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I need to calculate the mortality rates per 100,000 for various age groups (Ex: age 0-19), however the data I am given is broken down into smaller age categories (Ex: ages 0-4, 5-9, 10-14, 15-19). Can I simply add the rates together for each sub-age group to get the total mortality rate for 0-19 years?

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No. Suppose, for example, that the rate for each age group is 1 in 100,000. Then the overall rate should of course be 1 in 100,000, not 4 in 100,000. In general the overall rate is an average of the individual rates where each individual rate is weighted by the population of its age group. So you would compute the overall rate as (rate of group 1 * population of group 1 + rate of group 2 * population of group 2 + ...) / total population of all groups.

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