"How many people have ever lived on Earth?" is not a question amenable to an exact answer: we have only rough estimates of the number of people living a thousand years ago, much less six thousand years ago; besides, since we don't know when humanoids become *Homo sapiens*, we don't know when we should start counting. However, it's interesting to have an order-of-magnitude estimate, and it allows us to put our present-day population explosion into historical perspective. Mathematical demographer Nathan Keyfitz proposed two methods of calculating the number of people who have ever lived. In this page, I'll derive each of these methods, provide a spreadsheet which uses these methods to calculate the number of people who have ever lived, and give an updated Feb. 2011 estimate of the number of people who have ever lived.

The idea of this method is simple: to determine the number of humans who have ever lived, we find the number of humans who have ever been born. (This method comes from *Applied Mathematical Demography* by Nathan Keyfitz.) First, suppose we know the number of births at time a and time b and we want to find the total number of births in between. Let B(t) be a function giving the number of births at time t. Suppose ; i.e. the birth rate is proportional to the number of births, which makes sense because people make more people. (This exponential population model is central to both methods. Since we have so little information, we have to use a population model to fill in the blanks.) This differential equation has the solution . The total number of births from time a to time b is . We can find r by considering the ratio of the number of births at time b to the number of births at time a, ; taking the log of both sides, we have . Substituting this value of r into the expression for the total number of births from before, we find that the number of births between time a and time b is:

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Instead of B(t), a function giving the number of births at time t, we look at N(t), a function giving the total population at time t. Integrating this function from time a to time b does not give the number of people who lived in this interval--it gives the amount of person-time lived. We divide this person-time by average life expectancy to convert it into the number of people who lived in this interval. (This method comes from Keyfitz's paper "How many people have lived on the earth?", *Demography* **3**: 581-582 (1966). See also this similar calculation by Tom Ramsey.) We have . We repeat the steps above to find an expression giving the value of the integral of N(t) from time a to time b, and divide this figure in person-hours by the average life expectancy to find an expression for the number of people who lived in this interval, namely:

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The birth rate (method 1) and the population growth rate (method 2) are not constant over human history. However, they may be constant, or nearly constant, over shorter intervals. The strategy for using these methods to calculate how many people have ever lived is 1) to divide up human history into short intervals; 2) to use estimates of the birthrate and population at the edges of each interval to find the number of people who lived in each interval; 3) to sum up the number of people who lived in each interval to find the number of people who have ever lived. Below is an Excel spreadsheet calculating how many people have ever lived on Earth based on the intervals and population estimates in Carl Haub's article "How Many People Have Ever Lived on Earth?"

how-many-people-have-ever-lived.xlsxMy population estimates based on number of births (method 1) are consistently high compared to Carl Haub's. He may have been working with unrounded population, birth, or year numbers. Method 1 gives that the number of people who have ever been born as of Feb. 2011 is **113.5 billion**, versus **74.5 billion** from method 2. This means (using the method 1 number) that **6.1% of people who have ever lived** are alive in Feb. 2011. In summary, probably about 100 billion people have ever lived, and a sizable fraction, although by no means most, of those 100 billion people are alive today.

Will Vaughan. Last revision February 7, 2011.