I want to write a math formula that represents the average of non zero value e.g. if I have 4 numbers, one of them is zero, then the sum of the numbers will be divided by 3. Is it correct to say:
$$ \frac {\sum_{k=1}^{4} x_{k}} {|x_{k} > 0|} $$
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No, it's not meaningful notation as written; $k$ is an index bound to the sum in the numerator, but what does it mean in the denominator? A better way to write it would involve the cardinality of the set of non-zero values, e.g. $$\frac{\sum_{k = 1}^4 x_k}{|{k : x_k > 0}|}.$$ – Nov 13 '18 at 19:26
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Also, I really don't see a reason for the downvotes here; it's a reasonable question and involves a clearly stated and common misunderstanding.... – Nov 13 '18 at 19:28
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I think what you want is the following:
$$\frac{\sum_{k=1}^{4} x_{k}}{\sum_{k:x_{k} \neq 0} 1}.$$
You could alternatively define a set $S$ such that
$$S = \{k: x_{k} \neq 0\}$$
and then sum over this set so that you would have
$$\frac{\sum_{k=1}^{4} x_{k}}{\sum_{S} 1}.$$
Note that adding zeros to the sum will not change the numerator. However, you need to have at least one non-zero value $x_{k}$ for this to make any sense.
Jack Moody
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There is no need to. The set $S$ is defined to only include the values of $k$ from $1$ to $4$. – Jack Moody Nov 13 '18 at 19:36
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1The numerator can sum over all $k$ - adding in the $0$ values won't hurt. – Ethan Bolker Nov 13 '18 at 20:49
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That’s true @EthanBolker. Either way will work. I just updated the answer. – Jack Moody Nov 13 '18 at 21:05