# Adjusting census data - exposed to risk - question 6 tut 3

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When adjusting our census data to get the correct age label is it correct to look at the population at different ages at time t? Or should we always look at the population at different times and decide which applies to our age label?

For example, in question 6 we were given census data age x nearest (Px(t)) and needed age x last (Px'(t)) to match the death data.

Consider t = 01/01/05 only. We could look at data for the population aged x at different times and would arrive at the answer Px'(01/01/05) = 1/2 (Px(01/01/05)) or we could consider different ages at a fixed t - Px'(t) = 1/2 Px(t) + 1/2 Px+1(t) as assuming UBD, 1/2 of those aged x+1 nearest at time t would have been age x last at time t. Is my thinking here completely incorrect?

$$E_x^c=\int_{0}^{2}P_x^{last}(t)dt$$
$$E_x^c=\frac{1}{2}(P_x^{last}(01/01/05)+P_x^{last}(01/01/06))+\frac{1}{2}(P_x^{last}(01/01/06)+P_x^{last}(01/01/07)$$
$$P_x^{last} = \frac{1}{2}(P_x^{nearest}(t)+P_{x+1}^{nearest}(t))$$