What is the range for correlation coefficients between two variables?

• A. (-inf, inf)
• B. [-1,1]
• C. (-1,1)
• D. (0,1)

Answer: (B)

The range for a correlation coefficient reflects how strong and in what direction a linear relationship can be measured between two variables. The coefficient, often denoted r, is computed by standardizing the data and looking at the average product of the standardized scores. This value cannot exceed 1 in magnitude, and it can reach -1, 0, or 1 depending on the relationship. If all points line up on a straight line with positive slope, r is 1; with negative slope, r is -1; and if there’s no linear pattern, r is around 0. The mathematical reason is that this measure is like the cosine of the angle between the two standardized vectors, whose value is always between -1 and 1 inclusive. So the valid range is [-1, 1], making endpoints possible for perfect linear relationships. The other ranges miss either the negative side, exclude the endpoints, or both.