Given the percentage of each range of years, how to calculate the total average of years?

Question

For example, in a pet shop, 35% of cats is 1 ~ 2 years old, 40% is 2 ~ 4 years old and the remaining 25% is 4 ~ 6 years old. How can I know the average of cat's age in this pet shop?

https://i.stack.imgur.com/OnQdD.png

Answer 1

Assuming the question asks for the average age as a range (the same way the group ages are given), consider the case where cats in each group are at the minimum age, then the overall average is:

$$ 35\% \cdot 1 + 40\% \cdot 2 + 25\% \cdot 4 = 2.15 $$

Then, work out the same calculation for the maximum ages:

$$ 35\% \cdot 2 + 40\% \cdot 4 + 25\% \cdot 6 = 3.8 $$

Therefore the average cat age is between $\,2.15 \,\cdots\, 3.8\,$ years old.

Answer 2

The best you can do is an approximation. Assume all the cats between $1$ and $2$ years old are $1.5$ years old. Do the same for the other two groups. Then take the average, weighted by the percentages (or assume there are exactly $100$ cats, as @TobyMark suggests).