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As we all know that the formula for crude death rate is $$CDR = \frac{D}{P} \times 1000$$ where $D$ is the totla number of deaths(during a calender year) among residents of a community and $P$ is the number of person living in that community during that year .

I have read that if two population have different age distributions then it is not reasonable to compare the the $CDR$ of these two population. This comparison can be misleading.

I don't see how can the comparison based on $CDR$ can be misleading in such a case. If a population has more number of old people then the $CDR$ must be higher than the population (let's say of same strength) which comprises of more younger people. And simply based on this comparison we can tell which population has more old people and which one has more younger people. Then what is the problem in such comparison?

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If $_nm_x$ is the death rate at age group $x$ to $x+n$ and $_nP_x$ is the population in the same age group,the total deaths will be $\sum{_nm_x}\times{_nP_x}$,the sum is over all age groups.

Then $CDR$ is $m=\frac{\sum{_nm_x}\times{_nP_x}}{\sum{_nP_x}}$

Clearly,if the values $_nP_x$ are increased for older ages at the expense of the younger ages then not withstanding the constancy of the age rates of mortality,the $CDR$ will rise.The weights used in such an average are therefore important.

Consider the CDR's for two communities $A$ and $B$,

$m^a=\frac{\sum{_nm_x^{a}}\times{_nP_x^{a}}}{\sum{_nP_x^{a}}}$ and $m^b=\frac{\sum{_nm_x^{b}}\times{_nP_x^{b}}}{\sum{_nP_x^{b}}}$

Even when two communities have same mortality situations at different age groups,then $m^a$ and $m^b$ may be unequal simply beacuse the proportions $\frac{_nm_x^{a}}{_nP_x^{a}}$ and $\frac{_nm_x^{b}}{_nP_x^{b}}$ may not be the same beacuse the age distributions of the two communities may not be identical.Hence,$CDR$ can not be used to compare mortality situations in different places unless the populations of the places have identical age/sex distributions,a condition which is seldom fulfilled.

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