A positive value is one which is bigger than zero.
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A positive value for a correlation indicates a positive correlation; e.g. it has a positive slope.
No. The absolute value of a number is the value of the number ignoring the sign - it is always positive: The absolute value of a negative number is a positive number; The absolute value of a positive number is a positive number.
No. The absolute value is non-negative but, to be pedantic, that does not mean positive. The absolute value of 0 is 0 which is NOT positive.
The absolute value of a positive integer is positive. When taking the absolute value of any integer, one is essentially removing the sign (whether positive or negative), always leaving the remaining number positive.
It's positive by definition. That's what the absolute value is.