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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.
A positive value is one which is bigger than zero.
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.