Converting Central Death Rates To qx

Why a Central Death Rate Is Not a Probability

A one-year death probability and a central death rate both describe mortality over the age interval from \(x\) to \(x+1\), but they use different denominators. The probability \(q_x\) is the proportion of lives aged \(x\) who die before age \(x+1\). The central death rate \(m_x\), by contrast, is the number of deaths divided by the person-years lived in that interval. Since the denominator is exposure rather than the number alive at the start, \(m_x\) is a rate, not a probability.

To convert \(m_x\) to \(q_x\), an assumption is needed about when deaths occur within the year. This tutorial uses the uniform distribution of deaths (UDD) assumption: conditional on dying between ages \(x\) and \(x+1\), a life’s time of death is uniformly distributed across that year. UDD provides a direct relationship between the number of deaths and the total exposure that those lives contribute.

Definition: For the age interval \([x,x+1)\), let \(l_x\) be the number alive at exact age \(x\), and let \(d_x=l_x-l_{x+1}\) be the number who die during the interval. The one-year death probability is \(q_x=d_x/l_x\), and the one-year survival probability is \(p_x=1-q_x\). The central death rate is \(m_x=d_x/L_x\), where \(L_x\) is the total person-years lived during the interval.

These definitions distinguish the quantities even when a table reports them side by side. The denominator of \(q_x\) is the starting population, \(l_x\); the denominator of \(m_x\) is exposure, \(L_x\). Under UDD, a life who dies contributes, on average, half a year of exposure during the interval, while a life who survives contributes one full year.

Deriving the UDD Conversion

Suppose the interval begins with \(l_x\) lives, of whom \(d_x\) die during the year. Under UDD, the number still alive \(t\) years after age \(x\), for \(0\leq t\leq1\), is \(l_x-td_x\). The total exposure is the integral of this number over the year. Integrating gives the exposure formula and then the conversion from \(m_x\) to \(q_x\).

L_x=\int_0^1 (l_x-td_x)\,dt =l_x-\frac{d_x}{2}.

Since \(d_x=l_xq_x\), this can also be written \(L_x=l_x(1-q_x/2)\). Dividing deaths by exposure yields the central rate. Solving that equation for \(q_x\) gives the required conversion.

m_x=\frac{d_x}{L_x} =\frac{l_xq_x}{l_x(1-q_x/2)} =\frac{q_x}{1-q_x/2}, \qquad q_x=\frac{m_x}{1+m_x/2} =\frac{2m_x}{2+m_x}.

The corresponding one-year survival probability follows by subtracting the converted death probability from 1. Equivalently, substitute \(q_x\) into \(p_x=1-q_x\):

p_x=1-q_x =\frac{1-m_x/2}{1+m_x/2}.

The derivation explains the direction of the adjustment. Under UDD, some lives who die during the interval are still present for part of the year and contribute exposure. Thus, the exposure denominator is smaller than \(l_x\), and the central rate \(m_x\) is generally greater than the corresponding probability \(q_x\). The conversion corrects for that difference; it does not treat the rate as though it were already a probability.

Validity check: A one-year death probability must satisfy \(0\leq q_x\leq1\). Under the UDD conversion, this requires \(0\leq m_x\leq2\). If a stated central rate is greater than 2, it cannot be converted to a valid one-year probability under this assumption.

Applying the Conversion

Use \(m_x\) as a decimal rate, not as a percentage number. For example, a rate of 8% is entered as \(0.08\). Once \(q_x\) is calculated, obtain the survival probability as \(1-q_x\), or use the equivalent formula for \(p_x\). The two values should add to 1.

Worked Example: Converting a Central Rate to a Death and Survival Probability

A hypothetical mortality table gives a central death rate of \(m_{55}=0.08\) for the year of age from 55 to 56. Assuming UDD during this year, calculate the one-year death probability and survival probability. Give probabilities to four decimal places.

Method. Apply the UDD conversion to the central rate, then use the complement for survival. The stated \(0.08\) is already a decimal rate.

First calculate the denominator:

1+\frac{m_{55}}{2} =1+\frac{0.08}{2} =1+0.04 =1.04.

Then calculate the death probability:

q_{55}=\frac{0.08}{1.04} =0.076923\ldots \approx 0.0769.

The survival probability is:

p_{55}=1-q_{55} =1-0.076923\ldots =0.923076\ldots \approx 0.9231.

Check. The rounded probabilities sum to \(0.0769+0.9231=1.0000\). Using the unrounded value, \(q_{55}/(1-q_{55}/2)=0.08\), recovering the stated central rate. The common mistake to avoid is reporting \(0.08\) itself as the death probability.

Worked Example: Converting Deaths and Exposure into \(q_x\)

A hypothetical age interval has 36 deaths and 900 person-years of exposure. Assume that the deaths are uniformly distributed through the year and that these figures refer to the same interval. Calculate the central rate, the one-year death probability, and the one-year survival probability, giving probabilities to four decimal places.

Method. First compute the central rate from its definition, then convert that rate under UDD. This makes the exposure denominator explicit rather than treating the death count as a probability.

The central rate is:

m_x=\frac{36}{900}=0.04.

Convert this rate to a one-year death probability:

q_x=\frac{0.04}{1+0.04/2} =\frac{0.04}{1.02} =0.039215\ldots \approx 0.0392.

The corresponding one-year survival probability is:

p_x=1-0.039215\ldots =0.960784\ldots \approx 0.9608.

Check. The probabilities sum to \(1.0000\) when rounded as shown, and substituting the unrounded \(q_x\) into \(q_x/(1-q_x/2)\) gives \(0.04\). Dividing 36 deaths by 900 exposure gives a central rate, not the death probability; the conversion is still required.

Worked Example: Recovering a Life-Table Probability from Exposure

In a hypothetical life table, \(l_x=80{,}000\) lives are alive at exact age \(x\), and \(d_x=2{,}400\) die before age \(x+1\). Suppose the interval exposure is \(L_x=78{,}800\) person-years, consistent with UDD. Find \(m_x\), convert it to \(q_x\), and calculate \(p_x\). Give probabilities to four decimal places.

Method. Use deaths divided by exposure to obtain the central rate, then convert. The given counts also allow a direct check of the resulting probability.

The central rate is:

m_x=\frac{2{,}400}{78{,}800} =\frac{6}{197} \approx 0.03045685.

Now apply the conversion. Since \(m_x/2=3/197\), the denominator is \(1+3/197=200/197\), so:

q_x=\frac{6/197}{200/197} =\frac{6}{200} =0.03.

Therefore:

$$ p_x=1-0.03=0.97. $$

To four decimal places, \(q_x=0.0300\) and \(p_x=0.9700\). Check. Directly from the counts, \(d_x/l_x=2{,}400/80{,}000=0.03\), confirming the conversion. Also, under UDD, \(L_x=l_x-d_x/2=80{,}000-1{,}200=78{,}800\), exactly the stated exposure. The common mistake to avoid is using \(d_x/L_x\) as though that ratio were \(q_x\).

Worked Example: Combining Age-Specific Rates into a Survival Probability

A hypothetical table gives central rates \(m_{62}=0.02\), \(m_{63}=0.03\), and \(m_{64}=0.04\). Assume UDD separately in each year and that the annual mortality rates apply successively to a life aged 62. Calculate the probability of surviving from age 62 to age 65. Give the final probability to four decimal places.

Method. Convert each age-specific rate to its one-year survival probability. Multiply those conditional annual survival probabilities to obtain three-year survival. Do not convert an average of the rates: each rate applies to a different age interval.

At age 62:

p_{62}=\frac{1-0.02/2}{1+0.02/2} =\frac{0.99}{1.01} =\frac{99}{101} \approx 0.980198.

At age 63:

p_{63}=\frac{1-0.03/2}{1+0.03/2} =\frac{0.985}{1.015} =\frac{197}{203} \approx 0.970443.

At age 64:

p_{64}=\frac{1-0.04/2}{1+0.04/2} =\frac{0.98}{1.02} =\frac{49}{51} \approx 0.960784.

The probability of surviving all three successive years is:

{}_3p_{62}=p_{62}p_{63}p_{64} =\frac{99}{101}\cdot\frac{197}{203}\cdot\frac{49}{51} \approx 0.9139.

Check. Each annual survival probability is below 1, so their product is also below each individual factor, as expected. The answer is the probability of surviving each of the three intervals, not the survival probability for just one interval.

Common Mistakes and Practical Checks

The conversion is short, but errors often arise from confusing a rate with a probability or from applying the formula to a rate expressed in the wrong units. A reliable calculation identifies the interval and the definition of the supplied rate before substituting.

Using \(m_x\) as \(q_x\). The central rate divides deaths by exposure, while \(q_x\) divides deaths by the number alive at the start. Under UDD, use \(q_x=m_x/(1+m_x/2)\).
Entering a percentage as a whole number. A central rate of 4% is \(0.04\), not \(4\). Substituting 4 would produce an invalid probability under UDD.
Forgetting the assumption. The conversion \(q_x=m_x/(1+m_x/2)\) follows from UDD. It should not be used just because a central rate is given; the mortality-timing assumption must support it.
Mixing age intervals. Use \(m_x\) to calculate \(q_x\) for the interval beginning at age \(x\). For survival across several years with age-specific rates, convert each rate for its own interval and multiply the annual survival probabilities.
Failing to check feasibility. If \(m_x>2\), the formula gives \(q_x>1\). That is not a valid probability, so the input and the UDD assumption are incompatible.

A quick audit is to confirm that \(0\leq q_x\leq1\), that \(p_x=1-q_x\), and that \(q_x/(1-q_x/2)\) returns the original \(m_x\). These checks catch common arithmetic and interpretation errors.

Key takeaway: Under the uniform distribution of deaths assumption, exposure over one year is \(L_x=l_x(1-q_x/2)\). Therefore \(m_x=q_x/(1-q_x/2)\), \(q_x=m_x/(1+m_x/2)\), and the matching one-year survival probability is \(p_x=(1-m_x/2)/(1+m_x/2)\).

Check Your Understanding

For each problem, assume UDD over every stated one-year interval. Use decimal rates and give probabilities to four decimal places.

A hypothetical table gives \(m_{48}=0.06\). Find \(q_{48}\) and \(p_{48}\). (Answer: \(q_{48}=0.0583,\ p_{48}=0.9417\))
There are 54 deaths and 1,350 person-years of exposure in a hypothetical age interval. Find the central rate and then the one-year death probability. (Answer: \(m_x=0.04,\ q_x=0.0392\))
A hypothetical age interval has \(m_x=0.12\). Find the death probability and survival probability for the year. (Answer: \(q_x=0.1132,\ p_x=0.8868\))
Two successive age intervals have central rates \(m_x=0.01\) and \(m_{x+1}=0.02\). Find the probability of surviving both years. (Answer: \({}_2p_x=0.9704\))
A proposed one-year central death rate is \(m_x=2.4\). Can it yield a valid one-year death probability under UDD? (Answer: No; the formula gives \(q_x=1.0909\), which exceeds 1.)

What You'll Learn