UDD Interpolation Errors and Boundary Checks

Why UDD Calculations Need an Audit

Under the uniform distribution of deaths (UDD) assumption, the number of survivors is interpolated linearly between consecutive integer ages. That makes fractional-age calculations direct, but it also makes them sensitive to a basic input error: confusing the number of deaths $d_x$ with the probability of death $q_x$, or using the wrong survivor count as a denominator.

A reliable calculation begins by checking the integer-age life-table entries, then interpolating survivor counts, and finally forming a conditional probability from the appropriate ratio. This sequence also reveals whether a result is consistent at integer-age boundaries and lies in the probability range from zero to one.

Definition: For integer age $x$, $l_x$ is the number of lives in the life table surviving to age $x$, and $d_x=l_x-l_{x+1}$ is the number dying between ages $x$ and $x+1$. The one-year probability of death is $q_x=d_x/l_x$, provided $l_x>0$. Under UDD, for $0\le t\le1$, the interpolated survivor count is $$ l_{x+t}=l_x-t d_x=l_x(1-tq_x). $$

Keep the Three Life-Table Quantities Separate

The quantities $l_x$, $d_x$, and $q_x$ are related, but they are not interchangeable. The first is a count at the start of the year, the second is a count of deaths during the year, and the third is a probability conditional on survival to the start of the year. The identities connecting them are

d_x=l_x-l_{x+1}, \qquad q_x=\frac{d_x}{l_x}, \qquad l_{x+1}=l_x(1-q_x).

In particular, $d_x/l_{x+1}$ is not $q_x$: it divides deaths by the number surviving to the end of the year, rather than by the number exposed to death at the start. When $l_x>0$, the basic validity checks are $0\le d_x\le l_x$, $0\le q_x\le1$, and $l_{x+1}\ge0$. For a conditional survival probability, the number alive at the conditioning age must also be positive.

For a fractional age within one year, the count interpolation can be written either with deaths or with the one-year probability:

l_{x+t}=l_x-t d_x=l_x(1-tq_x).

These forms agree because $d_x=l_xq_x$. If a problem supplies $d_x$, use the count form directly or first compute $q_x=d_x/l_x$. Do not subtract a fraction of $q_x$ from $l_x$: $q_x$ is a probability, not a number of lives.

Why Linear Interpolation Holds Under UDD

UDD assumes that, conditional on being alive at age $x$, deaths are spread uniformly over the following year. Thus the fraction of the year’s deaths that have occurred by time $t$ is $t$, for $0\le t\le1$. Since the full-year death count is $d_x$, the count of deaths by that time is $t d_x$. Subtracting this count from those alive at age $x$ gives the interpolated survivor count:

Start with the count at age $x$.
There are $l_x$ lives alive at the start of the year.

Count deaths by time $t$.
UDD assigns the fraction $t$ of the year’s deaths to the first $t$ years of the interval, giving $t d_x$ deaths.

Subtract deaths from the initial count.
Therefore $l_{x+t}=l_x-t d_x$. Substituting $d_x=l_xq_x$ gives $l_{x+t}=l_x(1-tq_x)$.

The endpoint checks follow immediately. At $t=0$, the formula gives $l_{x+0}=l_x$. At $t=1$, it gives $l_{x+1}=l_x-d_x$, exactly the next integer-age count. The two forms of the formula therefore agree with the life table at both ends of the year.

Conditional Probabilities and Boundary Checks

If $0\le s\le t\le1$ and $l_{x+s}>0$, the probability of surviving from age $x+s$ to age $x+t$ is the later survivor count divided by the earlier one:

{}_{t-s}p_{x+s} =\frac{l_{x+t}}{l_{x+s}} =\frac{1-tq_x}{1-sq_x}.

The denominator reflects the population alive at the conditioning age, not the population alive at age $x$. For an interval that crosses an integer age, use the same principle: calculate the interpolated counts at the interval’s start and end, using the appropriate year for each, and take their ratio. The count at the integer boundary must agree whether it is calculated as the end of one year or the start of the next.

A valid survival probability cannot exceed one. Under valid life-table inputs, survivor counts are nonincreasing with age, so for $a\le b$ with $l_a>0$, $l_b/l_a$ must lie between zero and one. A result above one often indicates that the counts were reversed or that a fractional-age count was calculated with the wrong year’s deaths. A negative count or probability indicates an invalid input or a calculation error.

Worked Example: Auditing a Within-Year Conditional Probability

A hypothetical life table has $l_{54}=12{,}000$ and $d_{54}=720$. Assume UDD during this year. Find the probability that a life aged 54.25 survives to age 54.75. Check the endpoints and the probability.

Method. First calculate $q_{54}$, then interpolate the survivor counts and divide the count at the later age by the count at the conditioning age. This avoids treating $d_{54}$ as a probability or using $l_{54}$ as the conditional denominator.

The one-year death probability is

q_{54}=\frac{d_{54}}{l_{54}}=\frac{720}{12{,}000}=0.06.

At age 54.25, UDD gives

l_{54.25}=12{,}000-0.25(720)=12{,}000-180=11{,}820.

At age 54.75, the count is

l_{54.75}=12{,}000-0.75(720)=12{,}000-540=11{,}460.

Thus the conditional survival probability is

\frac{l_{54.75}}{l_{54.25}} =\frac{11{,}460}{11{,}820} =\frac{191}{197} =0.969543\ldots

To four decimal places, the answer is 0.9695. As a check, the conditional probability of dying over the interval is $(11{,}820-11{,}460)/11{,}820=360/11{,}820=0.030457\ldots$; the two probabilities sum to one. The interpolated counts also lie between $l_{54}=12{,}000$ and $l_{55}=12{,}000-720=11{,}280$.

Crossing an Integer-Age Boundary

When an interval crosses an integer age, the annual death count changes at that boundary. The count at the boundary is computed from the first year as $l_{x+1}=l_x-d_x$; the following year’s interpolation starts from that same value and uses $d_{x+1}$. It is incorrect to apply one year’s $d_x$ to fractional ages in the next year.

Worked Example: Checking Survival Across an Integer Boundary

A hypothetical life table has $l_{62}=9{,}000$, $d_{62}=450$, and $d_{63}=540$. Assume UDD in both years. Find the probability that a life aged 62.5 survives to age 63.25. Verify the count at age 63 from both sides of the boundary.

Method. Interpolate from age 62 to find the starting count, calculate $l_{63}$, then use the age-63 death count to find the ending count. The required conditional probability is the ending count divided by the starting count.

At age 62.5,

l_{62.5}=9{,}000-0.5(450)=9{,}000-225=8{,}775.

The integer-age boundary count is

l_{63}=9{,}000-450=8{,}550.

Using the next year’s deaths, the count at age 63.25 is

l_{63.25}=8{,}550-0.25(540)=8{,}550-135=8{,}415.

The survival probability is therefore

\frac{l_{63.25}}{l_{62.5}} =\frac{8{,}415}{8{,}775} =0.958974\ldots

To four decimal places, the answer is 0.9590. The boundary check succeeds because $l_{63}=8{,}550$ is both $9{,}000-450$, the count at the end of the age-62 year, and the starting count used for age 63.25. The probability is positive and less than one, as required. Using $d_{62}=450$ again to interpolate from age 63 to 63.25 would produce the wrong ending count.

Common Errors and a Full Input Audit

Several mistakes can produce a plausible-looking number while violating the life-table definitions. Before accepting a result, check the units and the direction of each calculation:

Using $d_x$ as if it were $q_x$. The count formula subtracts $t d_x$; the probability formula multiplies $l_x$ by $1-tq_x$. These are equivalent only when $q_x=d_x/l_x$.
Using $l_{x+1}$ in the denominator of $q_x$. The definition is $q_x=d_x/l_x$, not $d_x/l_{x+1}$.
Using the wrong year’s deaths. For age $x+t$, use $d_x$. For age $x+1+t$, use $d_{x+1}$.
Reversing a conditional ratio. Survival from an earlier age $a$ to a later age $b$ is $l_b/l_a$, not $l_a/l_b$.
Failing to check the domain. Require $0\le t\le1$ for interpolation within a single year and nonnegative, nonincreasing survivor counts.

Worked Example: Correcting a Misused Denominator

A hypothetical table has $l_{70}=4{,}000$ and $d_{70}=260$. An analyst calculates the one-year death probability as $260/l_{71}$. Assuming UDD, find the correct $q_{70}$ and the probability of surviving from age 70.3 to age 70.8. Identify and correct the analyst’s error.

Method. Calculate $l_{71}$ from the life-table identity, use $l_{70}$ as the denominator for $q_{70}$, and then interpolate both fractional-age counts. The error to avoid is conditioning on the end-of-year count when defining $q_{70}$.

First,

l_{71}=l_{70}-d_{70}=4{,}000-260=3{,}740.

The analyst’s calculation would give $260/3{,}740=0.06952\ldots$, but this is not $q_{70}$. The correct one-year death probability uses the initial count:

q_{70}=\frac{260}{4{,}000}=0.065.

Using either $d_{70}$ or $q_{70}$, the interpolated counts are

l_{70.3}=4{,}000-0.3(260)=3{,}922, \qquad l_{70.8}=4{,}000-0.8(260)=3{,}792.

Therefore,

{}_{0.5}p_{70.3} =\frac{3{,}792}{3{,}922} =0.966853\ldots

To four decimal places, the answer is 0.9669. The check $l_{71}/l_{70}=3{,}740/4{,}000=0.935=1-q_{70}$ confirms that the corrected $q_{70}$ agrees with the integer-age table. The fractional-age counts decrease from 3,922 to 3,792, so their ratio is also in the required probability range.

When a Formula Is Not Applicable

The UDD interpolation formula applies within a single integer-age year, where $0\le t\le1$, and requires life-table entries that are internally consistent. If $d_x<0$, then the supposed number of deaths is negative. If $d_x>l_x$, then $l_{x+1}<0$. Either condition indicates invalid inputs for this life-table calculation. If $l_x=0$, $q_x=d_x/l_x$ is undefined; and if the conditioning-age survivor count is zero, the conditional survival ratio is undefined as well.

For valid inputs, the endpoint identities and monotonicity checks are especially useful: $l_{x+0}=l_x$, $l_{x+1}=l_x-d_x$, and $l_{x+t}$ must not increase as $t$ increases. These checks do not replace the UDD assumption, but they can quickly expose a wrong fraction, a misplaced denominator, or a death count taken from the wrong year.

Key takeaway: Keep counts and probabilities distinct: $d_x=l_x-l_{x+1}$ and $q_x=d_x/l_x$. Under UDD, interpolate with $l_{x+t}=l_x-t d_x$, check both integer-age endpoints, and calculate conditional survival as later survivors divided by earlier survivors.

Check Your Understanding

For each problem, use the stated UDD assumption and check that the result is consistent with the life-table counts.

A hypothetical table has $l_{40}=8{,}000$ and $d_{40}=400$. Find $q_{40}$ and $l_{40.6}$. (Answer: $q_{40}=0.0500$, $l_{40.6}=7{,}760$.)
A hypothetical table has $l_{51}=10{,}000$ and $d_{51}=800$. Find the probability of surviving from age 51.25 to age 51.75. (Answer: $0.9592$.)
A hypothetical table has $l_{48}=6{,}000$, $d_{48}=300$, and $d_{49}=420$. Find the probability of surviving from age 48.4 to age 49.5. (Answer: $0.9337$.)
A hypothetical table has $l_{30}=2{,}500$ and $l_{31}=2{,}250$. An analyst reports $q_{30}=250/2{,}250$. Find the correct $q_{30}$ and the UDD probability of surviving from age 30.2 to age 30.7. (Answer: $q_{30}=0.1000$, survival probability $=0.9490$.)
For a hypothetical year, $q_x=0.20$. Under UDD, find the probability of surviving from age $x+0.5$ to age $x+1$, and state whether the starting survivor count is positive. (Answer: $0.8889$; yes, since $1-0.5q_x=0.90>0$.)

What You'll Learn