Reading the Survivor and Death Columns
A life table follows a cohort as its members age. Its \(l_x\) column records how many members are alive at exact age \(x\); its \(d_x\) column records how many die between exact ages \(x\) and \(x+1\). These are counts, not mortality probabilities. For example, \(d_{60}\) refers to deaths during the year of age from 60 to 61, not deaths among people who have already reached age 61.
The cohort may be an actual group followed over time or a hypothetical cohort defined by a life table. In either case, the entries must refer to the same cohort and the same starting basis. A life table often starts with a chosen number of lives, called its radix. Once the radix is fixed, the \(l_x\) and \(d_x\) entries describe how that cohort changes with age.
The subscripts identify ages, and the interval matters. The entries \(l_{47}\) and \(l_{48}\) are the numbers alive at the start and end of the year of age from 47 to 48. Their difference is the number of deaths during that interval. If a table lists ages in whole years, each row's \(d_x\) is ordinarily matched to the \(l_x\) and \(l_{x+1}\) entries for consecutive ages.
Why the Difference Gives the Death Count
Consider the \(l_x\) people alive at exact age \(x\). Over the next year, each is either alive at exact age \(x+1\), or has died before reaching that age. If the cohort has no entries or exits other than deaths, those two groups partition the people counted by \(l_x\). The first group has size \(l_{x+1}\), so the second has size \(l_x-l_{x+1}\).
This argument also identifies the assumptions behind the identity: the entries refer to the same cohort and time basis, and deaths are the only way a person counted at age \(x\) is absent from the count at age \(x+1\). In a hypothetical life table, the entries may be expected counts rather than counts of individually observed people. In that case, subtraction still gives the table's implied number of deaths, which need not be an integer.
A useful consistency check follows directly. Since a cohort cannot gain members between ages \(x\) and \(x+1\) under these assumptions, \(l_{x+1}\leq l_x\), so \(d_x\geq 0\). If a table has \(l_{x+1}>l_x\), check whether the entries were copied correctly or whether they refer to different bases. If \(l_{x+1}=l_x\), then \(d_x=0\) for that interval.
Worked Example: Finding Deaths in Consecutive Age Intervals
A hypothetical cohort has 12,000 members alive at exact age 40. The table gives \(l_{41}=11{,}820\) and \(l_{42}=11{,}550\). Find the deaths from age 40 to 41, the deaths from age 41 to 42, and the total deaths from age 40 to 42.
Method. Subtract consecutive survivor counts to obtain each interval's deaths, then compare the two-year total with the change in survivors from age 40 to age 42.
For the year of age from 40 to 41: $$ d_{40}=l_{40}-l_{41}=12{,}000-11{,}820=180. $$ Thus, 180 members die in this interval, and 11,820 survive to exact age 41.
For the year of age from 41 to 42: $$ d_{41}=l_{41}-l_{42}=11{,}820-11{,}550=270. $$ Thus, 270 members die in the second interval.
The total deaths over both intervals are $$ 180+270=450. $$ The direct endpoint check gives $$ l_{40}-l_{42}=12{,}000-11{,}550=450, $$ which agrees. Common mistake avoided: subtracting \(l_{42}\) from \(l_{41}\) to find deaths over the entire two years would count only deaths from age 41 to 42, not deaths from age 40 to 42.
Deaths Across Multiple Years
For a span of several consecutive age intervals, add the relevant \(d_x\) entries. The differences telescope: every intermediate survivor count appears once as an amount subtracted and once as an amount added. Therefore, deaths between exact ages \(x\) and \(x+n\) equal the starting survivor count minus the ending survivor count:
This result is a count identity. It says how many members of the cohort counted at age \(x\) have died by exact age \(x+n\), provided the cohort is followed on the same basis. It does not, by itself, express the deaths as a fraction of the people alive at age \(x\). Keeping the count interpretation clear helps prevent accidentally treating a number of deaths as a probability or a rate.
Worked Example: Reconciling Annual and Cumulative Deaths
A hypothetical table has \(l_{60}=50{,}000\), \(l_{61}=49{,}250\), \(l_{62}=48{,}610\), and \(l_{63}=47{,}890\). Calculate the deaths in each year of age from 60 to 63, the total deaths over the three years, and the survivors at age 63.
Method. Use the difference between consecutive \(l_x\) entries for each annual interval, then sum those deaths and check against the endpoint counts.
From exact age 60 to 61: $$ d_{60}=50{,}000-49{,}250=750. $$ From exact age 61 to 62: $$ d_{61}=49{,}250-48{,}610=640. $$ From exact age 62 to 63: $$ d_{62}=48{,}610-47{,}890=720. $$
The total deaths during the three intervals are $$ 750+640+720=2{,}110. $$ The endpoint calculation confirms this: $$ l_{60}-l_{63}=50{,}000-47{,}890=2{,}110. $$ The number of survivors at exact age 63 is the given \(l_{63}=47{,}890\). As a further check, survivors plus deaths over the period equal the initial cohort: $$ 47{,}890+2{,}110=50{,}000. $$ Common mistake avoided: the three annual deaths are not all \(l_{60}-l_{63}\); that endpoint difference is the total across all three intervals. Each annual count uses its own pair of consecutive survivor entries.
Scaling Counts to a Different Cohort Size
A life table may use a radix that differs from the size of a group in a question. If the group is assumed to follow the same mortality pattern as the table, its survivor counts can be obtained by scaling the table's counts in the same proportion. For example, if the table starts with \(l_x\) lives and has \(l_{x+1}\) survivors one year later, a group of \(N\) people at age \(x\) has the table-based survivor count \(N(l_{x+1}/l_x)\) at age \(x+1\). The corresponding deaths are the starting group size minus those survivors.
This is a proportional application of the table, not a claim that every actual group will experience exactly the table's predicted number of deaths. When the calculation produces a non-integer, it represents an expected or model-based count; a literal observed count of people must be an integer.
Worked Example: Applying Life Table Counts to a Smaller Group
A hypothetical life table has radix \(100{,}000\), with \(l_{35}=96{,}000\) and \(l_{36}=95{,}520\). A separate group has 5,000 members, all alive at age 35. Assuming this group follows the table's proportions, calculate its expected survivors at age 36 and its expected deaths between ages 35 and 36.
Method. Find the table's one-year survivor proportion, apply that proportion to the group's starting count, and subtract from the starting count to get deaths.
The fraction surviving the interval in the table is $$ \frac{l_{36}}{l_{35}}=\frac{95{,}520}{96{,}000}=0.995. $$ Applying that proportion to the group gives $$ 5{,}000(0.995)=4{,}975 $$ expected survivors at age 36.
The expected deaths are $$ 5{,}000-4{,}975=25. $$ Equivalently, the table has \(96{,}000-95{,}520=480\) deaths per \(96{,}000\) starting lives, and scaling gives $$ 5{,}000\left(\frac{480}{96{,}000}\right)=25. $$ The two methods agree, and \(4{,}975+25=5{,}000\), as required. Common mistake avoided: applying the table's raw count of 95,520 survivors to a group of 5,000 would ignore the difference in starting cohort sizes. Scale by the ratio, not by copying the table entry.
Interpreting the Entries Carefully
When reading or constructing a small section of a life table, label every interval explicitly. A compact table can help keep the age labels aligned:
| Interval | Survivors at start | Survivors at end | Deaths in interval |
|---|---|---|---|
| Age \(x\) to \(x+1\) | \(l_x\) | \(l_{x+1}\) | \(d_x=l_x-l_{x+1}\) |
| Age \(x+1\) to \(x+2\) | \(l_{x+1}\) | \(l_{x+2}\) | \(d_{x+1}=l_{x+1}-l_{x+2}\) |
| Age \(x\) to \(x+2\) | \(l_x\) | \(l_{x+2}\) | \(l_x-l_{x+2}=d_x+d_{x+1}\) |
The table also makes clear that \(d_x\) is attached to the interval beginning at age \(x\). It is not the number who die at exact age \(x\), nor the number alive at that age. Further distinctions among life table columns and mortality measures are separate questions; for this calculation, identify the interval and subtract the survivor count at its end from the survivor count at its start.
The most useful checks are simple: every death count must be nonnegative when the survivor column is nonincreasing; each annual death count must reconcile with its two adjacent survivor entries; and the sum of deaths over several consecutive intervals must equal the difference between the endpoint survivor counts. These checks catch reversed subtraction, misaligned ages, and accidental omission of an interval.
Check Your Understanding
For each problem, track the age interval and keep survivor counts distinct from deaths.
- A hypothetical table gives \(l_{52}=8{,}500\) and \(l_{53}=8{,}390\). Find the deaths from age 52 to 53. (Answer: 110 deaths)
- A cohort has \(l_{70}=24{,}000\), \(l_{71}=23{,}760\), and \(l_{72}=23{,}410\). Find the deaths in each year and the total deaths from age 70 to 72. (Answer: 240 deaths from 70 to 71, 350 from 71 to 72, and 590 in total)
- A table has \(l_{40}=98{,}000\) and \(l_{41}=97{,}510\). A group of 4,000 people is alive at age 40 and follows the table's proportions. Find the expected survivors and deaths at age 41. (Answer: 3,980 survivors and 20 deaths)
- A cohort has \(l_{25}=100{,}000\), \(l_{26}=99{,}000\), \(l_{27}=98{,}100\), and \(l_{28}=97{,}000\). Find each annual death count and verify the total deaths from age 25 to 28. (Answer: 1,000, 900, and 1,100 annual deaths; 3,000 total deaths)
- A table gives \(l_{55}=96{,}000\), \(l_{56}=95{,}520\), and \(l_{57}=94{,}800\). A group of 12,000 people at age 55 follows the table's proportions. Find the expected survivors at ages 56 and 57, deaths from 55 to 56, deaths from 56 to 57, and total deaths by age 57. (Answer: 11,940 survivors at 56, 11,850 at 57, 60 deaths from 55 to 56, 90 from 56 to 57, and 150 total deaths)