in 1994 , waldinger et al .   introduced the intravaginal ejaculation latency time ( ielt ) as a measure for the ejaculation time of heterosexual intercourse .
the ielt is defined as the time from the moment of vaginal penetration until the moment of intravaginal ejaculation . in case of ejaculation outside the vagina
the most accurate way to measure the ielt is the use of a stopwatch handled by the female partner . in a stopwatch study in a cohort of dutch men with lifelong premature ejaculation
, it was shown that the ielt had a skewed distribution and that 85% of men ejaculated within 1 minute after penetration .
in addition , in a meta - analysis of all drug treatment studies of premature ejaculation from 1943 to 2003 , it was shown that prospective real - time stopwatch measurement of the ielt during selective serotonin reuptake inhibitors ( ssri ) treatment led to a smaller , and therefore more valid , confidence interval of the ielt compared to retrospective questionnaire studies of the ielt . in 2005 ,
a prospective stopwatch study of 491 men of the general male population in 5 countries ( the netherlands , united kingdom , spain , turkey , and the united states ) also showed a skewed distribution to the right with a median ielt of 5.4 minutes ( range , 0.5544.1 minutes ) . using a blinded timer device ,
a second study was performed in 2009 in a new group of 474 men of the general population in the same countries .
this study also showed a remarkable similar skewed distribution to the right , with a median ielt of 6.0 minutes ( range , 0.152.7 minutes ) .
the precise and similar method of ielt measurement by a stopwatch on various intercourse events in the 3 aforementioned studies facilitates comparison of their ielt distributions .
however , a prerequisite for further research of the ejaculation time , is a better knowledge of the type of distribution that is formed by the ielt in different populations of men .
the purpose of the current study , was to investigate which type of well - known mathematical distribution fitted to the ielt distribution of men with lifelong pe and of men in the general male population .
by application of the statistical program easy fit professional ver . 5.6 ( mathwave technologies inc . ) , we analysed the ielt data of 2 previous stopwatch studies of the ielt in the general male population   ( fig .
1a , b ) and of the ielt data of a previous stopwatch study in a clinical cohort of men with lifelong premature ejaculation ( fig .
in addition , we investigated which of the wellknown mathematical probability distributions , fitted most accurately to the curves of the aforementioned ielt distributions . this fitness of the distribution is expressed by the goodness of fit ( gof ) , which is calculated by the kolmogorov - smirnov test ( ks test )   the gof therefore is a measure for the accuracy of the fitness .
the ks test is a non parametric statistical test of the equality of continuous , one - dimensional probability distributions . in the current study
it was used to compare the ielt distributions with various well - known theoretical reference probability distributions ( one - sample ks test ) .
the difference between the actual measured ielt data ( as previously published in the three stopwatch studies ) and the theoretical mathematical distribution models is measured with the kolmogorov smirnov test . in other words , the ks test measures the gof .
the smaller the difference between the theoretical distribution and the ielt distribution , the more accurate is the fitness of the theoretical distributions on the ielt distributions .
gof , measured by the ks test showed that the ielt distributions in the 2 studies of the ielt in the general male population ( the 2005 study and the 2009 study )   and in the ielt distribution of dutch caucasian men with lifelong pe   were a gamma - distribution , which is characterized by ( 1 ) a boundary on the left at zero , and therefore excluding negative ielt values , ( 2 ) a positively skewed shape , including high ielt values , and ( 3 ) a significant flexibility in the shape , allowing the gamma distribution any number of ielts , with reasonable accuracy .
of the multiple operationalized gamma - distributions , the lognormal distribution fitted most well on the gamma - ielt distribution in the general male population .
the combined lognormal distribution of men in the general male population ( combination of ielt study 2005 and 2009 )   is shown in fig .
the form of this lognormal distribution is mirrored by the mathematical formula , shown in fig .
3 . in contrast , the gumbel max distribution fitted most well on the gamma - distribution of the ielt in the cohort of men with lifelong premature ejaculation ( fig .
the form of this gumbel max distribution is mirrored by the mathematical formula , shown in fig .
according to kolmogorov smirnov the gof of the lognormal distribution for the 2005 study was 0.11708 , and for the 2009 study the gof was 0.06284 .
the gof of the combined lognormal distribution of both the 2005 and 2009 study was 0.057 . according to kolmogorov smirnov
the gof of the gumbel max distribution of the ielt distribution in the 1998 study of men with lifelong pe was 0.179 . by using the lognormal distribution , the various ielt values of all men in the general male population of the 2005 study ,
the 2009 study and their combination were calculated ( table 1 ) . by using the gumbel max distribution ,
the various ielt values of all men in the 1998 study of dutch caucasian men with lifelong pe   were calculated ( table 2 ) .
table 3 shows that in the general male population 1.08% of men ejaculates within 0 to 60 seconds after vaginal penetration ( in 2005 study , 1.23% ; in 2009 study , 0.99% ; in combined study , 1.08% ) .
in contrast , in the 1998 study of men with lifelong pe , 85.4% ejaculates within 1 minute , and 55.1% ejaculates within 30 seconds .
notably , in the general male population 7.1% of men ejaculates between 1 and 2 minutes .
in contrast in men with lifelong pe , 14% of men ejaculates within 1 and 2 minutes .
2 and 4 show that the ielt distribution of the cohort of men with lifelong pe is not represented in the ielt distribution of the men in the general male population .
this is also shown by the ielt data , as represented in table 3 , which shows that 85.4% of men ejaculates within 1 minute in men with lpe , whereas only 1.08% of men in the general male population ejaculates within 1 minute .
by investigating which theoretical mathematical distribution fitted most accurately on the ielt distributions of 3 previously published ielt distributions , it appeared that the 3 ielt distributions were gamma distributions , but that the type of gamma distribution of men in the general male population does not fit in the gamma distribution of men with lifelong pe .
therefore this study showed that men with lifelong pe not only differ from men in the general population in their ielt values , e.g. , men with lifelong pe have ielts of less than 1 minute , but also in the mathematical type , e.g. , curve of their ielt distribution . in order to investigate the ielt distributions , we investigated which of the well - known mathematically described distributions fitted most accurately to the ielt distribution of men in the general male population and of men with lifelong pe .
for that purpose we summarized the data of 2 prospective real - time stopwatch studies of caucasian men in the general population , performed in the same five countries ( the netherlands , united kingdom , spain , turkey , and the united states ) , to one set of ielt data , including 965 males , and compared this ielt distribution with the ielt distribution of a cohort of 117 dutch caucasian men with lifelong pe . in the current study , it was found by application of the ks test on various mathematical distributions that the ielt distribution of the males in the general male population most accurately fitted the lognormal distribution , which belongs to the family of gamma distributions .
however , the ielt distribution of males with lifelong pe did not fit to the lognormal distribution .
importantly , the ielt distribution of the males with lifelong pe most accurately fitted to the gumbel max distribution , which , on its turn , inaccurately fits to the ielt distribution of men in the general male population .
for example , whereas only a neglectable minority of men in the general male population ejaculates within 1 minute ( e.g. , 1.08% ) and within 30 seconds after vaginal penetration ( e.g. , 0.08% ) , the current study shows that the opposite occurs in men with lifelong pe . in these men , the majority ejaculates within 30 seconds ( e.g. , 55.1% ) and within 30 to 60 seconds ( e.g. , 30.3% ) . in other words , there are significantly more men with lifelong pe who ejaculate within 30 seconds than can be extrapolated from the probability density curve of the ielt distribution of men in the general population .
an advantage of the formula for the lognormal distribution , is that it enables the calculation of the number and percentage of men in a specific population with a specific ielt value .
similarly , based on the gumbel max distribution , the number and percentage of men with a specific ielt value can be calculated in a cohort of men with lifelong pe .
our calculations by means of the formula of the lognormal distribution ( table 1 ) showed a striking similarity of the percentages of males with a specific ielt value in the 2 separate studies of men in the general population , supporting the reliability and validity of stopwatch measurement of the ielt in an epidemiological study of the ielt .
notably , as the 2 general population ielt stopwatch studies , e.g. , the 2005 and 2009 studies , have been conducted in caucasian men , a limitation of the current study is that the formula may only be valid for caucasian men in western countries , whereas for african , middle east , and far east asian countries our formula remains to be investigated on the condition that similar ielt stopwatch studies in random samples of their general populations have been conducted .
one may argue that another limitation of the current study is the number of men with lifelong pe .
however , a simple calculation shows that in order to discover the gumbel max ielt distribution of men with lifelong pe in the lognormal ielt distribution in the general male population , one has to multiply the number of men with lifelong pe by 100 .
1c could be extracted out of a general male population of 11.000 men ( 1% = 1/100 of the population ) . as it is impossible to perform a stopwatch study of the ielt in 11.000 males
still , with a few hundreds of men with lifelong pe who have measured their ielt prospectively with a stopwatch it is argued that the current formula of the gumbel max ielt distribution can be become more accurate for application in a cohort of men with lifelong pe .
interestingly , based on the formula of the lognormal distribution that most accurately describes the ielt data of men in the general male population , it is possible to answer certain questions regarding the ielt of men in the general male population without the necessity of conducting realtime stopwatch measurements in very large samples of men .
for example , according to the statistics netherlands ( centraal bureau voor de statistiek or cbs ) , the netherlands currently has 16.982.433 inhibitants and 8.372.858 of them are males . of these males , 5.912.984 are aged between 18 and 70 years .
application of our lognormal formula shows that in the netherlands there are 70.305 men ( 1.19% ) who ejaculate within 1 minute , and 48.546 ( 0.82% ) men who ejaculate within 30 seconds .
in addition , using this formula , 502.012 ( 8.5% ) men ejaculate between 0 and 2 minutes , 2.419.002 men ( 40.9% ) who ejaculate between 1 and 5 minutes , and 1.971.330 ( 33.3% ) men who ejaculate between 5 and 10 minutes .
in addition , there are 1.112.291 men ( 18.8% ) who ejaculate between 10 and 20 minutes .
notably , after 20 minutes , the number and percentages decrease : 233.208 men ( 3.9% ) ejaculate between 20 and 30 minutes and 66.225 men ( 1.12% ) ejaculate between 30 and 40 minutes .
a minority of 40.633 ( 0.687% ) ejaculates with an ielt of more than 40 minutes .
the application of the formula , which describes the probability density of the ielt , shows that it may provide answers to questions that can not be measured by a stopwatch , due to the extremely large number of men that otherwise ought to be included in an epidemiological study , or just can not be measured as the ielt duration is too long and potentially painful for the female partner of a subject participating in a scientific study .
notably , it is questioned why the ielt distribution of the men with lifelong pe is so completely different from the ielt distribution of men in the general male population .
further research into this question is warranted and is presumably associated with genetic and epi - genetic factors . according to the recent classification by waldinger and schweitzer , there are 4 pe subtypes , e.g. , lifelong pe , acquired pe , subjective pe , and variable pe .
their prevalence in the general population differs significantly , as has been shown by serefoglu et al .   and gao et al . , with the lowest prevalence of lifelong pe .
notably , lifelong pe is characterized by a hypertonic state , consisting of premature ejaculation ( ejaculatio praecox ) , a facilitated erection ( erectio praecox ) and a facilitated penile detumescence ( detumescentia praecox ) , when the male is becoming involved in an erotic situation .
the current study shows that lifelong pe not only differs from the 3 other pe subtypes with regard to this triad of symptoms , but that lifelong pe has a different ielt and a different ielt distribution than males in the general population , of which men with variable and subjective pe are part of .
the separate state of lifelong pe among the 3 other pe subtypes with regard to the ielt and its ielt distribution has consequences for the methodology and design of drug treatment studies . as has recently been argued by waldinger , the method and design for drug treatment studies of men with lifelong pe who ejaculate within seconds , are not required for drug treatment studies of men with variable and subjective pe with normal ielt values .
the ielt distribution of men in the general population is most accurately fitted by the lognormal distribution , whereas the ielt distribution of men with lifelong pe is most accurately fitted by the gumbel max distribution .
as there are significantly more men with lifelong pe who ejaculate within 30 and 60 seconds than can be extrapolated from the probability density curve of the ielt distribution of men in the general population it is concluded that men with lifelong pe have a distinct ielt distribution that can only be retrieved from the general male population ielt distribution when thousands of men would participate in a stopwatch study of the ielt .
the cause of the difference in ielt distribution should be further investigated and may be due to genetic and epigenetic factors .
application of the mathematical formula of the lognormal ielt distribution is useful for epidemiological research of the ielt in large populations of men .
 purposeto find the most accurate mathematical description of the intravaginal ejaculation latency time ( ielt ) distribution in the general male population.materials and methodswe compared the fitness of various well - known mathematical distributions with the ielt distribution of two previously published stopwatch studies of the caucasian general male population and a stopwatch study of dutch caucasian men with lifelong premature ejaculation ( pe ) . 
 the accuracy of fitness is expressed by the goodness of fit ( gof ) . 
 the smaller the gof , the more accurate is the fitness.resultsthe 3 ielt distributions are gamma distributions , but the ielt distribution of lifelong pe is another gamma distribution than the ielt distribution of men in the general male population . 
 the lognormal distribution of the gamma distributions most accurately fits the ielt distribution of 965 men in the general population , with a gof of 0.057 . 
 the gumbel max distribution most accurately fits the ielt distribution of 110 men with lifelong pe with a gof of 0.179 . 
 there are more men with lifelong pe ejaculating within 30 and 60 seconds than can be extrapolated from the probability density curve of the lognormal ielt distribution of men in the general population.conclusionsmen with lifelong pe have a distinct ielt distribution , e.g. , a gumbel max ielt distribution , that can only be retrieved from the general male population lognormal ielt distribution when thousands of men would participate in a ielt stopwatch study . the mathematical formula of the lognormal ielt distribution is useful for epidemiological research of the ielt . 