 

131420021F02 Shoshone Co., ID 

142960010F01 Granite City, IL 

151780011F05 Hammond, IN 

243300038F01 St. Paul, MN 

262280011J05 Jefferson Co., MO 

270860002F02 Lewis & Clark Co.; 

Composite Avg. C.V.'s 

MT 

Source 

Type* 

SLS 

SLS 

PLS 

SLS 

LOM 

LOM 

PLS 

PLS 

n 

15 

15 

14 

13 

15 

10 

15 

26 

QUARTERLY 

Avg S.D. 

3.04 

7.08 

1.75 

1.54 

2.73 

7.97 

6.70 

3.86 

2.64 

9.49 

2.05 

1.83 

5.16 

16.81 

12.45 

2.89 

CV. 

0.87 

1.34 

1.17 

1.19 

1.89 

2.11 

1.86 

0.75 

n 

5 

5 

4 

6 

5 

3 

5 

9 

MONTHLY 

Avg. S.D. 

4.66 

13.07 

2.75 

2.02 

6.15 

6.13 

11.85 

4.39 

3.22 

14.61 

3.42 

2.47 

8.15 

5.35 

19.17 

2.85 

CV. 

0.69 

1.12 

1.24 

1.22 

1.33 

0.87 

1.62 

0.65 

1.40 1.09 

* LOM = Lead Oxide Manufacturer 

PCS = Primary Copper Smelter 

PLS = Primary Lead Smelter 

IS = Iron and Steel Plant 

SLS = Secondary Lead Smelter 
P.9 

Table 3. List of National Air Monitoring Stations (NAMS) Used to Determine 

the Monthly and Quarterly Coefficients of Variation for the Maximum 

Urban Concentration Sites. 

NAMS SITE QUARTERLY MONTHLY 

No. of Std. 

Location Obsn(s) Avg. Dev. CV . 

056400003F01 Riverside, CA 11 0.69 0.32 0.46 

057160001101 San Rafael, CA 16 0.55 0.23 0.42 

191870003F01 Jefferson Parrish, 11 0.42 0.26 0.62 

LA 

261040002G01 St. Louis Co., MO 16 0.51 0.40 0.78 

330660017F01 Buffalo, NY 10 0.85 0.60 0.71 

334680052F01 New York City, NY 12 0.92 0.29 0.32 

442340035G01 Memphis, TN 14 1.41 0.75 0.53 

451880023F01 Fort Worth, TX 15 0.90 0.62 0.69 

COMPOSITE AVERAGE C.V.'s 0.57 0.50 

No. of 

Obsn(s) 

5 

6 

4 

5 

6 

4 

4 

5 

Avg. 

0.76 

0.57 

0.44 

0.79 

0.69 

0.93 

2.10 

1.05 

Std. 

Dev. 

0.37 

0.26 

0.26 

0.58 

0.43 

0.41 

1.03 

0.19 

C V. 

0.49 

0.46 

0.59 

0.73 

0.62 

0.44 

0.49 

0.18 
P.10 

Table 4. Comparison of Confidence Intervals for Alternative Sampling 

Plans for Monthly and Quarterly Arithmetic Means for All Sites. 

1 1 PtKIUlJ 1 

» 1 MTH 1 UTH | 

1 IFPC -X UF IrtU.FPC -X IFPC -X OF l.tll_FPC -X 1 

1 1 MEAN 1 OF MEAM 1 MEAN 1 OF MEAN 1 

ITIM& PERIOD (PLAN 1 1 1 1 I 

I t YEA« lEb (7SX) *| 87.U l 93.9 i 3«,9 I 37.5 1 

1 IE6 1 66.V I 73.2 1 29.8 1 32.7 1 

1 IE3 175* j 1 12.2 1 *»9,3I 21.3 1 2«.<*l 

1 IE3 1 3<4.5I "2.2 1 1 to. 01 22.o | 

1 l£ 2 (75X) 1 2V.0 I 37.5 1 15.51 20.0 1 

1 IE2 1 23.i l 32.7 1 12.51 17.7 1 

1 IEU (75X) 1 11.11 2«.'VI 6.2 1 13." 1 

1 "*' IEU 1 0.0 1 22.I' I O.Oi 12.« l 

l i rtA* 3 IPLAN 1 1 1 1 1 

1 IE6 (75X) 1 3«.9 I 37.5 1 18.61 20.0 1 

1 lEb 1 29.8 1 32.7 1 16.11 17.7 1 

1 IE3 C7SX) 1 21.3 1 2«.V I 11.91 13.9 1 

1 IE3 1 Id.0 1 . 22.0 1 lu.l l 12.0 1 

1 IE2 (7SX) 1 15.5 1 20.0 1 8.7 1 11.3 1 

1 IE2 1 12.51 17.71 7.11 10.l l 

1 IEU 175X) 1 6.2 1 13.91 3.6 1 8.0 | 

1 IEU 1 0.0 1 12.11 O.OI 7.1 1 
P.11 

Table 5. Comparison of Confidence Intervals for Alternative Sampling 

Plans for Monthly and Quarterly Arithmetic Means for SourceOriented Sites with Maximum Quarterly Averages Less than the 

Standard. 

PfcWIUU 

MTM WlK 

IFPC -X UF l*0_FPC -X IFPC »X UF IwU.FPC -X 

I MEAN I OF MEAN | MEAN I (IF MEAN 

TIMfc PEK1UQ IPIAN 

» Vfc*K IE6 (7SX) 

I 

102.21 

I 

1U9.8I 

I 

50.91 So.oi 

I — 

_L£6_ 

I........ . 

IE3 t75X ) 

J-lmAmiL 

49.41 57.71 3 1 . H 

47.61 

36.31 

I — 

IE3 

T - i —— -

I E 2 175X ) 

40.31 19,41 26.21 

IE2 

3i.u l 

I -

IEU (75XJ 

27.01 

43.Bl 

36.21 

22.51 

lb.<j l 

I-- . 

IEU 

\1A9.±. 29,1L„ 

>t.... « 

I PLAN 

O.OI .2S.«I. . 

.....f . 

I 

o.oi 

32.1 1 

29,1 

25.8 1 

20.21 

18.0 1 

i rtAKS 

l ' 

IE6 (75X) 

I — 

l E b 

40.8 1 

I........ . 

I E 3 175X ) 

mh.m.V-m 

25.01 

43.*l 

36.21 

27.1 1 

23.5 1 

I — 

IE3 

I E 2 175X ) 

_21**LL 

miHimllL. 

2 b , 8J _ 

17.3 1 

I—• 

IE2 

16.1 1 

I' 

14.6 1 

2it*! . 12.7 1 

IEU (75X ) 

IEU 

10.41 

0.0* 

A5**L 

14.5 1 

b.2 l 

(I . 0 I 

29.11 

25.HI 

20.21 

>»•»' 

16.41 

l»l " 

11.6 1 

10.41 
P.12 

1 

t 

V 

Table 6. Comparison of Confidence Intervals for Alternative Sampling 

Plans for Monthly and Quarterly Arithmetic Means for SourceOriented Sites with Maximum Quarterly Averages Greater than 

the Standard. 

1 1 PEHiOU 1 

1 1 MTH 1 UTK 1 

• IFPC -X UF IhO.FPC -X IFPC -X UF MO_FPC -X 1 

1 1 MEAN 1 UF MEAN | MEAN 1 OF MEAN | 

IT1ME PfcKIUO IPLAN 1 1 1 1 I 

11 YEAH lEb (7SX) 1 161.4 1 173.41 82.0 1 89.0 1 

1 It o 1 123.5 1 135.3 1 7o.b l 77.5 1 

1 IE3 175X> 1 76.0 1 91.1 1 Su.o i S9 f 1 | 

1 IE3 1 63,7 1 70.0 1 42.7 1 S2.3 I 

1 IE2 (75X) 1 53.6 1 e9.3 l 36.7 1 47.4 1 

1 IE2 1 4*.7 1 60.4 1 29.7 1 41.9 1 

1 IEU 175XJ 1 20.6 1 46.0 1 14./ | 32.9 1 

1 IEU 1 O.oi 40.7 1 O.OI 29.3 1 

13 YfcAHS IPLAN 1 1 1 1 1 

1 lEb l75X) 1 64.5 1 oV.J I 44.1 1 47.4 | 

1 lEo 1 55.1 1 60.4 1 36.3 * 41.9 | 

1 IE3 175 U 1 39,4 1 46,0 1 28.2 1 32.9 | 

1 IE3 1 33.2 1 40.7 1 24.u l 29.3 1 

1 IE2 175X) 1 20.6 1 3o,V| 20.7 1 26.7 1 

1 IE2 1 23.1 1 32.7 1 16.9 1 23.9 1 

1 Ifcu (75XJ 1 11,51 25.o | 8.5 1 18.9 | 

1 IEU 1 O.oi 22.1 1 o.\)l lo.9 l 
P.13 

Table 7. Comparison of Confidence Intervals for Alternative Sampling Plans 

for Monthly and Quarterly Arithmetic Means for NAMS Maximum Urban 

Concentration Sites. 

PtHiUU 

MTM UTH 

IFPC -X UF IrtO.FPC -X IFPC -X UF IwU.FPC -X 

J ME>N I (IP MEAN | MEAN I OF MgAN 

TJMt, PfcHlUQ "»LAN J_ _L 

1 YEAN tEO C7SXJ 

I 

.UUl-L 

I 

79.5' 

I 

33.7 1 _ib^£ 

I — 

lEo 

.....t . 

S b t 7 l o2. 1 I ?«.« ' at.fc 

IE3 t7SXJ 35.8 1 41.6 1 20.6 1 24- 1 

I — 

g9.g | iS.H I 

.«•• • + . 

17.4 ' 21.3 

iI E 2 175* ) 2^.6 1 ..-.1-JLL- JJL.9J_ 19. 3 

I.. . 

ALi. 19.b l ^7. 7 I 12.1 ' 17.1 

iIEU C75X) 

I — . 

IEU 

mlmdmlm . 

o.o i 

21.1 1 6 . 0 1 

•+.... . 

I PL an 

10.71 

p....f . 

o.o i 

M." 

11. 9 

3 YEAK S I 

lEo C7SX) 

I-- . 

lEo 

I 

g*<.Ol 

I 

-UL**L . 

I 

lo.o ' 

.Hill ^l*T± ji.^ J 

| . ....... . 

' £ i C75XJ lb.l l til.I I U.*l 

19. 3 

17. 1 

13. 4 

IIE3 

I........ . 

I E 2 175X ) 

15.2 1 

..... t. 

A«*LL 9.8 1 

I — 

I t 2 

13.1 1 

lo.b l 

16.91 

15.0 1 

6.4 1 

b.9 l 

_Ll ^ 

10. 9 

9 . 7 

_Lku. LZ*?A__ 

lEu 

J>.AL 

»».• SLL .10.5 1 

3.4| . 

<>.l " 

_L*1 

_JU±