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Bringing Closure to the Plotting PositionControversyLasse Makkonen aa VTT Technical Research Centre of Finland , Espoo, FinlandPublished online: 30 Jan 2008.

To cite this article: Lasse Makkonen (2008) Bringing Closure to the Plotting Position Controversy,Communications in Statistics - Theory and Methods, 37:3, 460-467, DOI: 10.1080/03610920701653094

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Communications in Statistics—Theory and Methods, 37: 460–467, 2008Copyright © Taylor & Francis Group, LLCISSN: 0361-0926 print/1532-415X onlineDOI: 10.1080/03610920701653094

Extreme Value Analysis and Order Statistics

Bringing Closure to the PlottingPosition Controversy

LASSE MAKKONEN

VTT Technical Research Centre of Finland, Espoo, Finland

In this article, it is explicitly demonstrated that the probability of non exceedance ofthe mth value in n order ranked events equals m/�n+ 1�. Consequently, the plottingposition in the extreme value analysis should be considered not as an estimate, butto be equal to m/�n+ 1�, regardless of the parent distribution and the application.The many other suggested plotting formulas and numerical methods to determinethem should thus be abandoned. The article is intended to mark the end of thecentury-long controversial discussion on the plotting positions.

Keywords Cumulative distribution function; Extreme value analysis; Orderranking; Plotting positions; Probability paper.

Mathematics Subject Classification 62G30; 62G32.

1. Introduction

Plotting positions, i.e., the cumulative probabilities that are associated with order-ranked data, are essential in estimating the probability of extreme events. Duringthe long history of discussion on the plotting positions in scientific literature, morethan ten plotting formulas and some numerical methods have been presented. In1984, a review on the plotting positions was given by Harter (1984). In additionto presenting an extensive historical review on the subject, some conclusions weremade. They were:

1. The optimum choice of plotting positions depends upon the purpose of theinvestigation.

2. The optimum choice of plotting positions may also depend upon the distributionof the variable under consideration.

3. One may wish to avoid the difficulties associated with unbiased estimates byobtaining median unbiased estimates instead.

Received October 27, 2006; Accepted April 27, 2007Address correspondence to Lasse Makkonen, VTT Technical Research Centre of

Finland, Box 1000, 02044 VTT, Finland; E-mail: lasse.makkonen@vtt.fi

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Harter’s article was published more than two decades ago, but most of themodern literature and related engineering software adopt the same line of thinking.For example, conclusions 1 and 2 are promoted in the textbooks by Castillo (1988)and Jordaan (2005) and are reflected by presenting a number of optional plottingformulas, e.g., in the NIST Handbook of Statistical Methods (Anon, 2006) andMATLAB�. Also, conclusion 3 is often seen as a viable option (e.g., Castillo, 1988)and was recently promoted by Folland and Anderson (2002).

In the following, it is shown that conclusions 1–3 are all inappropriate. Thecorrect solution to the plotting position problem is outlined and the historicalorigins of the errors that have resulted in various plotting formulas are discussed.

2. Plotting Positions are Unique

The use of the plotting positions in the analysis of extremes originates fromCalifornia Department of Public Works in the late 1800s (Anon, 1923; Galton, 1899;Harter, 1984). Their plotting formula,

Pm = m/n

where m is the rank from the smallest �m = 1� to the largest �m = n� observationand n is the number of observations, is still the starting point of discussions onplotting positions in the modern literature, including text books. To give someexamples, Harter (1984) wrote that “The cumulative probability or cumulativedistribution function cdf of a sample of size n is usually defined as a step functionwhich jumps form �m− 1�/n to m/n”. Castillo’s (1988) version of this is that “Itis well known that the empirical cdf is a step function with steps taking the values0� 1/n� 2/n� � � � � 1�” According to Coles (2001), “For any one of the xm, exactly m ofthe n observations have a value less than or equal to xm, so an empirical estimateof the probability of an observation being less that or equal to xm is m/n”. Similarstatements are found throughout the literature.

The problem then identified by all authors is, following Harter (1984), “if theplotting position m/n is used, the largest value cannot be plotted, while if �m− 1�/nis used , the smallest value cannot be plotted, since the probabilities 1 and 0 are offscale for probability paper constructed for distributions unlimited in extent (−� to�)”. In order to avoid the problem with plotting the smallest or the largest pointby Pm = m/n, Hazen (1914) proposed

Pm = �m− 1/2�/n�

This is still in the modern literature presented as “a compromise plotting positionbetween m/n and �m− 1�/n” (e.g., Castillo, 1988; Jordaan, 2005) and is widely used.For example, Hazen’s formula is the default plotting position in MATLAB�.

Thus, starting from the first developments over a century ago, the optimumplotting position has been considered as a “compromise” or “a slight adjustment”(Coles, 2001) to the cumulative probability Pm = m/n, which cannot be used inpractice. It is no wonder then that the issue of determining the plotting positionshas become an area with apparent room for subjective assessment and that aplethora of plotting formulas have been presented (see Harter, 1984). Illustrativeof the situation are the comments by Langbein (1960) who considers the selection

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462 Makkonen

of plotting positions “like taking a stand on a political question” and by Jordaan(2005) who notes that “there appear to be almost as many opinions as there arestatisticians”.

However, the conclusion that m/n is the correct estimate for the cumulativeprobability of order ranked data is improper. The empirical cdf can be defined as astep function with steps taking the values 0� 1/n� 2/n� � � � � 1 only when the case x >xn is impossible. The number of possible outcomes of selecting the next observationwould then be n and the cumulative probability m/n. This, however, is not the casewhen the data are order ranked because the largest value of a random sample doesnot imply the largest possible value of the variable. On the contrary, the probabilityof x > xn is equal to that of x falling in any other rank xm−1 < x ≤ xm.

Taking the next observation of a variable x from an unbounded continuousdistribution may have the following outcomes in relation to n observationsx1� x2� � � � � xn, ranked in ascending order:

x ≤ x1� x1 < x ≤ x2� � � � � xn−1 < x ≤ xn and x > xn�

The number of possible outcomes of this sampling is n+ 1, not n. Therefore,the empirical cdf is a step function with steps 1/�n+ 1�, not 1/n, and theprobability of an observation being less than or equal to xm is m/�n+ 1�. Itis not m/n, as assumed originally, implied in the above-mentioned quotes, andsuggested by many other sources in the contents of analyzing and plottingorder-ranked data (e.g., http://planetmath.org/encyclopedia/EmpiricalDistributionFunction.html). This error has persisted in the literature probably due to a falsenotion of an analogy with an ogive, i.e., a cdf of grouped data. Only in rareconnection the cdf of order ranked data has been correctly defined (Coles, 2001,

Figure 1. Two definitions of the empirical cdf of order-ranked data x1� x2� � � � � xn as astep function �n = 4�. The step function marked with dashed lines (scale on the left) isthe commonly used definition with steps taking values of 0� 1/n� 2/n� � � � � 1 (see text). Thisdefinition incorrectly rules out the values that are larger than the largest value in a sample,i.e., cdf = 1 when x ≥ xn. The proper definition is the step function marked with solid lines(scale on the right) taking the steps of 1/�n+ 1�� 2/�n+ 1�� � � � � n/�n+ 1�.

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Plotting Positions 463

Definition 2.4) and even then with inappropriate justification. The commonly usedimproper definition of the empirical cdf of order ranked data as a step function, andthe proper one, are shown in Fig. 1.

Thus, the “compromises” or “adjustments” to the plotting positions have beenmade to an inadequate cumulative probability from the beginning (Hazen, 1914) upto this day. The starting point of the discussion on the plotting positions shouldhave been the exact cumulative probability of order ranked data, i.e.,

Pm = m/�n+ 1��

This plotting formula was first proposed by Weibull (1939) and can be usedto plot also the largest and smallest values. It was promoted as a plottingposition in the classical work by Gumbel (1958). It has been insufficiently justified,however. In addition to the arguments above, it will be shown later in thisarticle, specifically, that m/�n+ 1� is the only correct probability plotting position.Accordingly, plotting positions are unique and there is nothing subjective indetermining them. They certainly do not depend on the purpose of the investigation,so that conclusion 1, quoted in Sec. 1, is inappropriate.

3. Plotting Positions are Distribution-Free

Gumbel (1958) showed that m/�n+ 1� is the mean of the cdf of the mth observation(see Sec. 4). The beauty of the result derived by Gumbel is in that it is distribution-free. Nevertheless, conclusion 2, which claims that the plotting formula may dependon the distribution of the variable in question, is still generally made. The originof this line of thinking may be partly in the faulty notion, discussed above, thatthe whole issue is open to subjectivity. However, the main origin of distributiondependent plotting formulas in the literature may be identified as follows.

The plotting formulas are historically intended to be used for plotting onprobability paper where the scale P is transformed by the inverse function � of theanticipated parent cdf in order to obtain a linear fit that is convenient to extrapolate.The ordinate is then P transformed to the so called reduced variate and the abscissais the variable value x. As an example, for the anticipated Gumbel distribution��P� = − ln�− ln�P��, i.e., the transformation is strongly nonlinear.

When one considers sets of sample data then F�xm� is a random variable (seeSec. 4). In the classical Gumbel analysis, the probability Pm is estimated by the meanof F�xm�, i.e., E�F�xm��. A nonlinear transformation � is then applied to that mean,i.e., the plotted reduced variate is ��E�F�xm��.

Kimball (1960), Gringorten (1963), Cunnane (1978), and Harris (1996) haveargued that when the plotting involves a reduced variate then “a more correct”procedure would be to apply the transformation first, and plot the mean value ofthe reduced variate E���F�xm���. This results in a plotting formula

Pm = �−1m �E���F�xm�����

where �−1 is the inverse function of the transformation �. The plotting positionsbased on this approach, in contrast to the classical Gumbel analysis, depend onthe transformation � made and, hence, on the postulated probability distributionfunction of the extremes. The various distribution-tailored plotting formulas and

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methods presented in the literature reflect this situation, i.e., it is believed that theplotting positions depend on the underlying distribution when a reduced variateis involved in the analysis. Such an approach is included, e.g., in the probabilityweighted moments method (Hosking et al., 1985) and some applications of themaximum-likelihood method (e.g., Jones, 1997) and reflected on many other articles(e.g., Arnell et al., 1986; De, 2000; Guo, 1990; Harter, 1984; In-na and Nguyen,1989; Palutikof et al., 1999).

It is noteworthy that the above-mentioned transformation from F�xm� toE���F�xm��� is a different transformation compared to that of the classical Gumbelanalysis, in which the transformation is from E�F�xm�� to ��E�F�xm���. Thedifference arises because the result of taking a mean and making a nonlineartransformation depends on the order in which these operations are applied.Consequently, the linearity, shown by Gumbel (1958) to exist as a result of plotting��E�F�xm���, is lost when E���F�xm��� is being plotted. This linearity and a good fitcan be returned when one knows the underlining distribution (e.g., Barnett, 1975;Cunnane, 1978; Harris, 1996). However, this can only be done by manipulatingthe plotting positions, which thereafter no more represent the probabilities of non-exceedance of the original variable x because

�−1�E���F�x���� �= E�F�x���

when � is a nonlinear function.Therefore, the results of an analysis based on E���F�x���, instead of ��E�F�x���,

cannot be re-transformed to the non exceedance probability P and the purposeof the analysis will not be obtained. If, as is often done in the literature,�−1�E���F�xm���� is interpreted as E�F�xm�� = Pm, an error is made. Thus, thegenerally applied concept of distribution-specific plotting formulas in the extremevalue analysis arises from an improper process of transforming and re-transforming.The correct plotting formula Pm = m/�n+ 1� is independent of the underliningdistribution. Hence, conclusion 2 is inappropriate.

4. Cumulative Probability Equals Mean Plotting Position

The third conclusion in Harter’s (1984) review is based on the idea, suggested byBeard (1943), advocated by Johnson (1951), Benard and Bos-Levenbach (1953),and some Russian publications, and recently promoted in detail by Folland andAnderson (2002), that the median of F�xm� should be used as the plotting position.Folland and Anderson (2002) wrote “a natural estimate for the plotting position isthe median of its probability density distribution”. The use of the mode of F�xm�has also been considered as an option for the of plotting positions (e.g., Castillo,1988; Gumbel, 1939).

The background for the notion in the literature that some “natural estimate” ofPm needs to be chosen subjectively is as follows. Consider a variable x that has aprobability density function f�x� and cdf F�x�. Then a new variable F�xm� relatedto x by order ranking from the smallest �m = 1� to the largest �m = n� value willhave the probability density fm�F�xm�� given by

fm�F�xm�� =n!

��m− 1�!�n−m�!� �F�xm��m−1�1− F�xm��

n−m�

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Plotting Positions 465

where F�xm� is the cdf of the order-ranked values �0 ≤ F�xm� ≤ 1�. As shown alreadyby Gumbel (1958), the mean E of the variable F�xm� is, independently of the parentdistribution,

E�F�xm�� =m

n+ 1�

Thus, the plotting position Pm = m/�n+ 1� is the mean of the parent probabilitydistribution F�xm� associated with the rank m. It is the mean in the followingsense. When there are L independent sets of n values taken from the same parentdistribution, then there will be L individual mth ranked values of xm denoted by xm�L.The mean of F�xm� taken over the ensemble of L samples, i.e., E�F�xm�L��, converges,for a large L, to the value of m/�n+ 1�.

One may, however, derive distribution-free equations for the median and themode of F�xm� as well (Beard, 1943; Gumbel, 1939). The problem which has causeda lot of controversy and confusion then is: Which of these should be used asthe plotting positions, i.e., as the estimates of the cumulative probability Pm? Inaddition to those authors that actually use the median plotting position, manyothers consider it not obvious at all that the plotting position should be the meanE�F�xm��. Jordaan (2005), for example, notes that Gumbel’s distribution-free resultabove is “in fact hard to interpret”. Gumbel (1958) himself justified the use of hisresult for the mean position to be used as the plotting position by writing that “itfulfils the postulates” given elsewhere in his book. These five postulates have laterbeen disputed by many authors causing further notions of subjectivity of the issue.

The root of this problem is that in the literature fm�F�xm�� is commonly calledthe probability of observing the mth-order statistic so that F�xm� is the probabilitythat x takes a value less than or equal to a value xm associated with m. They areoften marked as “fm�Pm�” and “Pm”, so that Gumbel’s distribution-free result for themean is interpreted as “E�P�”. A quote from Castillo (1988) illustrates this popularline of thinking: “If the cdf F�x� were known, every value x would have only oneassociated P = F�x� value. In this case there is no doubt about which point shouldbe plotted on the probability paper because there is only one possibility. However,due to random character of the sample to the mth order statistic, xm, a whole set ofvalues (generally infinitely many) of P can occur.” In other words, the probabilityPm is considered as a random variable and the mean, mode or median of Pm asoptions for its estimate on a probability plot.

The inadequacy here is that when one considers sets of sample data, so thatF�xm� is a random variable, then F�xm� is no more a probability. Probability P is nota random variable but the limiting value of the relative cumulative frequency whenthe number of sample sets L goes to infinity. When F�xm� is considered a randomvariable, it represents, in each set L of n observations, the sample relative cumulativefrequency F�xm�L� = rL�x ≤ xm�. One may call the random variable F�xm�L� “asample estimate of probability”, but that is different from the true probability Pm.The classical definition of statistical probability P is

P�x ≤ xm� = limL→�

rL�x ≤ xm�

where lim denotes the limiting value in terms of statistical convergence. Thisfrequency interpretation of probability is, of course, in harmony with the axiomaticprobability theory via Bernoulli’s rule.

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Taking into account Gumbel’s (1958) result,

Pm = P�x ≤ xm� = limL→�

rL�x ≤ xm� = limL→�

F�xm�L� = E�F�xm�� = m/�n+ 1��

This result outlines three facts:

• The classical question of choosing the “natural estimate of Pm from itsprobability distribution fm�Pm�” is meaningless and cannot be answered sincePm is not a random variable.

• The probability Pm is E�F�xm�� = m/�n+ 1�.• For any analysis of the cdf, order-ranked sample data must be plotted at theirnon exceedance probability Pm, i.e., at m/�n+ 1�.

Thus, it is the mean of F�xm� that is the correct plotting position in the extremevalue analysis and conclusion 3, quoted in Sec. 1, is inadequate.

5. Conclusions

The cumulative probability Pm of non exceedance of the mth value in n orderranked observations equals m/�n+ 1�. This result is unique and independent of theparent distribution. The fundamental purpose of the extreme value analysis is toestimate the cdf by order-ranked sample data, so that the plotting position shouldbe considered not as an estimate, but to be equal to m/�n+ 1�. The numerous otherproposed plotting formulas and methods are based on inappropriate assumptionsand should be abandoned.

Acknowledgments

Thanks are due to M. Pajari for continuous and fruitful discussions and commentson the manuscript. This work was funded by the Environmental Cluster ResearchProgram, Ministry of Environment, Finland and Tekes.

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