To answer this question, let's first understand the concept of z-score and what it represents in the context of a scatterplot. A z-score is a statistical measure that describes a data point's relationship to the mean of a group of values, measured in terms of standard deviations from the mean.
A z-score can be positive or negative:
- A positive z-score indicates that the data point is above the mean.
- A negative z-score indicates that the data point is below the mean.
For a single variable, a larger absolute value of a z-score means that the data point is further from the mean. That is, a z-score close to 0 indicates a value near the mean, while a z-score with a large absolute value indicates a value that is far from the mean.
Now, for an observation with two variables, x and y, the z-scores for x and y (Zx and Zy) tell us how many standard deviations away from the mean x and y are, respectively. If we have a scatterplot of x and y, the center of the scatterplot would be located at the mean of x-values and the mean of y-values, which corresponds to the point (mean of x, mean of y).
When we look at the product of these two z-scores, ZxZy, we are effectively looking at a measure of how far an observation is from the mean in a multidimensional (two-dimensional, in this case) space. The absolute value of the product, |ZxZy|, helps us understand the distance without concern for the direction:
- If |ZxZy| is small, it means that at least one of the variables is close to its mean, and hence the observation is closer to the center of the scatterplot.
- If |ZxZy| is large, it means that both Zx and Zy are far from their respective means (either both are high, both are low, or one is high and the other is low), resulting in the observation being farther from the center of the scatterplot.
Thus, the larger the absolute value of the product ZxZy, the farther that point will be from the center of the scatterplot of x and y.
The correct answer is:
d. farther that point will be from the center of the scatterplot of x and y.
