Life Table Column Identities

Why Life Table Columns Must Agree

A life table presents related descriptions of mortality over an age interval. It may report how many lives are present at age $x$, how many die before age $x+1$, and the probability that a person aged $x$ dies or survives the year. These are not independent quantities. If they describe the same cohort and the same one-year interval, the survivor, death, and probability columns must satisfy exact identities.

Those identities are useful both for constructing entries and for auditing a table. A frequent error is to apply a one-year probability to the wrong number of lives, or to treat deaths during an interval as survivors at its end. Keeping the age interval and the denominator explicit prevents these mistakes.

Definition: For the one-year interval from exact age $x$ to exact age $x+1$, $\ell_x$ is the number of lives alive at age $x$, and $d_x$ is the number of those lives who die before reaching age $x+1$. The one-year mortality probability is $q_x$, and the one-year survival probability is $p_x$. When the entries are consistent, $q_x$ and $p_x$ describe the same one-year interval and the same lives represented by $\ell_x$.

A life table may use an initial cohort size called a radix, or it may report expected cohort counts rather than whole-number observed counts. Either way, the identities below apply when its counts and probabilities are defined consistently. In an observed data set, the ratio of recorded deaths to the starting count is an empirical proportion; it need not equal a separately specified model probability unless that probability was defined from those same table entries.

The Column Identities

The mortality probability $q_x$ is the probability that a life already present at age $x$ dies during the next year. The survival probability $p_x$ is the probability that the life reaches age $x+1$. The people counted in $\ell_x$ therefore divide into two groups over this interval: those counted in $d_x$, and those who survive to be counted in $\ell_{x+1}$.

Theorem: For a consistent life table over the one-year interval from age $x$ to age $x+1$, $$ d_x=\ell_x q_x,\qquad \ell_{x+1}=\ell_x p_x,\qquad p_x+q_x=1. $$ Consequently, $$ \ell_{x+1}=\ell_x-d_x,\qquad d_x=\ell_x-\ell_{x+1},\qquad q_x=\frac{d_x}{\ell_x},\qquad p_x=\frac{\ell_{x+1}}{\ell_x}, $$ provided $\ell_x>0$.

The probability identities also show why the survivor and death counts must reconcile. Since $p_x+q_x=1$, multiplying both sides by $\ell_x$ gives $\ell_xp_x+\ell_xq_x=\ell_x$. Substituting the corresponding table entries gives $\ell_{x+1}+d_x=\ell_x$. Thus, the number alive at the start of the interval equals the number who survive it plus the number who die in it.

Translate probabilities into counts.
Each of the $\ell_x$ lives has probability $q_x$ of dying during the year, so the table's death count is $d_x=\ell_xq_x$. The probability applies to the lives at age $x$, not to the survivors at age $x+1$.

Count those who survive.
Each of the same $\ell_x$ lives has probability $p_x$ of surviving to age $x+1$, giving $\ell_{x+1}=\ell_xp_x$.

Reconcile the two outcomes.
Death and survival are complementary outcomes for the year, so $q_x+p_x=1$. The corresponding counts must therefore satisfy $d_x+\ell_{x+1}=\ell_x$.

These equations can be rearranged according to which entries are known. For example, if $\ell_x$ and $d_x$ are given, calculate $q_x=d_x/\ell_x$, $p_x=1-q_x$, and $\ell_{x+1}=\ell_x-d_x$. If $\ell_x$ and $q_x$ are given, calculate $d_x=\ell_xq_x$, $p_x=1-q_x$, and $\ell_{x+1}=\ell_xp_x$. In each case, a final check should confirm that the two probability columns sum to one and that the counts balance.

Worked Example: Building the One-Year Columns from a Mortality Probability

A hypothetical life table has $\ell_{50}=10{,}000$. For the year from age 50 to age 51, it specifies $q_{50}=0.0125$. Find $d_{50}$, $p_{50}$, and $\ell_{51}$, then check the entries.

Method. Apply each one-year probability to the 10,000 lives present at age 50. The calculation assumes the supplied $q_{50}$ describes deaths over this same age interval.

First, the number dying is $$ d_{50}=\ell_{50}q_{50}=10{,}000(0.0125)=125. $$

The one-year survival probability is $$ p_{50}=1-q_{50}=1-0.0125=0.9875. $$

The number surviving to age 51 is $$ \ell_{51}=\ell_{50}p_{50}=10{,}000(0.9875)=9{,}875. $$

Check. The probabilities sum to $0.0125+0.9875=1$, and the counts balance because $125+9{,}875=10{,}000$. The answer is internally consistent. This method avoids the common mistake of calculating deaths using $\ell_{51}$, which counts survivors after the interval rather than lives exposed at its start.

Recovering Probabilities from Table Counts

When a table gives both the starting and ending survivor counts, the change in survivors determines the number of deaths during that year. Dividing deaths by the starting survivor count gives the one-year mortality probability. The starting count is the denominator because $q_x$ is conditional on being alive at age $x$.

Worked Example: Finding Rates from Consecutive Survivor Counts

A hypothetical table lists $\ell_{67}=4{,}800$ and $\ell_{68}=4{,}740$. Find $d_{67}$, $q_{67}$, and $p_{67}$.

Method. Use the difference between consecutive survivor counts to find deaths, then divide by the count at the start of the year to obtain the conditional probability.

The number of deaths during the interval is $$ d_{67}=\ell_{67}-\ell_{68}=4{,}800-4{,}740=60. $$

Therefore, the one-year mortality probability is $$ q_{67}=\frac{d_{67}}{\ell_{67}}=\frac{60}{4{,}800}=0.0125. $$

The survival probability is $$ p_{67}=1-q_{67}=1-0.0125=0.9875. $$

It can also be recovered directly from the survivor counts: $$ p_{67}=\frac{\ell_{68}}{\ell_{67}}=\frac{4{,}740}{4{,}800}=0.9875. $$

Check. Both ways of calculating $p_{67}$ agree, $q_{67}+p_{67}=1$, and $d_{67}+\ell_{68}=60+4{,}740=4{,}800$. The method avoids dividing the deaths by the ending count: $60/4{,}740$ would use the wrong denominator for a probability conditional on being alive at age 67.

Auditing a Table with Conflicting Entries

A table can contain entries that cannot all be true at once. An audit should compare all three forms of the information: mortality and survival probabilities, deaths, and the change in survivor counts. If two entries are treated as authoritative, the identities determine what the remaining entries must be. If a supplied entry conflicts, state which values are retained before proposing a correction.

Worked Example: Identifying an Inconsistent Row

A hypothetical row for age 40 reports $\ell_{40}=2{,}500$, $q_{40}=0.04$, $d_{40}=90$, and $\ell_{41}=2{,}410$. Audit the row. If $\ell_{40}$ and $q_{40}$ are the intended entries, find the corrected values.

Method. Calculate the death count and ending survivor count from the stated starting count and mortality probability, then compare them with the reported counts.

Using $\ell_{40}=2{,}500$ and $q_{40}=0.04$, the implied number of deaths is $$ d_{40}=2{,}500(0.04)=100. $$ The implied survival probability is $$ p_{40}=1-0.04=0.96, $$ so the implied number alive at age 41 is $$ \ell_{41}=2{,}500(0.96)=2{,}400. $$

The reported counts do balance with each other because $90+2{,}410=2{,}500$, but they do not agree with $q_{40}=0.04$. If the reported counts were retained instead, they would imply $q_{40}=90/2{,}500=0.036$, not $0.04$.

Correction under the stated condition. Retaining $\ell_{40}=2{,}500$ and $q_{40}=0.04$ requires replacing the reported death and survivor counts with $d_{40}=100$ and $\ell_{41}=2{,}400$.

Check. The corrected counts satisfy $100+2{,}400=2{,}500$, and the probabilities satisfy $0.04+0.96=1$. The trap is to check only whether the two reported counts add to the starting count. That check is necessary, but not sufficient: the probability column must agree with the counts too.

Common Mistakes and Practical Checks

The identities are simple, but their conditional meaning and time interval matter. Before calculating, identify the age at the beginning of the year, the age at the end, and which quantity is the denominator.

Using the ending count as the denominator. The mortality probability is $d_x/\ell_x$, not $d_x/\ell_{x+1}$. It is conditional on being alive at age $x$.
Mixing interval endpoints. $d_x$ counts deaths from age $x$ up to age $x+1$; it is not the number dying in the following interval.
Subtracting deaths from the wrong count. The survivor count at the end is $\ell_{x+1}=\ell_x-d_x$, not $\ell_x+d_x$.
Checking only one identity. A row can satisfy $d_x+\ell_{x+1}=\ell_x$ while conflicting with a stated $q_x$. Check both the count balance and the probability balance.
Rounding too early. If probabilities are reported to a limited number of decimal places, small apparent discrepancies may result from rounding. Use unrounded values for the identities when they are available.

Key takeaway: Within a consistent one-year life table row, the starting lives split into deaths and survivors: $d_x=\ell_xq_x$, $\ell_{x+1}=\ell_xp_x$, $q_x+p_x=1$, and $d_x+\ell_{x+1}=\ell_x$. These relationships provide both a calculation method and a complete basic consistency check.

Check Your Understanding

For each problem, use the one-year interval specified and check that the survivor and death entries balance.

A hypothetical table has $\ell_{55}=8{,}000$ and $q_{55}=0.015$. Find $d_{55}$, $p_{55}$, and $\ell_{56}$. (Answer: $d_{55}=120$, $p_{55}=0.985$, $\ell_{56}=7{,}880$.)
A row has $\ell_{72}=2{,}750$ and $d_{72}=55$. Find $q_{72}$, $p_{72}$, and $\ell_{73}$. (Answer: $q_{72}=0.02$, $p_{72}=0.98$, $\ell_{73}=2{,}695$.)
A hypothetical table lists $\ell_{61}=6{,}400$ and $\ell_{62}=6{,}240$. Find $d_{61}$, $q_{61}$, and $p_{61}$. (Answer: $d_{61}=160$, $q_{61}=0.025$, $p_{61}=0.975$.)
At age 45, $\ell_{45}=5{,}000$ and $q_{45}=0.018$. Find the number dying, the survival probability, and the number reaching age 46. (Answer: $d_{45}=90$, $p_{45}=0.982$, $\ell_{46}=4{,}910$.)
A row reports $\ell_{30}=12{,}000$, $q_{30}=0.01$, $d_{30}=150$, and $\ell_{31}=11{,}850$. If $\ell_{30}$ and $q_{30}$ are correct, what should $d_{30}$ and $\ell_{31}$ be? (Answer: $d_{30}=120$ and $\ell_{31}=11{,}880$.)

What You'll Learn