A life table (also called mortality table) is a tabular summary of the mortality experience of a population, showing for each age the probability of dying, the number of survivors and life expectancy. It is a powerful demographic and actuarial tool.
| Column | Symbol | Description |
|---|---|---|
| Age | \(x\) | Exact age |
| Survivors | \(l_x\) | Number alive at exact age \(x\) (starts at \(l_0\), the radix, usually 100 000) |
| Deaths | \(d_x\) | Deaths between ages \(x\) and \(x+1\); \(d_x = l_x - l_{x+1}\) |
| Probability of dying | \(q_x\) | \(q_x = d_x / l_x\) |
| Probability of surviving | \(p_x\) | \(p_x = l_{x+1}/l_x = 1 - q_x\) |
| Person-years lived | \(L_x\) | Average person-years lived between ages \(x\) and \(x+1\): \(L_x \approx (l_x + l_{x+1})/2\) |
| Total person-years lived above x | \(T_x\) | \(T_x = \sum_{y \ge x} L_y\) |
| Life expectancy at age x | \(e_x\) | \(e_x = T_x / l_x\) |
Each relation below is derived from the definitions in the table above, one step at a time.
Consider a person exactly aged \(x\). Over the next year exactly one of two things happens: the person survives to age \(x+1\), or the person dies before age \(x+1\). These two events are mutually exclusive and together cover every possibility, so their probabilities must add to \(1\). Writing \(p_x\) for the survival probability and \(q_x\) for the death probability,
\[ p_x + q_x = 1, \qquad\text{equivalently}\qquad p_x = 1 - q_x. \]Of the \(l_x\) people alive at exact age \(x\), those still alive one year later number \(l_{x+1}\). The people who are no longer alive are precisely the deaths between ages \(x\) and \(x+1\), which is what \(d_x\) counts. Subtracting the survivors from the starting group,
\[ d_x = l_x - l_{x+1}. \]Step 1. By definition the survival probability is the fraction of the age-\(x\) group that reaches age \(x+1\): \(p_x = l_{x+1}/l_x\). Multiplying both sides by \(l_x\),
\[ l_{x+1} = l_x\,p_x. \]Step 2. Substituting \(p_x = 1-q_x\) from Relation 1,
\[ l_{x+1} = l_x(1-q_x). \]By definition \(T_x = \sum_{y \ge x} L_y\) is the total person-years still to be lived by the group from age \(x\) onward. Split that sum by peeling off the first term:
\[ T_x = \sum_{y \ge x} L_y = L_x + \sum_{y \ge x+1} L_y. \]The remaining sum is exactly \(T_{x+1}\), so
\[ T_x = L_x + T_{x+1}. \]In words: the years lived from age \(x\) on = the years lived during the single year \([x, x+1]\) (that is \(L_x\)) plus all the years lived from age \(x+1\) on (that is \(T_{x+1}\)). This is why the \(T_x\) column is built up from the bottom of the table.
\(T_x\) is the total number of future years that will be lived, added up over all \(l_x\) people who are alive at age \(x\). The average number of future years per person is that total shared equally among the \(l_x\) survivors:
\[ e_x = \frac{\text{total future person-years}}{\text{number of survivors}} = \frac{T_x}{l_x}, \]which is the (complete) expectation of life at age \(x\).
With \(l_{30}=95\,000,\ l_{31}=94\,500\): Relation 2 gives \(d_{30}=500\); then \(q_{30}=500/95\,000=0.00526\) and \(p_{30}=1-0.00526=0.99474\) (Relation 1); and Relation 3 recovers \(l_{31}=95\,000\times0.99474=94\,500\). ✓
For ages 30 to 32: \(l_{30} = 95\,000,\; l_{31} = 94\,500,\; l_{32} = 94\,000\).
\(d_{30} = 500\), \(q_{30} = 500/95\,000 = 0.00526\), \(p_{30} = 0.99474\).
\(L_{30} = (95\,000 + 94\,500)/2 = 94\,750\).
If \(T_{30} = 4\,500\,000\) and \(l_{30} = 95\,000\), then \(e_{30} = 4\,500\,000 / 95\,000 = 47.37\) years (expected additional years of life for a 30-year-old).
where \(B\) = live births in the year, \(P\) = mid-year population.
Merits: simple, easy to interpret.
Demerits: affected by age-sex composition of population.
where the denominator is the mid-year female population aged 15–49 (reproductive ages).
where \(B_x\) = births to women in age group \(x\), \(P^{\text{f}}_x\) = mid-year female population in age group \(x\).
where \(n\) is the width of the age group (5 if data grouped in 5-year bands). Equivalent: TFR = average number of children a woman would have, surviving the full reproductive span, with current ASFRs.
Replacement-level TFR ≈ 2.1 in low-mortality settings.
City population 1 million; live births 18 000 in the year. CBR = 18 per 1 000.
Female population 15–49 = 250 000. GFR = 18 000/250 000 × 1 000 = 72 per 1 000 women.
Suppose ASFRs (per 1 000) by 5-year age group are:
| Age | 15–19 | 20–24 | 25–29 | 30–34 | 35–39 | 40–44 | 45–49 |
|---|---|---|---|---|---|---|---|
| ASFR | 40 | 120 | 140 | 80 | 40 | 10 | 2 |
Sum = 432. TFR = 432 × 5 / 1 000 = 2.16 children per woman — close to replacement.
Pearl's vital index measures the ratio of births to deaths:
\[ \text{Vital Index} \;=\; \dfrac{B}{D} \times 100. \]If > 100, population is growing; if < 100, declining.
Births = 18 000, Deaths = 8 000. Vital Index = 18 000/8 000 × 100 = 225 → population growing rapidly.
GRR is the average number of female babies that a cohort of women would bear during their reproductive lifetime if they experienced current ASFRs.
where \(f\) is the proportion of female births (≈ 0.487 worldwide), \(n\) is the age-group width.
Often computed simply as: \(\text{GRR} \approx \text{TFR} \times f\).
NRR allows for mortality of women before they complete their reproductive years. It is the average number of daughters that a woman would have, allowing for the chance she may die before age 49.
where \(L_x/l_0\) is the survival probability from the life table.
From Example 2 of fertility: TFR = 2.16. Female-births fraction = 0.487. GRR = 2.16 × 0.487 = 1.052 daughters per woman.
Continuing: suppose the average survival probability from birth to mid-reproductive age is 0.92.
NRR = GRR × 0.92 = 1.052 × 0.92 ≈ 0.968. Slightly below 1 → population just below replacement; long-term decline expected.
| Rate | Formula |
|---|---|
| CDR | (D / P) × 1 000 |
| ASDR | (Dx / Px) × 1 000 |
| CBR | (B / P) × 1 000 |
| GFR | (B / Pf15-49) × 1 000 |
| ASFRx | (Bx / Pfx) × 1 000 |
| TFR | \(\sum \text{ASFR}_x \times n\) / 1 000 |
| Vital Index | (B / D) × 100 |
| GRR | \(\sum \text{ASFR}_x \cdot f \cdot n\) / 1 000 |
| NRR | \(\sum \text{ASFR}_x \cdot f \cdot (L_x/l_0) \cdot n\) / 1 000 |