(a) The population 3, 4, 9, 14, and 20 has a mean of 10 and a standard deviation of 6.356. The z-scores for each of the five data values are

z₃ 1.101,

z₄ 0.944,

z₉ 0.157,

z₁₄ 0.629,

and z₂₀ 1.573.

Find the mean and the standard deviation of these z-scores.

mean
standard deviation

(b) The population 2, 6, 12, 17, 22, and 25 has a mean of 14 and a standard deviation of 8.226. The z-scores for each of the six data values are

z₂ 1.459,

z₆ 0.973,

z₁₂ 0.243,

z₁₇ 0.365,

z₂₂ 0.973,

and z₂₅ 1.337.

Find the mean and the standard deviation of these z-scores.

mean
standard deviation

(c) Use the results of part (a) and part (b) to make a conjecture about the mean and standard deviation of the z-scores for any set of data. a. The mean of the z-scores is always 1 more than the standard deviation.

b. The mean of the z-scores is always 1, and their standard deviation is 0.

c. The mean of the z-scores is always 0, and their standard deviation is 1.

d. The mean and standard deviation of z-scores are never the same.