The logarithmic normal distribution in relation to the epidemiology of drug abuse


FIGURE - Logarithmic normal curve fitted to drug use scorces of randomly selected Kansas residents
Correlation of mean licit and illicit drug usage for eleven population districts


Author: Diane McDERMOTT , James SCHEURICH
Pages: 13 to 19
Creation Date: 1977/01/01

The logarithmic normal distribution in relation to the epidemiology of drug abuse


Two issues previously raised in this journal, and central to the treatment and prevention of drug abuse are addressed in the present research. The first issue deals with the distribution of the individuals who use drugs and concerns the explanatory power and scientific basis for the assumption that an effective identification of the abuser as distinct from the user can be made. The second issue concerns causal epidemiological factors characteristic of the spread of drug abuse, and examines the relationship between the correlation of the use of licit and illicit drugs.

Smart, Whitehead and Laforest (1971) have noted that on the basis of clinical reports on drug abuse, the distribution of drug use in young persons "might be bi-modal, with most persons average or 'normal' in drug use and a smaller high point in the heavy or "abusing category'". Most drug control programmes designed in response to the epidemiological aspect of illicit drug use have tacitly assumed that such a bi-modal distinction between normal and abusing usage patterns could indeed be made. However, where distributions of drug use have been examined this has not generally been found to be an accurate description. De Lint and Schmidt (1968) first raised questions about the explanatory power of this bi-modal model through an analysis of the distribution of alcohol consumption in Ontario, Canada. In an extensive series of studies on drug use in Canada and Britain, and subsequently reported in the Bulletin on Narcotics, Smart and his associates have strongly suggested that a unimodal and logarithmic normal distribution may more aptly characterize illicit drug consumption (Smart, Whitehead and Laforest, 1971; Smart and Whitehead, 1972; and Smart and Whitehead, 1973). A log-normal distribution curve describes a population in which many persons use few drugs, where fewer are moderate users, and the fewest yet consume a great deal of drugs. The important feature of such a continuum of drug usage is that no clear cut distinction between the average drug user and the drug abuser can be made. Thus it is doubtful if prevention and treatment programmes targeted solely on assumed abusers can ever be maximally effective. A further implication of the log-normal distribution is related to epidemiological factors associated with drug abuse. If drug abusers do not form a distinct group that can be effectively "quarantined" for treatment, but instead represent one end of a continuum of total drug users, this strongly suggests that general drug use in a population is linked with specific drug abuse. In more vivid terms, the sanctioned "normal" use of some drugs may provide a "climate" conducive to the extended heavier use of all drugs.

Data from several sources lend additional support for this proposition. Sargent (1971), using four different national groups, found that degree of acceptance of alcohol and drug use was associated with alcoholism and drug abuse. Single, Kandel and Faust (1974) concluded that illegal use of drugs was associated with legal drug use. Fracchia, Sheppard and Merlis (1974) noted that the use of socially approved drugs creates a high risk for later drug abuse. Kandel (1975), in describing stages of drug use, has noted that illicit drug use begins with licit drug use. Consistent with Kandel's findings, Einstein, Hughes and Hindmarch (1975) stated that the "observed correlations indicated that subjects who consume socially acceptable drugs were more likely to try other, less acceptable drugs".

Another area of research supporting the idea of a climate which is conducive for or encouraging to drug abuse are those studies dealing with parent or sibling use of licit or illicit drugs. There is considerable research which suggests that familial drug use patterns, including the use of alcohol and tobacco, enhances the chances of drug abuse in children of that family. (Prescor, 1943; Smart, Fejer and White, 1970; Smart, 1970; Carney, 1972; Cox and Smart, 1972; Smart and Fejer, 1972; Adler and Lotecka, 1973; McKillip, Johnson and Petzel, 1973; Wiener and Egan, 1973; Hemminki, 1974; Lawrence and Velleman, 1974; Lerner, Linden and Burke, 1974; Stenmark, Wackwitz and Pelfrey, 1974; Cohen, 1975; Craig and Brown, 1975; Battegay, Ladewig, Muhlemann and Weldmann, 1976).

Research on the peer groups of drug users is also supportive of the milieu or drug climate proposition. Results from many studies suggest that the peer group may provide an atmosphere of drug use which can foster drug abuse (Bilodeau, 1971; Mercer, 1971; Adler and Lotecka, 1973; Kandel, 1973; McKillip, Johnson and Petzel, 1973; Lawrence and Velleman, 1974; Cohen, 1975; Battegay, Ladewig, Muhlemann and Weidmann, 1976). In a recent study of 500 drug users in the state of Kansas (McDermott, Poggio and Holloway, 1976) 44 per cent reported having been introduced to drugs or first motivated to use them by members of their peer group.

Most of the research cited here concentrated on smaller units of the social system such as family and peer groups, whereas the implications of the drug climate or milieu idea are most relevant for the general practices and drug using ethic of a society as a whole. Yet whether through the vehicle of acculturation, the socialization process, or psychological conditioning, it has long been recognized that the messages of a society are conveyed by the immediate social groups in an individual's experience. As noted above, most of the research concerning log-normal distributions of drug use previously published in the Bulletin on Narcotics have been based largely on Canadian and British school and college student samples. Smart and Whitehead (1973) specifically mention the need to replicate their observations through studies among drug users of other regions, nations and ages before such implications of the log-normal curve model would warrant a fuller theoretical treatment.

The present study has collected data on drug use from 2,130 randomly selected citizens of the United States residing in the centrally located midwestern state of Kansas. The data were analysed with respect to the shape of the distribution and the correlation between the use of licit and illicit drugs. The present data were collected as part of a larger statewide survey on drug use patterns funded by the state of Kansas.


Data collected for the research were obtained through a system of telephone interviews, with a total of 2,130 being completed. Respondents were selected at random by being drawn from telephone directories for counties throughout the state. The sampling design employed was a stratified random sampling procedure, based on estimates of the projected population size of the 105 counties in the state of Kansas for 1975 (from figures provided by the U.S. Dept. of Commerce, Bureau of the Census). Stratification in this case, was to ensure proportionate representation from each county. Three restrictions were placed on the sampling process: (1) only home residents were sampled, thus places of business and individuals at work were not sampled, (2) only one resident in each home would be interviewed, and, (3) data would be collected only from persons 10 years old or older.

The sample consisted of 69 per cent female, 30 per cent male and 1 per cent unidentified as to sex. Ninety-one per cent were of general Anglo descent, 4 per cent were negro, 1 per cent Mexican-American, and 4 per cent unknown. The educational level ranged from less than ninth grade (9 per cent) to post graduate school (0.5 per cent) with the largest numbers having high school degrees (42.2 per cent), or 2 to 4 years of college (38.8 per cent). The mean age of respondents was 40 years with the range from 10 to 89 years. Sixty-nine per cent of the sample were married while the next largest frequency was the never-been-married group with 22 per cent. The income distribution showed a slight positive skewness with 64 per cent reporting a less than average income. A wide variety of occupations were reported, however, 30 per cent of the respondents considered themselves to be housewives. The composite picture of a typical respondent shows a white, middle-class, middle to lower middle income woman, reasonably well educated, who is likely to be a housewife. Even though telephoning was done from 9 a.m. until 9 p.m., it is reasonable to assume that the characteristics of the sample were in part an artifact of the extensive daytime calls.

All respondents were assured of anonymity and confidentiality, and with the assistance of a public information programme conducted by the state government, co-operation was good. Cross-validation with a door-to-door sample showed no systematic variation in results. Given the sampling procedures used and the accuracy of the method of data collection, these results appear to be an accurate reflection of drug use in the state of Kansas.


In determining the shape of the distribution for this population, a drug use score for each respondent was obtained in the following manner. For each of nine drug categories (cough syrups, inhalents, analgesics, tranquillizers, sedatives, stimulants, cannabis products, hallucinogens, and narcotics) the question was asked "how frequently in the last twelve months have you used - - ?" Each answer was assigned to one of eight possibilities (not sure, not in past 12 months, 1-2 times a year, 5-6 times a year, once a month, four times a month, few times a week, once a day or more). These answer categories were weighted from 1 = not sure through 8 = once a day or more. Scores obtained on the nine categories were then summed for each respondent, yielding the individual's total drug use score.

The distribution of the resulting drug use scores was then compared with the expected values characteristic of the log-normal curve. A skewness test originally described by Croxton and Crowder (1955) was used to determine goodness of fit. These researchers concluded that if skewness measures are 0.20 or lower, log-normality has been achieved. The figure presents the distribution obtained for our sample; skewness measures 0.19.

FIGURE - Logarithmic normal curve fitted to drug use scorces of randomly selected Kansas residents

Full size image: 24 kB

Secondly, in order to consider factors possibly involved in the spread of drug abuse, a determination was made of the correlation between licit and illicit drug use. Since respondents were located in a variety of areas and not necessarily homogeneous, data were divided for the purposes of correlation on the basis of population density. The population categories were as follows: zero-5,000; 5,000-10,000; 11,000-17,000; 18,000-24,000; 25,000-28,000; 29,000-45,000; 46,000-55,000; 56,000-100,000; 100,000-200,000; 200,000-300,000; over 300,000.

Using the drug score method previously described it was possible to determine the extent of the correlational relationship between both licit and illicit drug use. A Pearson Product Moment Correlation Coefficient of 0.74 was determined for all population densities. These data are presented in the table.

Correlation of mean licit and illicit drug usage for eleven population districts

Population density

Mean licit drug usage

Mean illicit drug usage

0- 5,000
8.69 0.95
5,000- 10,999
7.71 0.62
11,000- 17,999
7.76 1.37
18,000- 24,999
8.02 1.11
25,000- 28,999
8.60 1.61
29,000- 45,999
7.98 0.78
46,000- 55,999
9.77 1.47
56,000- 99,999
8.66 1.13
10.19 5.62
9.29 1.96
300,000 and over
8.73 2.14
Pearson product moment correlation coefficient = 0.74 t = 3.31 PL.005 (one tailed test)


The results of the present study clearly show that for a randomly selected sample of individuals from the state of Kansas the distribution of drug use scores most nearly fits the logarithmic normal curve, with most persons reporting little use of drugs, some reporting moderate use, while a small number reported heavy drug use. It has been noted by previous researchers (Smart, Whitehead and Laforest, 1971), and is also noted here, that given a continuous distribution such as this, the most reliable and scientifically predictable way to reduce drug abuse is to reduce drug use in general.

Another finding of this study was that the use of illicit drugs was highly correlated with the use of licit drugs. The two results work together in support of the idea that an ethic which condones drug use of one variety, i.e. the high point on the log-normal curve, provides a climate or milieu conducive to drug abuse, on the low end of the log-normal curve. Continued testing of the general-izability of the unimodal and log-normal model on a variety of populations is called for, yet the evidence in support of this idea is mounting.

Such findings argue that one factor relevant in assessing the effectiveness of existing prevention and treatment programmes could be the limited explanatory power of a bi-modal division between the users and abusers of drugs. Since the traditional test of the scientific validity of an explanation has been its ability to affect the practical course of human events, it would behove nations with drug problems of an epidemiological nature to consider further research in this direction. A better understanding of why log-normal distributions exist in national communities could lead to a more accurate profile of the dimensions involved in drug abuse. A better theoretical fit between explanations of drug abuse and the structure of drug control programmes could in turn result in those programmes becoming more scientifically effective in combating this timely problem.


Adler, Peter T. and Lynn Lotecka, "Drug use among high school students: patterns and correlates", Inter. J. of the Addictions, Vol. 8 (3), 537-548, 1973.

Battegay, R., D. Ladewig, R. Muhlemann and M. Weidmann, "The culture of youth and drug abuse in some European countries", Inter. J. of the Addictions, Vol. 11 (2), 245-261, 1976.

Bilodeau, L. Drug use among students in the secondary schools and CEAEP's on the Montreal Island in 1969 and 1971. Research Division, Office for the Prevention and Treatment of Alcoholism and other Toxicomanias, Quebec, 1971.

Carney, P.A., M.W. Timms and R.D. Stevenson, "The social and psychological background of young drug users in Dublin", British J of Additions, Vol. 67 (3), 199-207, 1976.

Cox, Carole and Reginald G. Smart, "Social and psychological aspects of speed use: A study of types of speed users in Toronto", Inter. J. of the Addictions, Vol. 7 (2), 201-217, 1972.

Craig, Starlett R. and Barry S. Brown, "Comparison of youthful heroin users and nonusers from one urban community", Inter. J. of the Addictions, Vol. 10 (1), 53-64, 1975.

Croxton, F.R. and D.J. Crowder, Applied General Statistics. Englewood Cliffs, Prentice-Hall, 1955.

Einstein, Rosemarie, I.E. Hughes and I. Hindmarch, "Patterns of use of alcohol, cannabis and tobacco in a student population", British J. of Addictions, Vol. 70 (2), 145-150, 1975.

Fracchia, John, Charles Sheppard and Sidney Merlis, "Early cigarette smoking and drug use: some comments, data, and thoughts", Psychological Reports, Vol. 34(2), 371-374, 1974.

Hemminki, Elina, "Tobacco, alcohol, medicines and illegal drug taking", Adolescence, Vol. 9 (35), 421-424, 1974.

Kandel, Denise, "Stages in adolescent involvement in drug use", Science, Vol. 90 (4217), 912-914, 1975.

Lawrence, Theodore S. and J. David Velleman, "Correlates of student drug use in a suburban high school", Psychiatry, Vol. 37 (2), 129-136, 1974.

Lerner, Steven E., Ronald D. Linder and Elaine M. Burke, "Drugs in the junior high school: II", J. of Psychedelic Drugs, Vol. 6 (1), 51-56, 1974.

McDermott, Diane, John Poggio and William Holloway, "Kansas drug use survey: Phases I, II, and III", State of Kansas, 1976.

McKillip, Jack, James E. Johnson and Thomas P. Petzel, "Patterns and correlates of drug use among urban high school students", J. of Drug Education, Vol. 3 (1), 1-12, 1973.

Mercer, A.W., "Youth Centre Study", Report to the Borough of York, York University, Toronto, Ontario, 1971.

Prescor, M.F., "A statistical analysis of clinical records of hospitalized drug addicts", Public Health Reports, 1943, Supplement No. 143.

Sargent, Margaret J., "A cross-cultural study of attitudes and behaviour towards alcohol and drugs", British J. of Sociology, Vol. 22 (1), 83-96, 1971.

Schmidt, W. and J.E. DeLint, "Estimating the prevalence of alcoholism from alcohol consumption and mortality data", Quarterly Journal of Studies on Alcohol, 31, 957-964.

Single, Eric, Denise Kandel and Richard Faust, "Patterns of multiple drug use in high school", J. of Health and Social Behavior, Vol. 15 (4), 344-357, 1974.

Smart, Reginald G., "Some current studies of psycho-active and hallucinogenic drug use", Canadian J. of Behavioral Science, 2 (3), 232-245, 1970.

Smart, R.G. and D. Fejer, "Drug use among adolescents and their parents: closing the generation gap in mood modification", J. of Abnormal and Social Psychol., 79, 153-160, 1972.

Smart, R.G., D. Fejer and J. White, "The Extent of drug use in metropolitan Toronto schools: a study of changes from 1968 to 1970", Addiction Research Foundation, Toronto, Ontario, 1970.

Smart, Reginald G. and Paul Whitehead, "The consumption patterns of illicit drug use and their implications for prevention of abuse", Bulletin on Narcotics, XXIV: 1, 39-47, 1972.

Smart, Reginald G. and Paul C. Whitehead, "The prevention of drug abuse by lowering per capita consumption: distributions of consumption in samples of Canadian adults and British university students, Bulletin on Narcotics, XXV: 4, 49-55, 1973.

Smart, Reginald G., Paul Whitehead and Lucien Laforest, "The prevention of drug abuse by young people: an argument based on the distribution of drug use", Bulletin on Narcotics, XXIII: 2, 11-15, 1971.

Stenmark, David E., John H. Wackwitz, C. Michael Pelfrey and Frank Dougherty, "Substance use among juvenile offenders: relationships to parental substance use and demographic characteristics", Addictive Diseases, Vol. 1(1), 43-54, 1974.

Wiener, Jerry M. and lames H. Egan, "Heroin addiction in an adolescent population", J. of the Am. Academy of Child Psychiatry, Vol. 12 (1), 48-58, 1973.