Final answer:
Z-scores help determine how far and in what direction a value is from the mean in a normal distribution. Scores of -2.00 and -0.34 are left of the mean indicating values below average, with -2.00 being much further away. Scores of 1.25 and 3.50 are right of the mean indicating values above average, with 3.50 being significantly higher.
Step-by-step explanation:
Interpreting the location, direction, and distance of z-scores involves understanding their relationship with the mean of a normal distribution. A z-score indicates how many standard deviations an element is from the mean.
- -2.00: This z-score is two standard deviations to the left of the mean, indicating a value far below the average in the distribution.
- 1.25: This z-score is 1.25 standard deviations to the right of the mean, showing a value somewhat above the average but not extremely far.
- 3.50: Signifying a value 3.5 standard deviations to the right of the mean, this is a z-score far above the average, quite distant from the mean.
- -0.34: It is 0.34 standard deviations to the left of the mean, implying a value slightly below average, but relatively close to the mean.
The empirical rule, also known as the 68-95-99.7 rule, helps us to understand that approximately 68 percent of values lie within one standard deviation of the mean (z-scores -1 to 1), about 95 percent within two standard deviations (z-scores -2 to 2), and about 99.7 percent within three standard deviations (z-scores -3 to 3).