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U.U.D.M. Project Report 2011:18
Examensarbete i matematik, 15 hpHandledare och examinator: Sven Erick AlmJuni 2011
Department of MathematicsUppsala University
Approximating the Binomial Distribution by the Normal Distribution – Error and Accuracy
Peder Hansen
Approximating the Binomial Distribution by theNormal Distribution - Error and Accuracy
Peder HansenUppsala University
June 21, 2011
Abstract
Different rules of thumb are used when approximating the binomialdistribution by the normal distribution. In this paper an examinationis made regarding the size of the approximations errors. The exactprobabilities of the binomial distribution is derived and then comparedto the approximated value of the normal distribution. In addition aregression model is done. The result is that the different rules indeedgives rise to errors of different sizes. Furthermore, the regression modelcan be used in order to get guidance of the maximum size of the error.
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Acknowledgenment
Thank you Professor Sven Erick Alm!
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Contents
1 Introduction 4
2 Theory and methodology 42.1 Characteristics of the distributions . . . . . . . . . . . . . . . 42.2 Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . 52.3 Continuity correction . . . . . . . . . . . . . . . . . . . . . . . 62.4 Error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.5 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.5.1 Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . 82.5.2 Regression . . . . . . . . . . . . . . . . . . . . . . . . . 9
3 Background 10
4 The approximation error of the distribution function 114.1 Absolute error . . . . . . . . . . . . . . . . . . . . . . . . . . . 114.2 Relative Error . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
5 Summary and conclusions 17
3
1 Introduction
Neither is any extensive examination found, regarding the rules of thumbused when approximating the binomial distribution by the normal distribu-tion, nor of the accuracy and the error which they result in. The scope of thispaper is the most common approximation of a Binomial distributed randomvariable by the normal distribution. We let X ∼ Bin(n, p), with expectationE(X) = np and variance V (X) = np(1 − p), be approximated by Y , whereY ∼ N(np, np(1− p)). We denote, X ≈ Y .
The rules of thumb, is a set of different guidelines, minimum values orlimits, here denoted L for np(1− p), in order to get a good approximation,that is, np(1−p) ≥ L. There are various kinds of rules found in the literatureand any extensive examination of the error and accuracy has not been found.Reasonable approaches when comparing the errors are, the maximum errorand the relative error, which both are investigated.
The main focus lies on two related topics. First, there is a shorter section,where the origin of the rules, where they come from and who is the originator,is discussed. Next comes an empirical part, where the error affected by thedifferent rules of thumb is studied. The result is both plotted and tabled. Ananalysis of regression is also made, which might be useful as a guideline whenestimating the error in situations not covered here. In addition to the maintopics, a section dealing with the preliminaries, notation and definitions ofprobability theory and mathematical statistics is found. Each of the sectionswill be more explanatory themselves regarding their topics. I presume thereader to be familiar with some basic concepts of mathematical statisticsand probability theory, otherwise the theoretical part would range way tofar. Therefor, also proofs and theorems are just referred to. Finally there isa summarizing section, where the results of the empirical part are discussed.
2 Theory and methodology
First of all, the reader is assumed to be familiar with basic concepts inmathematical statistics and probability theory. Furthermore there are, asstated above, some theory that instead of being explicitly explained, only isreferred to. Regarding the former, I suggest the reader to view for instance[1] or [4] and concerning the latter the reader may want to read [7].
2.1 Characteristics of the distributions
As the approximation of a binomial distributed random variable by a normaldistributed random variable is the main subject, a brief theoretical introduc-tion about them is made. We start with a binomial distributed random
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variable, X and denote,
X ∼ Bin(n, p), where n ∈ N and p ∈ [0, 1].
The parameters p and n are the probability of an outcome and the numberof trials. The expected value and variance of X are,
E(X) = np and V (X) = np(1− p),
respectively. In addition, X has got the probability function
pX(k) = P (X = k) =
(n
k
)pk(1− p)n−k, where 0 ≤ k ≤ n,
and the cumulative probability function, or distribution function,
FX(k) = P (X ≤ k) =
k∑i=0
(n
i
)pi(1− p)k−i. (1)
The variable X is approximated by a normal distributed random variable,call it Y , we write,
Y ∼ N(µ, σ2), where µ ∈ R and σ2 <∞.
The parameters µ and σ2 are the mean value and variance, E(Y ) and V (Y ),respectively. The density function of Y is
fY (x) =1
σ√
2πe−(x−µ)
2/2σ2
and the distribution function is defined by,
FY (k) = P (Y ≤ x) =
∫ x
−∞
1
σ√
2πe−(t−µ)
2/2σ2dt. (2)
2.2 Approximation
Thanks to De Moivre, among others, we know by the central limit theo-rem that a sum of random variables converges to the normal distribution.A binomial distributed random variable X may be considered as a sum ofBernoulli distributed random variables. That is, let Z be a Bernoulli dis-tributed random variable,
Z ∼ Be(p) where p ∈ [0, 1],
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with probability distribution,
pZ = P (Z = k) =
{p for k = 1
1− p for k = 0.
Consider the sum of n independent identically distributed Zi’s, i.e.
X =n∑i=0
Zi
and note that X ∼ Bin(n, p). For instance one can realize that the proba-
bility of the sum being equal to k, P (X = k) =
(n
k
)pk(1 − p)n−k. Hence,
we know that when n → ∞, the distribution of X will be normal and forlarge n approximately normal. How large n should be in order to get a goodapproximation also depends, to some extent, on p. Because of this, it seemsreasonable to define the following approximations. Again, let X ∼ Bin(n, p)and Y ∼ N(µ, σ2). The most common approximation, X ≈ Y , is the onewhere µ = np and σ2 = np(1− p), this is also the one used here. Regardingthe distribution function we get
FX(k) ≈ Φ
(k − np√np(1− p)
), (3)
where FX(k) is defined in (1) and Φ is the standard normal distributionfunction. We extend the expression above and get that,
FX(b)− FX(a) = P (a < X ≤ b) ≈ Φ
(b− np√np(1− p)
)− Φ
(a− np√np(1− p)
).
(4)
2.3 Continuity correction
We proceed with the use of continuity correction, which is recommended by[1], suggested by [4] and advised by [9], in order to decrease the error, theapproximation (3) will then be replaced by
FX(k) ≈ Φ
(k + 0.5− np√np(1− p)
)(5)
and hence (4) is written as
FX(b)− FX(a) = P (a < X ≤ b) ≈ Φ
(b+ 0.5− np√np(1− p)
)−Φ
(a+ 0.5− np√np(1− p)
).
(6)
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This gives, for a single probability, with the use of continuity correction, theapproximation,
pX(k) = FX(k)−FX(k−1) ≈ Φ
(k + 0.5− np√np(1− p)
)−Φ
((k − 1) + 0.5− np√
np(1− p)
)(7)
and further we note that it can be written
FX(k)− FX(k − 1) ≈k+0.5∫k−0.5
fY (t)dt. (8)
2.4 Error
There are two common ways of measuring an error, the absolute error andthe relative error. In addition another usual measure of how close, so tospeak, two distributions are to each other, is the supremum norm
supA|P (X ∈ A)− P (Y ∈ A)|.
However, from a practical point of view, we will study the absolute errorand relative error of the distribution function. Let a denote the exact valueand a the approximated value. The absolute error is the difference betweenthem, the real value and the one approximated. The following notation isused,
εabs = |a− a| .
Therefor, the absolute error of the distribution function, denoted εFabs(k),
for any fixed p and n, where k ∈ N : 0 ≤ k ≤ n, without use of continuitycorrection, is
εFabs(k) =
∣∣∣∣∣FX(k)− Φ
(k − np√np(1− p)
)∣∣∣∣∣ . (9)
Regarding the relative error, in the same way as before, let a be the exactvalue and a the approximated value. Then the relative error is defined as
εrel =
∣∣∣∣a− aa∣∣∣∣ .
This gives the relative error of the distribution function, denoted εFrel(k),
for any fixed p and n, where k ∈ N : 0 ≤ k ≤ n, without use of continuitycorrection, is
εFrel(k) =
εFabs(k)
FX(k),
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or equivalently, inserting εFabs(k) from (9),
εFrel(k) =
∣∣∣∣∣FX(k)− Φ
(k − np√np(1− p)
)∣∣∣∣∣FX(k)
.
2.5 Method
The examination is done in the statistical software R. The software providespredefined functions for deriving the distribution function and probabilityfunction of the normal and binomial distributions. The examination is splitinto two parts, where the first part deals with the absolute error of theapproximation of the distribution function and the second part concerns therelative error. The conditions under which the calculations are made, arethose found as guidelines in [4]. The calculations will be made with the helpof a two-step algorithm. At the end of each section a linear model is fitted tothe error. Finally, an overview, where a table and a plot of how the value ofnpq, where q = 1− p, affects the maximum approximation error for differentprobabilities are presented.
2.5.1 Algorithm
The two step algorithm below is used. The values of npq mentioned in theliterature are, in all cases said to be equal or larger than some limit, heredenoted L. The worst case scenario, as to speak, is the case where they areequal, that is, npq = L. Therefor equalities are chosen as limits. We knowthat n ∈ N, which means that p must be semi-fixed if the equality shouldhold, this means that the values of p are adjusted, but still remain close tothe ones initially chosen. The way of doing this is a two-step algorithm. Firsta reasonable set of different initial probabilities, pi’s are chosen, whereafterthe corresponding ni values, which in turn will be rounded to ni, are derived.These are used to adjust pi to pi so that the equality will hold.
1. (a) Chose a set P of different initial probabilities, pi ∈ [0, 0.5], wherei ∈ N : 0 < i <
∣∣∣P∣∣∣.(b) Derive the corresponding ni ∈ R+ so that nipi(1− pi) = L,
(c) continue by deriving ni ∈ N, in order to get a integer,
ni(pi) := min{n ∈ N : npi(1− pi) ≥ L}. (10)
Now we got a set of ni ∈ N, denote it N.
2. Chose a set P so that for every pi ∈ P,
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nipi(1− pi) = L.
The result is that we always keep the limit L fixed. Let us take a lookat an example. Let L = 10, use continuity correction and the initial P =0.1(0.1)0.5,
Exemplifying table of algorithm valuesi 1 2 3 4 5pi 0.1 0.2 0.3 0.4 0.5ni 111.11 62.50 47.62 41.67 40.00ni 112 63 48 42 40pi 0.099 0.198 0.296 0.391 0.500
Different rules of thumb are suggested by [4]. Using approximation (3)the authors say that np(1 − p) ≥ 10 gives reasonable approximations andin addition, using (5), it may even be sufficient using np(1 − p) ≥ 3. Theinvestigation takes place under three different conditions,
• np(1− p) = 10 without continuity correction, suggested in [4],
• np(1− p) = 10 with continuity correction, suggested in [2],
• np(1− p) = 3 with continuity correction, suggested in [4].
The investigation of the rules is made only for pi ∈ [0, 0.5] due to symmetry.As we see, np(1 − p) simply gets the same values for p ∈ [0, 0.5] as forp ∈ [0.5, 1]. So, for every p, ni(pi) is derived, this in turn, means that weget ni(pi) + 1 approximations. For every ni(pi), and of course pi as well, wedefine the maximum absolute error of the approximation of the distributionfunction,
MFabs= max{εFabs
(k) : 0 ≤ k ≤ ni(pi)}, (11)
and in addition the maximum relative error
MFrel= max{εFrel
(k) : 0 ≤ k ≤ ni(pi)}. (12)
The results are both tabled and plotted.
2.5.2 Regression
Beforehand, some plots where made which indicated that the maximum ab-solute error could be a linear function of p. Regarding the relative maximumerror, a quadratic or cubic function of p seemed plausible. Because of that,a regression is made. The model assumed to explain the absolute error is
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Mε = α+ βp+ εl, (13)
where Mε is the maximum error, α is the intercept, β the slope and εl theerror of the linear model. For the relative error, the two additional regressionmodels are,
Mε = α+ βp+ γp2 + εl (14)
and
Mε = α+ βp+ γp2 + δp3 + εl. (15)
3 Background
In the first basic courses in mathematical statistics, the approximations (3)and (5) are taught. Students have learned some kind of rules of thumb theyshould use when applying the approximations, myself included, for examplethe rules suggested by Blom [4],
np(1− p) ≥ 10,
np(1− p) ≥ 3 with continuity correction.
Any motivation why the limit L is set to be L = 10 and L = 3 respec-tively is not found in the book. On the other hand, in 1989 Blom claimsthat the approximation ”gives decent accuracy if npq is approximately largerthan 10” with continuity correction [2]. Further, it is interesting, that Blomchanges the suggestion between the first edition of [3] from 1970, where itsays, similarly as above, that it ”gives decent accuracy if np(1 − p) is ap-proximately larger than 10” with continuity correction, and in the secondedition from 1984 the same should yield, but now instead without use ofcontinuity correction, the conclusion is that there has been some fuzzinessregarding the rules. Neither have I, nor my advisor Sven Erick Alm, foundany examination of the accuracy of these rules anywhere else. With Blom [4]as starting-point, I begun backtracking, hoping that I could find the sourceof the rules of thumb. It is worth mentioning that among authors, slightlydifferent rules have been used. For instance Alm himself and Britton, presenta schema with rules for approximating distributions, in which np(1− p) > 5with continuity correction is suggested [1]. Even between countries, or froman international point of view, so to speak, differences are found. Schaderand Schmid [10] says that ”by far the most popular are”
np(1− p) > 9
10
and
np > 5 for 0 < p ≤ 0.5,
n(1− p) > 5 for 0.5 < p < 1,
which I am not familiar with and I have not found in any Swedish literature.In the mid-twentieth century, more precise 1952, Hald [9] wrote,
An exhaustive examination on the accuracy of the approxi-mation formulas has not yet been made, and we can thereforeonly give rough rules for the applicability of the formulas.
With these words in mind, the conclusion is that there probably does not ex-ist any earlier work made about the accuracy of the approximation. However,Hald himself made an examination in the same work for npq > 9. Furtherhe also points out that in cases where the binomial distribution is very skew,p < 1
n+1 and p > nn+1 , the approximation cannot be applied. Some articles
have been found that briefly discuss the accuracy and error of the distribu-tions. Mainly, the focus of the articles lies on some more advanced methodof approximating than (3) or (5). An update of [2] has been made by Enger,Englund, Grandell and Holst in 2005, [4]. The writers have been contactedand Enger was said to be the one that assigned the rules. Hearing this mademe believe that the source could be found. However, Enger could not recallfrom where he had got it [6]. That is how far I could get. Nevertheless, theexamination remains as interesting as beforehand.
Discussing rules for approximating, one can not avoid at least mentioningthe Berry-Esseen theorem. The theorem gives a conservative estimate, in thesense that it gives the largest possible size of the error. It is based upon therate of convergence of the approximation to the normal distribution. TheBerry-Esseen theorem will not be further examined here, but there are severalinteresting articles due to that the theorem is improved every now and then,most recently in May 2010 [11].
4 The approximation error of the distribution func-tion
The errors of the approximations,MFabsandMFrel
, defined in (11), and (12)respectively, are plotted and tabled. The cases that are examined are thosementioned earlier, suggested by [4].
4.1 Absolute error
We examine the absolute maximum errors of the approximation of the distri-bution function, MFabs
defined in (11), here in the first part. In addition to
11
that a regression is made, defined in (13), to see if we might find any lineartrend.
Case 1: npq = 10, without continuity correction
First, the case where L = 10 = npq, without continuity correction. P, theset of different initial probabilities is chosen to be pi = 0.01(0.01)0.50. Thismeans that we use 50 equidistant pi. The smallest probability is p1 = 0.0100and it has the largest error MFabs
= 0.0831. MFabsdecreases the closer to
0.5 we get, which is natural since the binomial distribution tends to be skew.The points make a pattern which is a bit curvy, but still the points are closeto the straight line in Figure 1. Another remark made, is that the distancebetween the probabilities decreases the closer to 0.5 we get. The fact thatthere are several ni rounded to the same value of ni, which in turn givesequal values on pi, makes several MFabs
the same, and plotted in the samespot. So they are all there, but not visible due to that reason. Next we tryto fit a linear model for MFabs
. The result is
MFabs= 0.0836− 0.0417p+ εl.
The regression line is the straight line in Figure 1. The slope of the lineshows that the size of MFabs
changes moderately. Note that the sum of theerrors of the regression line,
∑|εl|, is relatively small, the result should be
somewhat precise estimates of MFabsfor probabilities which are not taken in
consideration here.
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0.0 0.1 0.2 0.3 0.4 0.5
0.06
50.
070
0.07
50.
080
Probabilities
Max
err
or
Figure 1: Maximum absolute error for npq = 10 without continuity correc-tion. The straight line is the regression line, MFabs
= 0.0836− 0.0417p.
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Case 2: npq = 10, with continuity correction
Under these circumstances MFabsdecreases and is about four times smaller
than without continuity correction. The regression line,
MFabs= 0.0209− 0.0416p+ εl, (16)
also has got a four times smaller intercept than in the first case. What isinteresting is that, the slope is approximately the same in both cases, thisin turn, means that for every pi = 0.01(0.01)0.50, it holds that MFabs
also isfour times smaller. This can be seen in Figure 2.
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0.0 0.1 0.2 0.3 0.4 0.5
0.00
00.
005
0.01
00.
015
0.02
0
Probabilities
Max
err
or
Figure 2: Maximum absolute error for npq = 10 with continuity correction.The straight line is the regression line, MFabs
= 0.0209− 0.0416p+ εl.
Case 3: npq = 3, with continuity correction
Finally we take a look at the last case, regarding the absolute error, whereL = 3 = npq and continuity correction is used. The plot is seen in Figure 3.P is the same as above. In this case the regression line is
MFabs= 0.0373− 0.0720p+ εl.
The largest error, MFabs= 0.0355 appears at p1 = 0.0100 and is about twice
the size compared to the largest MFabsfor L = 10. The slope of the line is
more aggressive here, which in turn results in errors, one order of magnitudeless than in the Case 1 for probabilities close to 0.5. Also here the sum of
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discrepancy from the regression line is relatively small which should resultin fairly good estimations of MFabs
.
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0.0 0.1 0.2 0.3 0.4 0.5
0.00
50.
010
0.01
50.
020
0.02
50.
030
0.03
5
Probabilities
Max
err
or
Figure 3: Maximum absolute error for npq = 3, with continuity correction.The straight line is the regression line, MFabs
= 0.0373− 0.0720p.
4.2 Relative Error
Here, the maximum relative error of the approximation of the distributionfunction, MFrel
, defined in (12) is examined. The regression models (14) and(15) are both tested.
Case 1: npq = 10, without continuity correction
In the first case we perform the calculations under, L = 10 = npq with-out continuity correction. The result is shown in Figure 4. As we see MFrel
increases very rapidly. The smallest value of MFrel, 16.97317 is at p1. The
largest 138.61756 at p50. As we see in Table 4, it is k = 0 that gives thelargest error. For other values of k the error is much smaller. Furthermorewe note that MFrel
is very large. If we look at a specific example wherep = 0.2269, which means that n = 57, then X ∼ Bin(57, 0.2269). Let Xbe approximated, according to (3), by Y ∼ N(12.933, 3.162078). We getthat P (X ≤ 1) = 7.55 · 10−6 and P (Y ≤ 1) = 8.04 · 10−5. Under thesecircumstances we get,
MFrel=|P (X ≤ 1)− P (Y ≤ 1)|
P (X ≤ 1)= 9.64.
14
The result is shown in Table 4. So the relative error is, as we also can see,large, for small k and small probabilities. The regression curves, defined in(14) and (15) are,
MFrel= 14.66 + 69.86p+ 416.14p2 + εl
and
MFrel= 21.53− 92.26p+ 1246.60p2 − 1136.07p3 + εl
respectively. We note that there are not any larger differences in accuracydepending on the choice of model. Naturally, the discrepancy of the secondmodel is lower.
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0.0 0.1 0.2 0.3 0.4 0.5
2040
6080
100
120
140
Probabilities
Max
err
or
Figure 4: Maximum relative error for npq = 10 without continuity correction.The solid line is the regression curve, MFrel
= 14.66 + 69.86p+ 416.14p2 andthe dashed line, MFrel
= 21.53− 92.26p+ 1246.60p2 − 1136.07p3.
Case 2: npq = 10, with continuity correction
We continue by looking at the same case as above, but here continuity correc-tion is used. This gives somewhat remarkable results,MFrel
is actually abouttwo times larger than without continuity correction. Let us study the samenumeric example as above, except that we use continuity correction. We gotp = 0.2269 which again means that n = 57, then X ∼ Bin(57, 0.2269). Welet X be approximated, according to (5), by Y ∼ N(12.933, 3.162078). Itresults in, P (X ≤ 1) = 7.55 · 10−6 and P (Y ≤ 1 + 0.5) = 0.000150. Under
15
these circumstances we get,
MFrel=|P (X ≤ 1)− P (Y ≤ 1 + 0.5)|
P (X ≤ 1)= 18.84,
which fits the values in Table 5. MFabsgets dramatically worse when we
use continuity correction than without. Hence, also MFrelbecomes worse.
In Figure 5 one can judge that the results gets worse as we get closer toprobabilities near 0.5. The regression curves, defined in (14) and (15) are,
MFrel= 34.9− 69.8p+ 1597.1p2 + εl
and
MFrel= 37.4− 127.3p+ 1891.8p2 − 403.2p3 + εl,
respectively. Looking at Figure 5, we see that the difference between the twomodels is insignificant.
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0.0 0.1 0.2 0.3 0.4 0.5
5010
015
020
025
030
035
0
Probabilities
Max
err
or
Figure 5: Maximum relative error for npq = 10 with continuity correction.The solid line is the regression curve, MFrel
= 34.9 − 69.8p + 1597.1p2 andthe dashed line, MFrel
= 37.4− 127.3p+ 1891.8p2 − 403.2p3.
Case 3: npq = 3 with continuity correction
Here, in the last case npq = 3 and continuity correction is used, see Fig-ure 6. This gives the curves of regression, defined in (14) and (15),
MFrel= 0.473 + 2.204p+ 2.123p2 + εl
16
and
MFrel= 0.514 + 1.155p+ 7.858p2 − 7.885p3 + εl,
respectively. As we see MFrelactually get the smallest value here, where
npq = 3 and continuity correction is used. As well as in the two other casesregarding the relative error the difference between the quadratic and cubicregression model is minimal.
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0.0 0.1 0.2 0.3 0.4 0.5
0.5
1.0
1.5
2.0
Probabilities
Max
err
or
Figure 6: Maximum relative error for npq = 3 with continuity correction.The solid line is the regression curve, MFrel
= 0.473 + 2.204p+ 2.123p2 andthe dashed line, MFrel
= 0.514 + 1.155p+ 7.858p2 − 7.885p3.
5 Summary and conclusions
The three different rules of thumbs that are focused on turned out to giveapproximation errors of different sizes. Regarding the absolute errors, thelargest difference is found between the case where L = 10 without continuitycorrection and L = 10 with continuity correction. The largest error decreasesfrom ∼ 0.08 to about ∼ 0.02, which is approximately four times smaller, arelatively large difference. Letting L = 3 and using continuity correctionwe end up with the largest error ∼ 0.035, closer to the latter case, but stillbetween them. When using this common and simple way of approximating,depending on the problem, different levels of tolerance usually are accepted.A common level in many cases may be 0.01. If we look deeper, we see thatthe probabilities for getting such a small MFabs
differs from between therules of thumb. Using npq = 10 without continuity correction does not evenreach to the 0.01 level of accepted accuracy. Comparing this to the other
17
two cases which in contrast reach the 0.01 level for probabilities ∼ 0.25in the same case as above but in addition with continuity correction, andfor probabilities ∼ 0.35 in the case where npq = 3. Further, it would beinteresting to investigate how the relationship between k and n affects theerror. In addition, another interesting part would be some tables indicatinghow large n should be in order to get sufficiently small errors, for differentprobabilities.
Concerning the relative errors I would say that the applicability may besomewhat uncertain, due to the fact that MFrel
is very large for small valuesof k but rapidly decrease. This fact, I may say, make the plots look a bit ex-treme and there are other values of k that give much better approximations.Judging by Tables 4, 5 and 6 indeed this seems to be the case. We knowthat the approximation is motivated by the central limit theorem, however,what we also know, is that it does not hold the same accuracy for smallprobabilities, that is, the tails of the distributions. This is also the directreason why the accuracy gets worse when using continuity correction, it putsextra mass on the already too large approximated value. In a similar waywe get the explanation why the relative error increases when the value ofnpq changes from 10 to 3, (as one maybe would expect the opposite), themean value of the normal distribution, np, gets closer to 0 which in turngives additional mass. The conclusion is, one should remember that due tothe fluctuations depending on k, of the relative errors, what we also can seein Tables 4, 5 and 6, that the regression model also provides conservativeestimates of the errors. As a natural alternative, and most likely better,Poisson approximation is recommended for small probabilities. Like in theprevious case concerning the absolute errors, some more exhaustive exami-nation of the relative error would be interesting. How large should n be toget acceptable levels of the error, for instance 10% or 5% and so on.
References
[1] Alm S.E. and Britton T., Stokastik - Sannolikhetsteori och statistikteorimed tillämpningar, Liber (2008).
[2] Blom, G., Sannolikhetsteori och statistikteori med tillämpningar (Bok C),Fjärde upplagan, Studentlitteratur (1989).
[3] Blom, G., Sannolikhetsteori med tillämpningar (Bok A), Studentlitter-atur (1970,1984)
[4] Blom G., Enger J., Englund G., Grandell J. and Holst L., Sannolihetsteorioch statistikteori med tillämpningar, Femte upplagan, Studentlitteratur(2008).
18
[5] Cramér H., Sannolikhetskalkylen, Almqvist & Wiksell/Geber Förlag AB(1949).
[6] Enger J., Private communication, (2011).
[7] Gut A., An Intermediate Course in Probability, Springer (2009).
[8] Hald A., A History of Mathematical Statistics from 1750 to 1930. Wiley,New York (1998).
[9] Hald A., Statistical Theory with Engineering Applications, John Wiley &Sons, Inc., New York and London (1952).
[10] Schader M. and Schmid F., Two Rules of Thumb for the Approximationof the Binomial Distribution by the Normal Distribution,The AmericanStatistician, 43, 1989, 23-24.
[11] Shevtsov I. G., An Improvement of Convergence Rate Estimates in theLyapunov Theorem, Doklady Mathematics, 82, 2010, 862-864.
Tables
Regarding the plotted probabilities, that is the set P, only the maximumerror is plotted. One can not tell from which k the error comes from, neithercan one tell if the error is of similar size for other values of k. To get a moredetailed picture this section contains tables both for the absolute errors andthe relative errors. It would have been possible to table all the errors forall values of k, but due to the fact that the cardinality of N at times, thatis for small probabilities, is relatively large, it would have taken too muchplace. This made me table only the 10 values of k which resulted in thelargest errors. The columns in the tables, that contains the values of k isin descending order. What this means is that the first value of k in eachcolumn is the maximum error that is plotted. On the side of every columnof k, there is a column where the corresponding error is written. These twosub columns, got a common header which tells the value of p in the specificcase.
19
p=
0.01
0.02
0.03
0.03
990.
0499
0.05
970.
0698
0.07
990.
0893
0.09
910.
109
0.11
960.
129
0.13
810.
1487
0.15
840.
1696
0.17
920.
1899
kεFabs
kεFabs
kεFabs
kεFabs
kεFabs
kεFabs
kεFabs
kεFabs
kεFabs
kεFabs
kεFabs
kεFabs
kεFabs
kεFabs
kεFabs
kεFabs
kεFabs
kεFabs
kεFabs
100.
0831
100.
083
100.
0828
100.
0824
100.
0819
100.
0813
100.
0805
100.
0795
110.
0793
110.
0794
110.
0793
110.
079
110.
0786
110.
078
110.
0771
110.
0761
120.
0763
120.
0763
120.
0761
90.
0808
90.
0795
90.
078
110.
0768
110.
0776
110.
0782
110.
0787
110.
0791
100.
0784
100.
0772
100.
0758
100.
0741
120.
0743
120.
075
120.
0757
120.
0761
110.
0747
110.
0733
110.
0715
110.
0738
110.
0749
110.
0759
90.
0765
90.
0748
90.
073
90.
071
90.
0689
120.
0698
120.
0711
120.
0723
120.
0734
100.
0724
100.
0706
100.
0684
130.
0665
130.
0683
130.
0696
130.
0709
80.
0672
80.
065
80.
0628
120.
0621
120.
0638
120.
0654
120.
067
120.
0685
90.
0669
90.
0647
90.
0623
90.
0597
130.
0614
130.
0631
130.
0649
100.
0662
100.
0636
100.
0611
100.
0583
120.
0569
120.
0587
120.
0604
80.
0605
80.
0581
80.
0557
80.
0532
130.
0516
130.
0536
130.
0555
130.
0575
130.
0596
90.
0573
90.
0549
90.
0521
140.
051
140.
0536
140.
0557
140.
058
70.
0465
70.
0442
70.
0419
130.
0435
130.
0455
130.
0475
130.
0496
80.
0507
80.
0483
80.
0458
80.
0433
140.
0422
140.
0443
140.
0464
140.
0488
90.
0494
90.
0463
90.
0436
150.
0418
130.
0377
130.
0396
130.
0416
70.
0396
70.
0373
70.
035
70.
0328
140.
0337
140.
0356
140.
0377
140.
0398
80.
0406
80.
0382
80.
0359
80.
0332
150.
0342
150.
0368
150.
0391
90.
0406
60.
0253
60.
0235
140.
0243
140.
026
140.
0278
140.
0297
140.
0317
70.
0305
70.
0285
70.
0264
70.
0244
150.
0258
150.
0277
150.
0296
150.
032
80.
0308
80.
0281
80.
0259
160.
0263
140.
021
140.
0226
60.
0217
60.
026
0.01
846
0.01
6815
0.01
715
0.01
8615
0.02
0215
0.02
1915
0.02
377
0.02
227
0.02
047
0.01
8716
0.01
816
0.01
9816
0.02
1916
0.02
398
0.02
3515
0.00
9215
0.01
0315
0.01
1515
0.01
2815
0.01
4115
0.01
556
0.01
526
0.01
386
0.01
256
0.01
1216
0.01
1816
0.01
3316
0.01
4716
0.01
627
0.01
697
0.01
527
0.01
3417
0.01
2517
0.01
42
0.19
790.
2066
0.21
630.
2269
0.23
890.
2454
0.25
980.
2678
0.27
640.
2857
0.29
590.
307
0.31
940.
3333
0.34
920.
3679
0.39
090.
4219
0.5
kεFabs
kεFabs
kεFabs
kεFabs
kεFabs
kεFabs
kεFabs
kεFabs
kεFabs
kεFabs
kεFabs
kεFabs
kεFabs
kεFabs
kεFabs
kεFabs
kεFabs
kεFabs
kεFabs
120.
0757
120.
0751
120.
0742
130.
0736
130.
0737
130.
0737
130.
073
130.
0724
130.
0715
140.
0714
140.
0715
140.
0711
140.
0701
150.
0695
150.
0693
150.
0676
160.
0675
170.
0662
200.
0627
130.
0717
130.
0725
130.
0731
120.
073
120.
0712
120.
0701
140.
0698
140.
0705
140.
0711
130.
0702
130.
0685
150.
0675
150.
0688
140.
0682
160.
0656
160.
0675
170.
0645
160.
0629
190.
0614
110.
0711
0.06
8111
0.06
614
0.06
5314
0.06
7214
0.06
8112
0.06
7212
0.06
5412
0.06
3315
0.06
4315
0.06
613
0.06
6213
0.06
3216
0.06
2914
0.06
5214
0.06
0315
0.06
3218
0.06
2621
0.05
814
0.05
9714
0.06
1514
0.06
3411
0.06
3411
0.06
0211
0.05
8415
0.05
915
0.06
0815
0.06
2612
0.06
0712
0.05
7816
0.05
716
0.06
130.
0593
170.
0554
170.
0601
180.
0552
150.
0533
180.
0544
100.
0561
100.
0536
100.
0508
150.
0511
150.
0541
150.
0557
110.
0541
110.
0516
110.
0489
160.
0514
160.
0541
120.
0543
120.
0502
170.
0508
130.
0542
180.
048
140.
0526
190.
0532
220.
0487
150.
0438
150.
046
150.
0484
100.
0476
100.
044
100.
042
160.
0442
160.
0464
160.
0488
110.
0459
110.
0425
170.
0428
170.
0466
120.
0453
180.
0418
130.
0475
190.
0424
200.
0408
170.
0434
90.
0384
90.
036
90.
0333
160.
0353
160.
0384
160.
0402
100.
0376
100.
0352
170.
0337
170.
0364
170.
0394
110.
0388
110.
0347
180.
0366
120.
0396
190.
0343
130.
0387
140.
0402
230.
0373
160.
0281
160.
0301
160.
0325
90.
0304
90.
0273
90.
0256
170.
0292
170.
0314
100.
0326
100.
0298
100.
0269
180.
0286
180.
0322
110.
0301
190.
0282
120.
0329
200.
0293
210.
0282
160.
0311
80.
0217
80.
0199
170.
0191
170.
0213
170.
024
170.
0256
90.
022
90.
0202
180.
0206
180.
0229
180.
0255
100.
0238
100.
0205
190.
0235
110.
0252
200.
022
120.
025
130.
0269
240.
026
170.
0156
170.
0172
80.
0179
80.
0159
80.
0138
180.
0142
180.
017
180.
0187
90.
0182
90.
0163
190.
0146
190.
017
190.
0199
100.
0171
200.
017
110.
0198
210.
0183
220.
0177
150.
02
Tab
le1:
Tab
leof
the10
largesterrors,ε F
absan
dwhich
kis
comes
from
,foreverypi,
unde
rnpq
=10
witho
utcontinuity
correction
.
20
p=
0.01
0.02
0.03
0.03
990.
0499
0.05
970.
0698
0.07
990.
0893
0.09
910.
109
0.11
960.
129
0.13
810.
1487
0.15
840.
1696
0.17
920.
1899
kεFabs
kεFabs
kεFabs
kεFabs
kεFabs
kεFabs
kεFabs
kεFabs
kεFabs
kεFabs
kεFabs
kεFabs
kεFabs
kεFabs
kεFabs
kεFabs
kεFabs
kεFabs
kεFabs
110.
0201
110.
0211
0.01
9811
0.01
9511
0.01
9211
0.01
8711
0.01
8211
0.01
7711
0.01
7112
0.01
6512
0.01
6312
0.01
6112
0.01
5812
0.01
5412
0.01
4912
0.01
4412
0.01
3613
0.01
3313
0.01
3110
0.02
100.
0193
100.
0185
100.
0176
100.
0167
120.
0164
120.
0165
120.
0166
120.
0165
110.
0164
110.
0157
110.
0148
110.
0139
130.
0133
130.
0134
130.
0135
130.
0134
120.
013
120.
0121
120.
0149
120.
0153
120.
0156
120.
0159
120.
0162
100.
0157
100.
0147
100.
0137
100.
0126
130.
0121
130.
0125
130.
0129
130.
0131
110.
0131
110.
0121
110.
0111
110.
0099
140.
0114
0.01
049
0.01
419
0.01
299
0.01
189
0.01
066
0.01
036
0.01
0213
0.01
0713
0.01
1213
0.01
1710
0.01
1610
0.01
0410
0.00
927
0.00
847
0.00
8314
0.00
8614
0.00
9114
0.00
9711
0.00
8811
0.00
775
0.01
145
0.01
15
0.01
076
0.01
045
0.00
9813
0.01
016
0.01
60.
0098
60.
0096
60.
0093
60.
0089
60.
0085
60.
0082
140.
008
70.
0081
70.
0079
70.
0076
70.
0074
70.
007
60.
0104
60.
0105
60.
0105
50.
0103
130.
0095
50.
0094
50.
0089
50.
0084
70.
0082
70.
0083
70.
0084
70.
0084
100.
0082
60.
0078
60.
0073
60.
0069
80.
0066
80.
0067
80.
0068
40.
0091
40.
0086
130.
0082
130.
0089
90.
0094
90.
0083
70.
0077
70.
008
50.
008
50.
0075
50.
0071
140.
0069
140.
0075
100.
0071
80.
0061
80.
0064
60.
0064
60.
006
60.
0055
160.
0076
130.
0075
40.
0081
40.
0076
40.
0071
70.
0074
90.
0071
170.
0066
170.
0064
170.
0062
140.
0062
50.
0065
50.
0061
80.
0058
100.
0059
180.
0053
180.
0052
180.
005
150.
0055
170.
0071
160.
0073
160.
0071
170.
007
70.
007
170.
0068
170.
0067
90.
006
180.
0058
180.
0058
170.
006
170.
0058
180.
0057
50.
0057
180.
0055
100.
0048
190.
0047
150.
0048
180.
0047
130.
0068
170.
0071
170.
007
160.
0069
170.
0069
40.
0066
40.
0061
180.
0058
40.
0053
140.
0055
180.
0058
180.
0057
170.
0055
180.
0056
50.
0052
50.
0047
50.
0043
190.
0047
190.
0046
0.19
790.
2066
0.21
630.
2269
0.23
890.
2454
0.25
980.
2678
0.27
640.
2857
0.29
590.
307
0.31
940.
3333
0.34
920.
3679
0.39
090.
4219
0.5
kεFabs
kεFabs
kεFabs
kεFabs
kεFabs
kεFabs
kεFabs
kεFabs
kεFabs
kεFabs
kεFabs
kεFabs
kεFabs
kεFabs
kεFabs
kεFabs
kεFabs
kεFabs
kεFabs
130.
0128
130.
0125
130.
012
130.
0114
140.
0108
140.
0106
140.
0102
140.
0099
140.
0095
140.
0089
150.
0085
150.
0082
150.
0077
150.
007
160.
0064
160.
0056
170.
0046
180.
0032
147e
-04
120.
0114
140.
0108
140.
0109
140.
0109
130.
0107
130.
0102
130.
009
150.
0086
150.
0087
150.
0086
140.
0083
140.
0074
160.
0067
160.
0067
150.
0059
170.
0051
160.
0043
170.
0032
277e
-04
140.
0106
120.
0106
120.
0097
120.
0087
150.
008
150.
0082
150.
0085
130.
0083
130.
0075
130.
0066
160.
0064
160.
0066
140.
0064
140.
0052
170.
005
150.
0045
180.
0036
190.
0024
156e
-04
110.
0068
80.
0067
150.
007
150.
0075
120.
0074
120.
0068
90.
0054
90.
0053
160.
0057
160.
006
130.
0057
130.
0046
100.
0042
170.
0046
140.
0038
110.
0033
110.
0028
160.
0022
266e
-04
80.
0067
150.
0065
80.
0066
80.
0063
80.
0061
80.
0059
80.
0054
160.
0053
90.
0052
90.
005
90.
0048
90.
0045
170.
0041
100.
004
100.
0036
180.
0032
150.
0027
120.
0022
286e
-04
70.
0067
70.
0064
70.
006
70.
0056
90.
0054
90.
0054
120.
0053
80.
0051
80.
0048
80.
0044
100.
0043
100.
0043
90.
0041
90.
0036
110.
0034
100.
0031
120.
0026
130.
0019
136e
-04
150.
006
110.
0058
90.
0049
90.
0052
70.
005
70.
0048
160.
0049
120.
0044
100.
004
100.
0042
80.
004
170.
0036
130.
0034
110.
0033
90.
003
120.
0024
100.
0023
110.
0019
254e
-04
60.
0051
60.
0047
110.
0048
190.
0042
190.
004
160.
004
70.
0041
70.
0038
120.
0036
200.
0033
200.
0031
80.
0035
80.
0031
80.
0025
180.
0026
90.
0023
190.
0019
100.
0013
164e
-04
190.
0046
90.
0046
190.
0044
60.
0038
200.
0037
190.
0039
200.
0036
100.
0037
70.
0034
70.
0031
170.
003
200.
0028
110.
003
210.
0024
210.
0021
140.
0022
90.
0015
200.
0011
124e
-04
180.
0045
190.
0045
60.
0043
200.
0037
160.
0036
200.
0037
190.
0035
200.
0035
200.
0034
210.
0028
210.
0028
210.
0027
210.
0026
130.
0021
220.
002
220.
0018
130.
0015
140.
001
294e
-04
Tab
le2:
Tab
leof
the10
largesterrors,ε
Fabsan
dwhichkiscomes
from
,for
everypi,un
dernpq
=10
withcontinuity
correction
.
21
p=
0.01
0.01
990.
0297
0.03
950.
0493
0.05
90.
0685
0.07
950.
089
0.09
780.
1086
0.11
720.
1273
0.13
94
kεFabs
kεFabs
kεFabs
kεFabs
kεFabs
kεFabs
kεFabs
kεFabs
kεFabs
kεFabs
kεFabs
kεFabs
kεFabs
kεFabs
20.
0356
20.
0343
30.
0332
30.
033
30.
0328
30.
0325
30.
0322
30.
0318
30.
0314
30.
031
30.
0304
30.
033
0.02
943
0.02
863
0.03
353
0.03
342
0.03
312
0.03
182
0.03
052
0.02
922
0.02
792
0.02
652
0.02
522
0.02
392
0.02
242
0.02
122
0.01
980
0.01
890
0.02
450
0.02
420
0.02
390
0.02
360
0.02
330
0.02
290
0.02
250
0.02
210
0.02
160
0.02
120
0.02
070
0.02
020
0.01
962
0.01
816
0.01
176
0.01
166
0.01
146
0.01
124
0.01
114
0.01
164
0.01
214
0.01
264
0.01
314
0.01
354
0.01
44
0.01
434
0.01
474
0.01
524
0.00
94
0.00
964
0.01
014
0.01
066
0.01
116
0.01
096
0.01
076
0.01
056
0.01
036
0.01
016
0.00
986
0.00
966
0.00
946
0.00
95
0.00
95
0.00
865
0.00
825
0.00
775
0.00
727
0.00
717
0.00
77
0.00
77
0.00
77
0.00
697
0.00
697
0.00
687
0.00
681
0.00
757
0.00
727
0.00
727
0.00
717
0.00
717
0.00
715
0.00
685
0.00
635
0.00
585
0.00
535
0.00
481
0.00
531
0.00
591
0.00
677
0.00
671
0.00
471
0.00
358
0.00
38
0.00
38
0.00
38
0.00
38
0.00
38
0.00
31
0.00
361
0.00
445
0.00
425
0.00
385
0.00
328
0.00
38
0.00
318
0.00
31
0.00
241
0.00
139
0.00
19
0.00
11
0.00
171
0.00
288
0.00
38
0.00
38
0.00
38
0.00
38
0.00
35
0.00
259
0.00
19
0.00
19
0.00
19
0.00
110
3e-0
41
8e-0
49
0.00
19
0.00
19
0.00
19
0.00
19
0.00
19
0.00
19
0.00
19
0.00
1
0.14
640.
1542
0.16
290.
1727
0.18
380.
1965
0.21
130.
2288
0.25
0.27
640.
311
0.36
130.
5
kεFabs
kεFabs
kεFabs
kεFabs
kεFabs
kεFabs
kεFabs
kεFabs
kεFabs
kεFabs
kεFabs
kεFabs
kεFabs
30.
0281
30.
0276
30.
0269
30.
0262
30.
0252
30.
0241
30.
0227
30.
0209
30.
0186
40.
016
40.
0146
40.
0112
90.
0024
00.
0185
00.
018
00.
0175
00.
0169
40.
0164
40.
0166
40.
0167
40.
0168
40.
0166
30.
0155
30.
0114
10.
0079
20.
0024
20.
0172
20.
0161
40.
0159
40.
0161
00.
0161
00.
0153
00.
0143
00.
0131
00.
0116
10.
0108
10.
0099
50.
0068
100.
0015
40.
0154
40.
0156
20.
0149
20.
0135
20.
012
20.
0104
10.
0106
10.
0109
10.
011
00.
0098
00.
0076
30.
0057
10.
0015
60.
0088
60.
0086
10.
0088
10.
0093
10.
0098
10.
0102
20.
0085
20.
0063
70.
0055
70.
005
50.
0061
00.
0047
30.
0015
10.
0079
10.
0084
60.
0083
60.
008
60.
0077
60.
0072
60.
0067
60.
006
60.
0051
50.
0048
70.
0042
20.
0043
80.
0015
70.
0066
70.
0066
70.
0065
70.
0064
70.
0063
70.
0062
70.
006
70.
0058
20.
0039
60.
0039
80.
0024
70.
0027
68e
-04
80.
003
80.
003
80.
003
80.
003
80.
003
80.
0029
80.
0029
80.
0029
50.
0036
80.
0026
60.
0024
80.
0017
58e
-04
50.
0021
50.
0017
50.
0012
99e
-04
99e
-04
99e
-04
50.
0015
50.
0025
80.
0028
20.
0012
20.
0017
95e
-04
76e
-04
90.
001
90.
001
99e
-04
57e
-04
102e
-04
57e
-04
99e
-04
99e
-04
99e
-04
99e
-04
98e
-04
62e
-04
46e
-04
Tab
le3:
Tab
leof
the10
largesterrors,ε
Fabsan
dwhichkiscomes
from
,for
everypi,un
dernpq
=3
withcontinuity
correction
.
22
p=
0.01
0.02
0.03
0.03
990.
0499
0.05
970.
0698
0.07
990.
0893
0.09
910.
109
0.11
960.
129
0.13
810.
1487
0.15
840.
1696
0.17
920.
1899
kεFrel
kεFrel
kεFrel
kεFrel
kεFrel
kεFrel
kεFrel
kεFrel
kεFrel
kεFrel
kεFrel
kεFrel
kεFrel
kεFrel
kεFrel
kεFrel
kεFrel
kεFrel
kεFrel
016
.973
2017
.746
8018
.566
6019
.428
4020
.343
5021
.301
5022
.335
4023
.436
70
24.5
15025
.714
4026
.986
8028
.440
4029
.814
6031
.217
5032
.939
10
34.6
254
036
.684
038
.551
90
40.7
878
13.
5794
13.
7342
13.
8973
14.
0678
14.
2478
14.
435
14.
636
14.
8487
15.
0558
15.
2848
15.
5263
15.
8005
16.
0582
16.
3197
16.
6388
16.
9494
17.
3262
17.
6659
18.
0701
21.
1123
21.
1665
21.
2233
21.
2825
21.
3447
21.
4092
21.
4781
21.
5507
21.
621
21.
6985
21.
7799
21.
8718
21.
9578
22.
0447
22.
1502
22.
2524
22.
3758
22.
4865
22.
6176
30.
3227
30.
3473
30.
373
30.
3998
30.
4278
30.
4568
30.
4877
30.
5202
30.
5516
30.
5861
30.
6222
30.
6629
30.
7009
30.
7391
30.
7855
30.
8302
30.
884
30.
9322
30.
989
70.
2216
70.
2213
70.
2209
70.
2203
70.
2195
70.
2186
70.
2176
80.
2178
80.
2184
80.
219
80.
2194
80.
2197
80.
2199
80.
2199
40.
2364
40.
2592
40.
2867
40.
3112
40.
3401
80.
2097
80.
2111
80.
2125
80.
2137
80.
2149
80.
216
80.
2169
70.
2163
70.
2149
70.
2133
70.
2114
70.
2092
90.
2094
40.
2127
80.
2197
80.
2193
80.
2187
80.
2179
90.
2187
60.
2064
60.
2036
60.
2007
60.
1975
60.
194
90.
1944
90.
1968
90.
1991
90.
2012
90.
2034
90.
2055
90.
2076
70.
207
90.
2111
90.
2129
90.
2145
90.
2161
90.
2174
80.
2169
90.
1817
90.
1843
90.
1869
90.
1895
90.
1919
60.
1904
60.
1864
60.
1821
60.
1779
100.
17510
0.17
8210
0.18
164
0.19
317
0.20
477
0.20
187
0.19
8810
0.19
6910
0.19
9710
0.20
2710
0.14
5710
0.14
8910
0.15
2210
0.15
5510
0.15
8810
0.16
210
0.16
5410
0.16
8710
0.17
186
0.17
316
0.16
814
0.17
3710
0.18
4610
0.18
7410
0.19
0710
0.19
367
0.19
517
0.19
167
0.18
745
0.14
545
0.13
95
0.13
225
0.12
5111
0.12
1711
0.12
5311
0.12
911
0.13
2811
0.13
6411
0.14
014
0.15
286
0.16
236
0.15
6811
0.15
5311
0.15
9511
0.16
3311
0.16
7711
0.17
1511
0.17
57
0.19
790.
2066
0.21
630.
2269
0.23
890.
2454
0.25
980.
2678
0.27
640.
2857
0.29
590.
307
0.31
940.
3333
0.34
920.
3679
0.39
090.
4219
0.5
kεFrel
kεFrel
kεFrel
kεFrel
kεFrel
kεFrel
kεFrel
kεFrel
kεFrel
kεFrel
kεFrel
kεFrel
kεFrel
kεFrel
kεFrel
kεFrel
kεFrel
kεFrel
kεFrel
042
.539
5044
.554
3046
.895
6049
.648
4052
.929
7054
.817
9059
.225
70
61.8
19064
.736
10
68.0
41071
.815
90
76.1
68081
.239
8087
.225
2094
.395
1010
3.13
77011
4.01
9012
7.79
4013
8.61
761
8.38
51
8.74
541
9.16
181
9.64
85110
.224
7110
.554
5111
.319
71
11.7
67112
.267
7112
.832
2113
.473
3114
.208
1115
.058
9116
.055
8117
.240
51
18.6
718
120
.433
5122
.628
61
24.1
341
22.
7193
22.
8352
22.
9685
23.
1235
23.
306
23.
412
3.65
012
3.78
972
3.94
542
4.12
012
4.31
762
4.54
282
4.80
212
5.10
412
5.46
042
5.88
732
6.40
722
7.04
462
7.40
283
1.03
293
1.08
283
1.14
31.
2063
31.
2841
31.
3283
31.
4299
31.
4887
31.
5541
31.
6273
31.
7097
31.
8032
31.
9105
32.
0347
32.
1804
32.
3537
32.
5626
32.
8144
32.
9154
40.
3625
40.
3878
40.
4168
40.
4504
40.
4897
40.
5119
40.
5631
40.
5926
40.
6253
40.
6619
40.
703
40.
7495
40.
8026
40.
8639
40.
9355
41.
0201
41.
1211
41.
2405
41.
2619
90.
2196
90.
2204
90.
2212
90.
2218
90.
2224
90.
2226
90.
2227
90.
2227
100.
2229
100.
2245
50.
2381
50.
2637
50.
293
50.
3267
50.
3659
50.
412
50.
4665
50.
5294
50.
5199
80.
216
80.
2148
80.
2133
100.
2124
100.
2152
100.
2166
100.
2197
100.
2212
90.
2225
90.
2222
100.
2262
100.
2279
100.
2297
100.
2315
100.
2334
110.
2371
110.
2443
110.
2554
120.
3121
100.
2049
100.
2072
100.
2097
80.
2114
80.
209
80.
2075
80.
2039
110.
2049
110.
2079
50.
2154
90.
2217
90.
2209
110.
2222
110.
2265
110.
2314
100.
2357
100.
2388
120.
25111
0.31
127
0.18
4111
0.18
2211
0.18
5911
0.18
9911
0.19
4411
0.19
6811
0.20
28
0.20
178
0.19
9211
0.21
1111
0.21
4511
0.21
829
0.21
999
0.21
859
0.21
6812
0.22
4912
0.23
5510
0.24
4913
0.30
2
Tab
le4:
Tab
leof
the10
largesterrors,ε F
relan
dwhich
kis
comes
from
,foreverypi,
undernpq
=10
witho
utcontinuity
correction
.
23
p=
0.01
0.02
0.03
0.03
990.
0499
0.05
970.
0698
0.07
990.
0893
0.09
910.
109
0.11
960.
129
0.13
810.
1487
0.15
840.
1696
0.17
920.
1899
kεFrel
kεFrel
kεFrel
kεFrel
kεFrel
kεFrel
kεFrel
kεFrel
kεFrel
kεFrel
kεFrel
kεFrel
kεFrel
kεFrel
kεFrel
kεFrel
kεFrel
kεFrel
kεFrel
029
.720
8031
.197
3032
.770
9034
.434
70
36.2
120
38.0
840
40.1
172
042
.297
044
.445
40
46.8
510
49.4
206
052
.378
20
55.1
954
058
.092
10
61.6
752
065
.214
10
69.5
729
073
.564
50
78.3
871
16.
4725
16.
7619
17.
0685
17.
3907
17.
7327
18.
0908
18.
4774
18.
8893
19.
2928
19.
7419
110
.218
81
10.7
642
111
.280
51
11.8
083
112
.457
21
13.0
941
113
.873
51
14.5
828
115
.434
42
2.29
232
2.39
262
2.49
842
2.60
912
2.72
612
2.84
82
2.97
892
3.11
782
3.25
322
3.40
322
3.56
172
3.74
212
3.91
22
4.08
492
4.29
642
4.50
32
4.75
452
4.98
222
5.25
423
0.97
073
1.01
663
1.06
483
1.11
513
1.16
813
1.22
323
1.28
223
1.34
453
1.40
513
1.47
213
1.54
253
1.62
243
1.69
753
1.77
353
1.86
633
1.95
663
2.06
63
2.16
473
2.28
224
0.42
474
0.44
914
0.47
474
0.50
144
0.52
954
0.55
864
0.58
984
0.62
274
0.65
474
0.68
994
0.72
694
0.76
894
0.80
814
0.84
794
0.89
634
0.94
334
1.00
024
1.05
144
1.11
215
0.16
655
0.18
045
0.19
515
0.21
045
0.22
655
0.24
335
0.26
125
0.28
025
0.29
865
0.31
895
0.34
035
0.36
455
0.38
735
0.41
025
0.43
825
0.46
545
0.49
825
0.52
785
0.56
299
0.04
516
0.04
686
0.05
536
0.06
426
0.07
366
0.08
356
0.09
416
0.10
546
0.11
646
0.12
866
0.14
146
0.15
66
0.16
986
0.18
376
0.20
086
0.21
736
0.23
746
0.25
556
0.27
718
0.04
399
0.04
479
0.04
429
0.04
369
0.04
289
0.04
199
0.04
089
0.03
949
0.03
810
0.03
727
0.03
737
0.04
617
0.05
447
0.06
297
0.07
337
0.08
357
0.09
67
0.10
737
0.12
086
0.03
878
0.04
28
0.03
988
0.03
7510
0.03
7110
0.03
7310
0.03
7510
0.03
7510
0.03
749
0.03
6310
0.03
6810
0.03
6210
0.03
5610
0.03
4810
0.03
3610
0.03
2411
0.03
0611
0.03
038
0.03
5410
0.03
5310
0.03
5910
0.03
6410
0.03
688
0.03
498
0.03
218
0.02
911
0.02
7811
0.02
847
0.02
979
0.03
449
0.03
2111
0.03
0411
0.03
0711
0.03
0811
0.03
0810
0.03
0610
0.02
8911
0.02
98
0.19
790.
2066
0.21
630.
2269
0.23
890.
2454
0.25
980.
2678
0.27
640.
2857
0.29
590.
307
0.31
940.
3333
0.34
920.
3679
0.39
090.
4219
0.5
kεFrel
kεFrel
kεFrel
kεFrel
kεFrel
kεFrel
kεFrel
kεFrel
kεFrel
kεFrel
kεFrel
kεFrel
kεFrel
kεFrel
kεFrel
kεFrel
kεFrel
kεFrel
kεFrel
082
.198
9086
.619
5091
.804
4097
.966
1010
5.40
31010
9.72
78011
9.95
26012
6.05
3013
2.99
07014
0.94
88015
0.16
82016
0.97
31017
3.81
13018
9.32
27020
8.46
03023
2.72
78026
4.70
54030
9.51
34038
2.97
11116
.103
8116
.876
11
17.7
77118
.841
21
20.1
171
120
.855
21
22.5
896
123
.617
91
24.7
821
126
.110
91
27.6
424
129
.427
41
31.5
357
134
.066
91
37.1
685
141
.072
71
46.1
771
53.2
711
64.8
288
25.
4669
25.
7115
25.
9954
26.
3291
26.
727
26.
9561
27.
4919
27.
8079
28.
1642
28.
5694
29.
0342
29.
5734
210
.207
22
10.9
642
11.8
862
13.0
394
214
.537
52
16.6
055
219
.961
33
2.37
383
2.47
873
2.60
013
2.74
233
2.91
113
3.00
793
3.23
353
3.36
63
3.51
53
3.68
393
3.87
693
4.10
013
4.36
123
4.67
183
5.04
853
5.51
743
6.12
333
6.95
553
8.30
44
1.15
944
1.21
344
1.27
584
1.34
874
1.43
54
1.48
444
1.59
924
1.66
654
1.74
194
1.82
734
1.92
464
2.03
684
2.16
774
2.32
34
2.51
074
2.74
364
3.04
364
3.45
434
4.12
075
0.59
025
0.62
145
0.65
735
0.69
935
0.74
95
0.77
745
0.84
335
0.88
195
0.92
525
0.97
45
1.02
975
1.09
385
1.16
865
1.25
75
1.36
385
1.49
625
1.66
635
1.89
915
2.27
886
0.29
396
0.31
316
0.33
526
0.36
126
0.39
196
0.40
946
0.45
036
0.47
426
0.50
16
0.53
146
0.56
596
0.60
576
0.65
226
0.70
726
0.77
366
0.85
66
0.96
26
1.10
746
1.34
667
0.13
147
0.14
357
0.15
767
0.17
417
0.19
387
0.20
57
0.23
147
0.24
687
0.26
427
0.28
397
0.30
657
0.33
257
0.36
297
0.39
97
0.44
287
0.49
737
0.56
767
0.66
477
0.82
678
0.04
198
0.04
958
0.05
848
0.06
898
0.08
158
0.08
888
0.10
598
0.11
68
0.12
748
0.14
058
0.15
548
0.17
288
0.19
338
0.21
778
0.24
758
0.28
488
0.33
348
0.40
18
0.51
6411
0.02
9211
0.02
8511
0.02
7411
0.02
612
0.02
4712
0.02
449
0.03
489
0.04
129
0.04
869
0.05
719
0.06
79
0.07
869
0.09
239
0.10
899
0.12
949
0.15
539
0.18
959
0.23
789
0.32
28
Tab
le5:
Tab
leof
the10
largesterrors,ε
Frelan
dwhichkiscomes
from
,for
everypi,un
dernpq
=10
withcontinuity
correction
.
24
p=
0.01
0.01
990.
0297
0.03
950.
0493
0.05
90.
0685
0.07
950.
089
0.09
780.
1086
0.11
720.
1273
0.13
94
kεFrel
kεFrel
kεFrel
kεFrel
kεFrel
kεFrel
kεFrel
kεFrel
kεFrel
kεFrel
kεFrel
kεFrel
kεFrel
kεFrel
00.
514
00.
533
00.
5523
00.
5721
00.
5924
00.
6132
00.
634
00.
6588
00.
6809
00.
7019
00.
7283
00.
7501
00.
7764
00.
8088
20.
0856
20.
0842
20.
0828
20.
0813
20.
0797
20.
078
20.
0762
20.
0741
20.
0721
20.
0702
20.
0678
20.
0657
20.
0631
10.
0634
30.
0524
30.
0527
30.
053
30.
0533
30.
0536
30.
0538
30.
054
30.
0541
30.
0542
30.
0542
30.
0542
30.
0542
30.
0541
20.
0599
10.
0243
10.
0189
10.
0132
40.
0133
40.
0141
40.
0148
40.
0155
10.
0181
10.
0247
10.
031
10.
039
10.
0455
10.
0535
30.
0538
60.
0121
60.
012
40.
0126
60.
0117
60.
0115
60.
0113
60.
0111
40.
0163
40.
017
40.
0177
40.
0184
40.
0191
40.
0198
40.
0206
40.
0112
40.
0119
60.
0118
50.
0085
50.
008
50.
0075
10.
0108
60.
0109
60.
0107
60.
0105
60.
0103
60.
0101
60.
0098
60.
0095
50.
0099
50.
0094
50.
009
10.
0075
70.
0072
70.
0072
70.
0071
70.
0071
70.
0071
70.
007
70.
007
70.
0069
70.
0069
70.
0068
70.
0073
70.
0073
70.
0072
70.
0072
80.
003
10.
0046
50.
007
50.
0064
50.
0059
50.
0054
50.
0048
50.
0043
50.
0037
80.
003
80.
0031
80.
0031
80.
0031
80.
0031
10.
0015
80.
003
80.
003
80.
003
80.
003
80.
003
80.
003
80.
003
80.
003
50.
0029
90.
001
90.
001
90.
001
90.
001
90.
001
90.
001
90.
001
90.
001
90.
001
90.
001
90.
001
90.
001
90.
001
9
0.14
640.
1542
0.16
290.
1727
0.18
380.
1965
0.21
130.
2288
0.25
0.27
640.
311
0.36
130.
5
kεFrel
kεFrel
kεFrel
kεFrel
kεFrel
kεFrel
kεFrel
kεFrel
kεFrel
kεFrel
kεFrel
kεFrel
kεFrel
00.
8281
00.
8499
00.
8748
00.
9036
00.
9373
00.
9773
01.
0255
01.
085
01.
1605
01.
2602
01.
3994
01.
6142
02.
0641
10.
0692
10.
0759
10.
0836
10.
0925
10.
1029
10.
1153
10.
1304
10.
1491
10.
173
10.
2048
10.
2498
10.
3203
10.
4769
20.
0579
20.
0557
30.
0531
30.
0526
30.
052
30.
0512
30.
053
0.04
843
0.04
593
0.04
23
0.03
542
0.04
362
0.12
273
0.05
363
0.05
342
0.05
32
0.04
992
0.04
622
0.04
172
0.03
612
0.02
94
0.02
634
0.02
684
0.02
664
0.02
43
0.02
40.
0211
40.
0216
40.
0222
40.
0228
40.
0234
40.
0241
40.
0249
40.
0256
20.
0196
20.
0068
20.
0122
30.
0226
40.
0031
60.
0093
60.
0091
60.
0088
60.
0085
60.
0081
60.
0077
60.
0071
60.
0064
70.
0056
50.
0061
50.
0081
50.
0099
90.
0024
70.
0068
70.
0067
70.
0066
70.
0066
70.
0065
70.
0063
70.
0062
70.
006
60.
0055
70.
0051
70.
0043
70.
0029
50.
002
80.
003
80.
003
80.
003
80.
003
80.
003
80.
003
80.
0029
50.
003
50.
0044
60.
0043
60.
0026
80.
0018
80.
0016
50.
0025
50.
002
50.
0014
99e
-04
99e
-04
99e
-04
50.
0018
80.
0029
80.
0028
80.
0027
80.
0024
95e
-04
100.
0015
Tab
le6:
Tab
leof
the10
largesterrors,ε
Frelan
dwhichkiscomes
from
,for
everypi,un
dernpq
=3
withcontinuity
correction
.
25
