I have 15 base-4 numbers (0-3) representing 0%, 33%, 66%, and 99% which I want to average. Is it a valid statement that if I convert the base-4 numbers to their associated % and find the average, that final average will represent the average-%?
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$\begingroup$ '$\%$' represents '$\div 100$' or '$\div (1210)_4$'. $\endgroup$– Shuri2060Commented Jul 30, 2017 at 15:44
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$\begingroup$ Do you have the original wording of the exercise ? It´s hard to understand what the sense of this exercise is. $\endgroup$– callculus42Commented Jul 30, 2017 at 15:55
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$\begingroup$ Perhaps more precise is this: Start with a length 15 word in the alphabet $\{0,1,2,3\}$. Let $f(x) = x / 3$ for any rational $x$. Let $x_1, \ldots, x_{15}$ represent the 0-3 digits of the word. Is it true that $\frac{1}{15} \sum_{i=1}^{15} f(x_i) = f(\frac{1}{15} \sum_{i=1}^{15} x_i)$? Less formally, if we convert each 0-3 to a ratio and then average the results, do we get the same thing as averaging the 0-3's and then converting to a ratio? If OP really meant x-> 33x/100 (instead of my x/3 interpretation), the same calculation works. $\endgroup$– Hugh DenoncourtCommented Jul 30, 2017 at 19:26
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If I understand your question correctly, you have a mapping from $(0,1,2,3)$ to $(0\%,33\%,66\%,99\%)$, respectively, and this is a linear mapping. That implies that the mapping preserves averages. For example, if you had instead, three numbers $(1,2,3)$, the average is $2$ which maps to $66\%$ and this is also the average of $(33\%,66\%,99\%)$.
