Griffith Feeney's Demography Website

Time-plotting Life Cycle Events

May 30, 2009

Given a population, a class of events that may occur to members of this population, and a cohort of persons born to this population at some time T, let Q and M denote, respectively, the average number of events per person in the cohort (“quantum”) and the mean age at which these events occur (“tempo”). For non-recurrent events—events an individual may experience at most once—, Q is the proportion of persons who experience the event.

Let t denote the time at which members of the cohort reach age M. The point (t,Q) plotted on coordinate axes representing time (horizontal axis) and mean events per person (vertical axis) is the “time plot” of Q. The “time plot” of M is defined in the same way, with the vertical axis representing the average age at which the event occurs.

Q and/or M may be plotted for a series of cohorts, in which case the time plots show one point for each cohort. When cohorts are defined by a time period, rather than by a point in time, the mid-point of the period may be used as the reference time.

Data for time plotting may come from any source, but the power of the method is the possibility of generating time plots from data collected at a point in time, with birth cohorts specified by age group.

Heuristic Rationale

The idea of time plotting is that the cohort statistics Q and M may be interpreted as estimates of the corresponding period statistics as of the time at which members of the cohort reach the mean age at which the event in question occurs. A thorough development of this idea is beyond the scope of this note, but a simple example will convey the rationale for time plotting.

Let the event be attainment of literacy and suppose that people who achieve literacy do so on their 10th birthday. The period measure of social production of literate persons t will thus be the proportion of persons at exact age 10 years who are literate. No one will be literate at younger ages, and proportions literate at age x > 10 years will represent production of literate persons at time t - (x - 10).

Intuitively, if the distribution of the ages at which literacy is attained clusters closely around age 10, a time plot of literacy data will provide a reasonable estimate of period production of literate persons.

Example: Literacy in China

A time plot of literacy data from China’s 1982 population census (reference time mid-year) is shown below. The data have been tabulated from the 1% sample provided by IPUMS International (

Time-plot of Male and Female Literacy in China:
Census of 1982

China 1982 Literacy Time Plot

Here are a few key observations, of the many that might be made.

(1) We are looking at nearly a century of changes in literacy in China. This is possible because older literate persons attained their literacy, on the average, a long time ago.

(2) Male literacy was higher than female literacy throughout the period, though the gap declined dramatically over the period.

(3) Literacy for both males and females rose dramatically from 1920 through 1970, before as well as after the success of the Communist revolution in 1949. The revolution appears to have had no impact on the trend of male literacy. Female literacy rose very rapidly from 1944, five years prior to the success of the revolution.

(4) The rising trend reversed during and after the “Great Leap Forward” years 1958-61. Female literacy recovered far more rapidly than male literacy, albeit from a lower level.


As always, we must ask what part of what we see reflects reality and what part reflects data and methodology issues. The plot suggests that the quality of reporting of age and literacy in the 1982 census was generally good. We can be sure that literacy in China was not uniformly attained at exact age 10 years, however, the impact of departs from this supposition needs to be assessed. A distribution of ages of at attainment of literacy clustering around age 10 years would probably have little effect, but adult literacy programs post-1949 could have a large effect. We can be sure also that mortality risks are higher for illiterate persons, so proportions literate are progressively biased upward for older ages. Estimating the magnitude of this bias is another matter. The simplest way to address these and other doubts is to add data from a subsequent (or earlier) census. 1990 census data for China is available from IPUMS International for the interested reader.

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