August 22, 1966 M20-2

CHAPTER 4. TABLE OF THREE STANDARD DEVIATION CONTROL LIMITS

FOR PERCENT IN ERROR

4.01 PURPOSE OF THE TABLE

Appendix E contains a table of \"Three Standard Deviation Control
Limits\" com­puted for various sample sizes and percentages in error.
This table was designed to aid in determining whether a process is or is
not in a state of statistical control and, therefore, when to look for
assignable causes of variation. These control limits may also be used in
the manner of \"confidence limits\" to measure the precision of sample
estimates of any type of population percentage, and to decide whether a
sample statistic differs \"significantly\" from another sample
statistic.

4.02 CONTROL LIMITS USED IN THE TABLE

\"Standard deviation\" is the name given to the single most universally
used measure of statistical variation. In a \"normal\" distribution of
sample statistics, a range of three standard deviations above and below
the distribution average will include about [99.7%]{.underline} of the
values which can occur because of chance sampling variation. The control
limits [shown in]{.underline} [the table are exactly]{.underline} three
standard deviations (rounded to two [decimal place]{.underline}
[midpoint]{.underline} of [any given samp]{.underline}le-size interval.
There ore, these control limits represent a little less than three
standard deviations for samples in the lower part of each sample-size
interval, and a little more than three standard deviations for samples
in the upper part of each sample-size interval.

4.03 WHAT THE TABLE SHOWS

a\. The table shows for an assumed true population percentage,
representing the underlying average error rate of the process being
reviewed, and for a certain sample size, the upper and lower \"limits\"
which bound an interval or band of percentage values within which sample
percentages may fall simply due to chance causes or random variation.
The fact that a given sample percentage falls within this band of
percentage values is not in itself evidence that the process average
actually is or remains the value it was assumed to be. Any given sample
percentage, especially where samples are small, conceivably could have
come from any of a large number of different populations, each with a
different percent in error. The presence of a single sample point within
the control limits appropriate for a particular population percentage,
and a particular sample size, means only that the sample
[could]{.underline} - have come from such a population. On the basis of
this sample percentage alone, there is no reason to doubt that the true
population-percentage (true average percent in error for the process) is
not the value that it was assumed to be. A sample percentage falling
outside either limit, however, is good evidence that the process average
has changed from or is different from the value originally assumed to
represent the process average.

b\. In ordinary control chart usage it is possible after a period of
experience to estimate closely the true process average or underlying
error rate. Once the pattern of consecutive sample points on the control
chart indicates that the process has achieved stability, small
independent samples can be accumulated to form larger samples, and thus
provide estimates of increasingly greater precision. On reaching this
stage it then becomes possible to interpret a point falling outside the
control limits as evidence of a real change in the process average. In
the early stages of control chart application, a point outside the
control limits might not necessarily indicate a true \"out of control\"
situation. It might indicate nothing more than that an incorrect
assumption had been made as to the magnitude of the true process error
rate. It would not necessarily indicate that an assignable cause-­a cause
other than chance\--was operating. Once the true process average has
been estab­lished with some precision, a point outside the control
limits, or particular patterns or sequences of points within the control
limits which could occur by chance alone about as rarely as a point
outside the control limits, are indications of a shift in the process
average and consequently of the presence of an assignable cause or
causes.

c\. The table also shows for each percent in error covered that
increasing sample size is accompanied by decreasing amounts of sampling
error. While the decreases in error that take place between any sample
of a given size and a larger sample of specified size vary in absolute
amount from percentage to percentage, their relative change is the

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same for all percentages. Sampling error varies inversely with the
square root of the sample size. If sampling error is to be reduced to
one-half its initial amount, the sample must be four times its original
size; to reduce sampling error to one-third its initial amount, the
sample must be nine times as large; to reduce sampling error to
one-fourth its initial amount, the sample must be 16 times as large; and
so on.

d\. The reduction in error as sample size becomes larger is reflected by
the table in the narrowing of the interval between the control limits.
Thus, as sample size increases, more precise estimates of population
values can be made. [Note, however, that sample sizes]{.underline} often
must be increased substantially to get even small [reductions in
sampling error. This]{.underline} [frequently is an important
con]{.underline}sideration [in determining the sample size to be
used]{.underline}. For example, a sample size of 750 for 2.00% \"in
error\" gives the interval between 3.53% (the UCL) and 0.47% (the LCL)
as the range within which almost all sampling variation may be expected
to take place. If the sample size is doubled to 1,500, the range of
sampling variation is reduced only slightly, to the interval between
3.08% and 0.92%. In many situa[tions, it may be neither economical nor
necessary to increase the sample size, and with it]{.underline} the
[related costs of sampling]{.underline}, only to achieve a slight
increase in [precision]{.underline}.

4.04 ASSUMPTIONS IN USING THE TABLE TO INTERPRET SAMPLE RESULTS

Use of the table in interpreting results obtained through sampling
assumes that certain conditions exist, as follows:

a\. The population or process to be sampled is d e f i n a b 1 e, errors
are clearly identified, and there is a feasible means of accurately
determining the percentage in error in any sample. In estimating from a
sample, the average quality level of the process, it is also assumed
that the process sampled has an underlying \"true\" though unknown error
rate.

b\. Samples drawn from the population or process are assumed to be
simple random samples, as described in paragraph 1.10b(l). The table is
not applicable to judgment or other nonrandom samples.

c It is assumed that the sample has been drawn from the actual
population it is intended to represent. A random sample from the wrong
population may be more misleading than a judgment sample from the real
population of interest. Samples taken under circumstances which reduce,
increase, or otherwise alter the probability of selection of certain
items or groups of items in the population could involve such bias that
in effect, the samples would represent a population other than the one
intended. This may occur, for example, when:

\(1\) Samples are drawn at \"convenient\" times or places and a
collection pattern develops which introduces consistent bias by denying
to some items any possibility of being selected into the sample.

\(2\) Samples are drawn in a manner which permits clusters of like items
improperly to get into the sample.

4.05 DETERMINING WHETHER A PROCESS IS IN STATISTICAL CONTROL

a\. One common use of the table is to test whether a sample percentage,
obtained through a random sampling procedure, is compatible with some
specific assumption or hypothesis as to the magnitude of the true
population percentage. This is, in effect, what is done when over a
period of time successive random samples are drawn from the output of a
work process to see by observing the succession of sample error rates
whether the process is or remains in statistical control. This continual
testing of the process is done by means of a control chart. The method
used is outlined below.

b\. A long-run process average (underlying error rate) is estimated as
closely as possible from sampling or from experience, or a standard for
the operation is established. A horizontal central line representing
this process average or the standard is drawn across the control chart.
The central line also represents the assumption or hypothesis which will
be tested through successive sampling.

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August 22, 1966 M20-2

c\. Based on the sample size established and used in periodic
independent sampling from the process, lines representing the upper and
lower control limits are drawn on the chart at predetermined distances
above and below the central line. These control limits may be obtained
directly from the table by referring to the limits shown for the sample
size and the particular percent in error assumed to represent the
process average or taken as the established standard.

d\. -As samples are taken, a series of individual sample percentages are
related on the chart to the control limits and to the central line to
test whether or not there may have been a change in the process average
or a departure from the established standard. If there is no indication
from the sample percentages of a change in the process average, the
process is regarded as \"in statistical control.\" if a change is
indicated to have taken place, the process is regarded as no longer \"in
control.\" -Action is then necessary to find and eliminate the factor(s)
that caused the \"out of control\" situation to arise. Note that in this
type of application, the control limits are based on the assumed process
average, or the established standard, and not upon the fluctuating
values of the independent sample estimates.

4.06 DETERMINING SAMPLING ERROR

a\. Another but a somewhat different use of the table is to approximate
the sampling error that accompanies a given sample estimate of a
particular population percentage. This percentage may be the percentage
occurrence of any attribute or characteristic of the population, whether
or not it is an error rate or other quality characteristic. Examples
are: percentage of work processed, or not processed, within a specified
period of time; percent­age of payments exceeding, or not exceeding, a
stated amount; or percentage of veterans or employees falling within a
specified age grouping.

b\. Sampling error may be interpreted as a range or interval of
percentage values around the sample estimate within which the true
population percentage may with confidence be expected to lie. For this
purpose, the control limits of the table are used not for I I control\"
purposes but as a measure of the sampling variability or error
associated with a particular sample estimate. The interval between the
control limits thereby serves to indicate the precision of the sample
estimate.

c\. As an example of this use of the table, suppose that a work sampling
study is made at a field station to estimate the average daily use of
electric typewriters. Assume that in this study a random time sample of
400 observations taken over a period of a month shows that the electric
typewriters were in use 2O% of the time. Since this percentage is a
sample estimate, and since sample estimates may be expected to vary from
sample to sample, it cannot be concluded that the true population
percentage is precisely 20%. By reference to the table it can be
determined that:

\(1\) A large range of percentages can be virtually eliminated as the
possible true population percentage; i.e., those percentages above the
UCL and those below the LCL on either side of the 20% sample estimate.

\(2\) The true percentage of the population almost surely lies within
the interval between the control limits, a relatively narrow range of
percentages.

d\. In this instance the table shows control limits of 14% and 26% for a
percent in error of 20% and a sample size of 400 observations.
Therefore, it can be determined almost with certainty that the true
percentage of time the typewriters were in use during the month was at
least 14% and not greater than 26%. In terms of average hours of use in
an 8-hour workday, it may be determined from the sample data that the
typewriters were in use somewhere between 14% and 26% of 8 hours, or
between 1 and 2 hours per day, on the average.

e\. The table may also be used in the same manner to [estimate the error
rate of]{.underline} [a population of work]{.underline} units
[not]{.underline} covered [by a formal SQC plan]{.underline}. To
illustrate, assume that 100 typed letters are randomly selected for
quality review and seven of these letters (7.0%) are \"in error.\" By
referring to the table for a sample size of 100 and 7.0% \"in error, \"
it is seen that the UCL is 14.65% and the LCL is 0.00%. This indicates
that a random sample

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M20-2 August 22, 1966

of 100 letters yielding an error rate of 7.0% [could]{.underline} come
from a population having a true error rate that might lie anywhere from
slightly above zero (you know from the sample that some letters are \"in
error\") to 14.65%. This \"interval estimate\" is the best estimate that
can be made based on this single sample of 100 letters.

f\. However, if a larger sample should be taken, or if a series of
samples of 100 letters are selected monthly and cumulated, a more
precise estimate of the quality level may be obtained. This is done by
referring to the table for the three standard deviation control limits
applicable to the larger sample size and the sample percentage in error.
For example, if a cumulative 12-month sample of 1,200 letters yields an
error rate of 4.0%, the table indicates that this sample in all
likelihood came from a population having a true error rate that lies
somewhere between 2.30% and 5.70%.

g\. Note that in these applications the control limits taken from the
table are based on the percentage estimate derived from an individual
sample, and not upon a known or assumed population percentage or \"error
rate,\" as in the case of an SQC chart. Different samples would result
in a number of different sample percentages, each with its own unique
pair of sampling \"limits.\" From statistical sampling theory it is
known that where a normal distribution of sample percentages is
appropriate, about 99.7% of all possible sample per­centages derived from
samples of a given size would be within an interval of three standard
deviations of the sampling distribution marked off on either side of the
true population percentage. Therefore, when a percentage is obtained
from a properly drawn random sample, the true population percentage will
in all likelihood be within three standard deviations of the sample
percentage; i.e., within the percentage \"limits\" appropriate for the
sample percentage obtained and the sample size used.

4.07 DETERMINING SAMPLE SIZE

a\. Another use of the table is as an aid in determining the sample size
needed to achieve a specified degree of precision in the percentage
estimate expected to be derived from a proposed sampling study. To
estimate the size of sample needed, it is first necessary that at least
a rough approximation be made as to the magnitude of the percentage
expected. The reason for this is that the sample size required is a
function of this magnitude. Two percentages that differ widely, as for
example 10% and 50%, will require widely different sample sizes for the
same degree of precision.

b\. When management or the intended user of the data specifies the
degree of pre­cision desired, there should be a clear indication of
whether a relative or an absolute measure of precision is wanted. For
example, an allowable relative error of 5%, when the true population
percentage is 10%, means that the sample percentage obtained must fall
between 9.5% and 10.5% (10% ± . 05 x 10%). Where the true population
percentage is 50%, the ob­tained sample percentage would have to fall
between 47.5% and 52. 5% (50% ± .05 x 50%).

c\. Conversely, if absolute rather than relative error is desired, all
that is required to meet this error limit where the population
percentage is 10% is to insure by the size of the sample that the sample
percentage will fall between 5% and 15%. Similarly, where the population
percentage is 50%, the sample percentage would have to fall between 45%
and 55%. It should be noted that an absolute error limit of 5% can be
obtained with much smaller samples than are required to obtain a
relative error limit of 5%. For example, a sample of 320 items would be
enough form [absolute]{.underline} error limit of 5% where the
population percentage is 10%, but a sample of over 32,000 items would be
needed to meet the [-relative]{.underline} error limit of 5%.

d\. Therefore, a question which management must resolve before proper
sample sizes can be determined is which particular estimates are most
critical and what are the maximum limits of sampling error (minimum
precision), expressed in absolute terms, that can be tolerated with
respect to the various estimates to be obtained from the proposed
sampling study. The most critical estimates may be defined as those on
which the most important decisions in terms of either policy or cost are
to be based. As a rule, it will be necessary to provide a sample size
large enough to achieve whatever precision is needed for each of the
most critical estimates, and then accept whatever sampling error results
from that sample size where less critical sample estimates are
concerned.

e\. After these matters are decided, refer to the table of \"Three
Standard Deviation Control Limits\" for guidance in determining the
sample size required. In this type of application of the table, the
\"percent in error\" indicators of quality level represent the

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August 22, 1966 M20-2

population percentage to be estimated, and the control limits represent
limits of sampling error to be associated with the expected sample
estimate. For a simple illustration, suppose there is only one critical
percentage to be estimated in a proposed sampling study, and that the
population percentage is determined to be about 15%. Assume further that
management decides that a sampling error limit of about 2 percentage
points above and below the sample percentage will give acceptable
precision (in absolute terms).

f\. Reference is then made to the columns of control limits (UCL and
LCL) in the table under the \"percent in error\" of 15.0%. Look down the
two columns of control limits until limits that deviate about 2
percentage points above and below 15% are identified. In this instance,
the limits closest to a 2-point deviation are the UCL of 17.08 and LCL
of 12.92. Therefore, it may be concluded that a sample size of 2,650 or
more will be needed (2,650 is the middle number in the sample size
interval of 2,300-2,999 upon which the control limits of 17.08 and 12.92
are based).

g\. It is important to recognize that a [truly precise determination of
sampling]{.underline} error [can be made in advance only when
it]{.underline} is possible to determine before the sample is [drawn
exactly what]{.underline} the true population percentage is. If this
determination could be made, then of course there would e no need for
sampling, since the information expected from a sample would already be
known. In real situations, sampling is used to obtain information not
already available. It is advisable to make a preliminary estimate of the
true population percentage based on experience or judgment, as discussed
above, since both this population percentage magnitude and the magnitude
of the allowable error influence the sample size required for a given
degree of precision. The true population percentage is most likely to be
something smaller or larger than the estimated percentage. Therefore,
the sampling error desired can only be approximated in advance. A more
precise estimate of sampling error can be made only after the sample has
been drawn and the sample results are obtained.

4.08 HOW TO USE THE TABLE WHEN QUALITY LEVEL (PERCENT IN ERROR) EXCEEDS
50%

a\. It is possible to use the table limits directly in computing control
(sampling) limits for certain percentages in excess of 50%. This may be
done whenever a \"percent in error\" specifically covered in the table
is the \"complement\" of a percentage for which control (sampling)
limits are desired. For this purpose, a \" complement\" of a percentage
is defined as the difference between that percentage and 100%; i.e., the
proportion needed to complete the \"whole.\" Each percentage of a pair
whose sum is 100% is a \"complement\" of the other.

b\. To illustrate this use of the table, suppose the three standard
deviation upper and lower sampling error limits are desired for \"80%\"
when samples of 400 units are taken. Although the table does not contain
\"percent in error\" columns for percentages over 50%, the
\"complement\" of 80%, which is 100% minus 80% or 20%, is covered by the
table. When the sample size is 400, and the \"percent in error\" is 20%,
the UCL is 26% and the LCL is 14%.

c\. To obtain comparable three standard deviation control limits for
80%, the complement\" of 20%, subtract the UCL and LCL for 20% from
100%. When 26%, the UCL for 20%, is subtracted from 100%, this gives 74%
as the LCL for 80%. When 14%, the LCL for 20%, is subtracted from 100%,
this gives 86% as the UCL for 80%. Limits for other percentages for
which their \"complements\" appear in the table may be determined in
similar manner.

4.09 USE OF INDEPENDENT SAMPLES AND TABLE TO VERIFY REPORTED QUALITY
LEVELS

a\. General. Independent samples from the same source as random samples
drawn by the supervisor can be used to help verify quality levels
reported by supervisors. The table may be used to determine the
applicable sampling limits for checks as to whether the results of an
independent sample show:

\(1\) That the standard is being met.

\(2\) That the quality level reported is not significantly different
from the independent

sample result.

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b\. [Check to Determine if Standard is Met:]{.underline}

**[EXAMPLE]{.underline}**

Supervisor\'s

12-Month Cum- Independent

ulative Report Sample Data

Sample Size 1.200 60

Standard 3.00% 3.00%

UCL 4.48% 9.61%

LCL 1.52% 0.00%

Percent in Error

Yielded by Sample 3. 50% 8.33%

[INTERPRETATION]{.underline}

The \"percent in error\" yielded by the supervisor\'s 12-month
cumulative sample and the independent sample result fall within expected
ranges of sampling variation, as indicated by the applicable \"control\"
limits. Therefore, the differences between the sample results and the
standard percent in error could be due solely to chance causes. It is
con­cluded that the evidence is not inconsistent with the assumption that
the reported quality level meets the standard.

c\. [Check to Verify the Quality Level Reported by the
Supervisor:]{.underline}

**[EXAMPLE]{.underline}**

Supervisor\'s

12-Month Cum- Independent

ulative Report Sample Data

Sample Size 1,200 60

Standard 3.00% 3.00%

UCL 4.48%\* 4.85%\*\*

LCL 1.52%\* 0.00%\*\*

Percent in Error

Yielded by Sample 1.00% 8.33%

\*Control limits based on standard of 3.00%.

\*\*Control limits based on 1.00% \"in error\" yielded by supervisor\'s
12-month cumulative sample size.

[INTERPRETATION]{.underline}

The supervisor\'s 12-month sample result (1.00%) falls below the LCL
(1.52%) for the standard of 3.00%, indicating an apparent quality level
significantly better than the standard. The independent sample result
(8.33%) falls above the UCL (4.85%) for the 1.00% \"in error\" reported
by the supervisor. This indicates that the actual quality level is not
as good as that reported by the supervisor. Note that in the previous
example the independent sample result was not significantly different
from the standard. Therefore, the excep­tional condition reflected by
checking the independent sample result against the quality level
reported is that it questions the reported quality level as being
significantly better than the standard. This is serious and warrants
further investigation because it casts doubt on the reporting accuracy.

The supervisor\'s 12-month cumulative sample\--the process as it
operated during the 12-month period in which the process average was
reported as 1.00% \"in error.\" If the independent sample was drawn only
from recent work, which conceivably may have differed significantly from
the process as it operated during the complete 12-month period, this
contention would not necessarily be justified.

**4-6**

August 22, 1966 M20-2

NOTE: In using the table to determine control limits for this purpose,
there is a slight loss of precision but not enough to be concerned about
if the independent sample is small relative to the supervisor\'s
cumulative sample size. This slight loss in precision is caused by the
fact that the supervisor\'s findings represent an approximation to the
true proportion \"in error.

4.10 REVIEW OF CASES OR WORK UNITS PREVIOUSLY QUALITY CHECKED BY THE
SUPERVISOR

When identical cases or work units previously quality checked by the
supervisor are reviewed, the purpose of the review is to judge the
accuracy and reliability of the supervisor\'s quality checks and
reports. The primary objective is to determine whether or not the
supervisor identified and recorded the substantive errors and the bulk
of the total defects found as a result of both reviews. The
interpretation of findings and their significance is a judgment matter.
In this kind of a situation the purpose of the review is not to estimate
the percent \"in error.\" Therefore, use of the table is not necessary.

4.11 DETERMINING COMPLIANCE WITH STANDARDS OF TIMELINESS

a\. General. Timeliness standards may be expressed in terms of either
\"the minimum percentage of work units to be processed within a
specified time limit (hours or days),\" or \"the maximum percentage of
work units allowed a processing time greater than a specified time
period (hours or days).\" In effect, these two ways are \"two sides of
the same coin.\" Whichever way the standard is expressed, compliance
with the standard may be tested by sample observations of either the
indicated standard percent or its complement (the indicated percent
subtracted from 100).

b\. Standard Expressed as \"Percent of Work Units not Expected To Be
Processed [Within a Specified Time Limit.\"]{.underline} The table of
\"Three Standard Deviation Control Limits\" in appendix E may often be
used directly when a given standard is expressed as the \"per­centage of
work units not expected to be processed within a specified time limit.\"
Ordi­narily, such a percentage value will be small enough to be contained
within those covered by the table, which range from 0.1% to 50.0%. The
allowable percentage of work units [which]{.underline} [may]{.underline}
be expected to require more than some [\"standard\"]{.underline} time
established for the [other units]{.underline} is similar to an
acceptab[le \"error rate\" or \"percent in]{.underline} error.\"

\(1\) The test to be applied here is to determine whether an observed
sample percentage exceeds the [upper control]{.underline} limit for the
relevant sample size of a \"percent in error\" equal to the allowable
percentage of work units requiring more than the specified elapsed time
to process. If a sample percentage is greater than the indicated upper
limit, it may be concluded with confidence that more than the allowable
proportion of work units are requiring excessive time for processing.
Hence, the timeliness standard is not being met.

\(2\) Sample. Assume that in processing a given type of action, it is
pre-scribed that no more than 15% of the work units processed shall
require in excess of 4 workdays\' processing time. Suppose a random
sample of 200 work units is drawn, of which 40 units, or 2O%, required
more than 4 workdays to process. Although this sample percent is in
excess of 15%, the \"standard,\" it is not in excess of 22.58%, the
upper limit of the range within which sample percents may be expected to
fall when samples of 200 units are drawn from a process in which the
true underlying propor­tion is 15%. Since a sample result of 20% could
have occurred by chance even when the true proportion of work units
requiring excessive processing time is 15% (or perhaps even less), the
\"standard\" is considered as being met.

c\. Standard Expressed as \"Percentage of Work Units To Be [Processed
Within]{.underline} a [Specified Time.\"]{.underline} Where the
timeliness standard is expressed as the \"percentage of work units to be
processed within a specified time,\" the indicated percentages
ordinarily will be greater than 50%. Since the table of \"Three Standard
Deviation Control Limits\" in appendix E does not cover percentages
larger than 50%, the limits shown cannot be used directly for

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M20-2 August 22, 1966

testing compliance with the \"standard\" percentage. The table limits
can be used indirectly, however, by obtaining the limits to
[complements]{.underline} of standard percentages which exceed 50%,
whenever such complements are to be found in the table. The control
limits for such complements may then be used as sampling limits for
appropriate timeliness tests.

\(1\) Such tests may be made in either of two ways. One way is to find
the [upper limit]{.underline} of sampling variability for [the
complement]{.underline} of the \"standard\" percentage (the standard
being the percentage of work units to be processed within a stated time
period) and then determine, as in the preceding example, whether or not
a sample percentage of work units not processed within this time period
exceeds this limit. The second way is to find the [lower]{.underline}
sampling limit to be applied to the standard percentage itself, and
observe whether or not the sample percentage of work units processed
[within]{.underline} the specified time falls below this limit.

\(2\) Example: Assume that for a particular activity or work process,
timeliness is considered satisfactory only if 90% or more of the work
units are processed within 5 workdays. Assume also that in a random
sample of 100 work units drawn from this process 85 units, or 85%, were
processed within 5 workdays. It must then be determined if a sample
finding of 85% here is consistent with the hypothesis that the true
level of the process is actually as good or better than the standard;
i.e., that 90% or more of the work units are processed within 5
workdays.

\(a\) One way to determine whether or not this hypothesis is tenable is
to use the table of \"Three Standard Deviation Control Limits\" in the
following manner. Subtract the \"standard,\" 90%, from 100% to obtain
its complement, 10%. The latter figure, 10%, represents the percentage
of cases which the timeliness standard permits, as a maximum, to require
more than 5 workdays of processing time. If this percentage is exceeded,
noncompliance with the standard is indicated. But since a sample
percentage of work units requiring more than 5 workdays may exceed 10%,
even when 10% or less of all units being processed actually require more
than 5 workdays, the normal limits of sampling error must be taken into
account. The table in appendix E shows that for a sample size of 100 the
upper limit of sampling variability when the true process level is 10%,
is 19%. The sample percentage of work units not completed within 5
workdays is 100% minus 85%, or 15%. Since 15% does not exceed this
[upper]{.underline} limit of 19%, there is insufficient reason to doubt
that the timeliness standard is being met. Compliance with the standard
is therefore presumed.

\(b\) Another way to use the table to test compliance in this same
situa­tion is as follows:

Subtract 19%, the [upper control]{.underline} limit for 10% in error
(10% being the complement of the standard, 90%, from 100%. The
difference, 81%, represents the lower sampling or control limit for the
standard, 90%, when samples of 100 work units are drawn. Since the
sample finding that 85% of the work units were completed within 5
workdays does not fall below this obtained limit, 81%, compliance with
the standard (90% within 5 days) is presumed. A sample percentage less
than 81%, however, would be a strong indication that the standard was
not being met.

d\. Table of \"Minimum [Acceptable Sample Percentages for Specified
Percentages]{.underline} [of Work Units Processed.\"]{.underline} In the
situation described in the last example above, and in many similar
situations where the timeliness standard is a percentage value of 50% or
higher, it may be easier to refer to the table in appendix D to test for
compliance with a timeliness standard. This table of \"Minimum
Acceptable Sample Percentages for Specified Percentages of Work Units
Processed,\" hereafter referred to as the \"Timeliness Table,\" was
adapted from M21-4, appendix D, section V, table 4. This source table
was developed to aid in making faster and easier determinations, from
sample data, of compliance or noncompliance with established timeliness
\"guidelines\" or standards for Adjudication Divisions.

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August 22, 1966 M20-2

\(1\) This timeliness table is basically an extension of the table of
\"Three Standard Deviation Control Limits\" shown in appendix E. In the
timeliness table, eight selected percentage values between 50% and 98%
representing timeliness \"standards,\" are covered. In view of the
purpose of this table, only [lower]{.underline} \"control\" or sampling
limits are shown; upper limits are omitted. The timeliness table also
shows a single sample size instead of a range of sample sizes for a
given limit, as shown in the basic table of control limits. Each sample
size listed in the timeliness table is comparable to the
[midpoint]{.underline} of a sample size interval in the other table
shown in appendix E.

e\. Use [of Timeliness]{.underline} Table. The timeliness table in
appendix D is intended to be used to test compliance in situations where
a specified \"standard\" percentage of work units are to be processed
within a specified time (hours or days). In the table, percentage values
representing standards are shown in columns (B) through (I) as the
column headings. The minimum acceptable sample percentages which, for a
given sample size, warrant a presumption of compliance with the
standard, are shown under the standard percentage to which they pertain.
Sample sizes for which the acceptable sample percentages were calculated
are listed in column (A).

\(1\) Compliance with a given standard percentage is presumed when, in a
sample drawn at random from a work process, the percentage of work units
processed within the specified time is equal to or greater than the
minimum acceptable sample percentage shown under the appropriate column
(B) through (I) for a sample size equal to or larger than that of the
sample drawn. Noncompliance with a standard is indicated if the sample
percentage is less than the minimum acceptable sample percentage shown
for a sample size equal to or smaller than that of the sample drawn. on
rare occasions, an observed sample percentage may fall between the two
adjacent sample percentages shown in the table for the listed sample
sizes next below and next above the size of the sample drawn. In such
cases, a judgment estimate should be made of the minimum percentage that
will be acceptable for compliance. Straight line arithmetic
interpolation may be used, if desired, to aid in making this judgment.

\(2\) Example: Assume again the same situation described in the last
example above (subpar. c(2)) where the \"standard\" required that 90% or
more of the work units be processed within 5 workdays. In testing
compliance, a random sample of 100 work units was drawn, of which 85 or
85% had been completed within 5 workdays. By referring in the timeliness
table to column (G), which shows the limits for a \"90%\" standard, it
is seen, as before, that the minimum acceptable sample percentage is
81.00% for a sample size of 100. Since the sample finding of 85% is
greater in value than 81%, compliance with the standard is presumed.

\(a\) If the sample percentage for the same size sample were any value
less than 81%, noncompliance with the standard would be indicated. If a
sample percentage as large as 85% were obtained with a sample size
between 101 and 12, compliance would be presumed because the sample
value is greater than 81.78%, the limit corresponding to the next larger
listed sample size, 120.

\(b\) [Other Evidence of Noncompliance]{.underline}. The foregoing
discussion of uses of the tables in appendixes D and E or testing
compliance with standards for timeliness has been entirely in terms of
the interpretation of results obtained when a single sample percentage
is considered. A sample percentage outside a given limit is taken to
indicate noncompliance. When a succession of independent samples are
taken, other evidence of noncompliance may appear in the patterns that
result from a sequence of sample percentages. For example, when an
established timeliness standard approximates the true underlying
timeliness (quality) level of a given \"process\" and successive sample
percentages are plotted on a control chart, the \"out of control\" plot
patterns discussed in paragraph 5.04 are equally applicable in testing
for compliance with timeliness standards.

4-9

M20-2 August 22, 1966

4.12 MATHEMATICAL FORMULAS USED TO DEVELOP THE TABLE OF THREE STANDARD
DEVIATION CONTROL LIMITS

a\. Persons using the table do not have to make mathematical
computations to determine the applicable control limits for a given
\"percent in error\" and sample size. The formulas on which the table
data are based are given here mainly for information purposes.

b\. For simplicity and brevity, the table groups sample sizes into
intervals. In the formulas, the sample size (N) is the midpoint of the
respective sample size intervals. Therefore, the control limits are
simply very close approximations rather than precise calculations for
sample sizes falling in the upper and lower range of each interval.

c\. The table alone is fully adequate for most uses. If more precise
control limits for sample sizes near the extremes of an interval are
needed, the formulas below may be used to compute them. However,
straight-line interpolation between limits shown in the table is
relatively simple and works almost as well as the formulas.

FORMULAS

\<I\<ADVANCED,20M20241.RPL\>\> Double - click here To display
MATHEMATICAL FORMULAS \</D\>

N = Sample size

P = Percent in error

UCL (Upper Control Limit) P + 3 P (100-P)

N

LCL (Lower Control Limit) P + 3 100

N

EXAMPLE

Where N = 400

and P = 4%

UCL = 4 + 3 [(4) (9 6)]{.underline} = 4 + 3 9 6 4 + 2. 94 = 6. 94%

400

LCL = 4 - 3 [(4) (96)]{.underline} = 4 - 3 .96 4 - 2.94 = 1.06%

400

4 - 10