If an entire population of N = 20 scores is transformed into z-scores, the set of 20 z-scores will have ∑ X = 0. true or false?

Answer 1

True

The z score is how many standard deviations a score is from the mean. With a normal distribution, about half the z scores are positive and half negative. Add them up, they equal zero, ignoring rounding errors.

It's the same as calculating the average grade in school. If there were 20 grades, the sum of all grades divided by 20 would equal the average. Z-scores just transform the average grade to zero. Then the below average grades are negative and the above average are positive. The sum of all such transformed grades = 0.

It's similar to finding the best fitting regression line through some data. The sum of errors will equal zero. Some data will fall above the line, some below. On average, the negative and positives cancel, and the expected error is zero.