Which of the following statements about the standardized z-score of a value of a variable X, which has a mean of m and a standard deviation of s, is/are TRUE? A. The z-score has a mean equal to 0 B. The z-score has a standard deviation equal to 1 C.The z-score tells us how many standard deviation units from the original observation fall away from the mean. D.The z-score tells us the direction the observation falls away from the mean. E. All of the above statements about the z-score are true.

The z score basically tells us that how standard deviation units far our observation fall from the mean value of the data we have which may be positive or negative which further means that our option C and D are correct as they are indicating the direction of our observation falls away from mean.

Now I will show how the mean and standard deviation of z-score is 0 and 1 respectively.

Consider value of variable X from population be {4,8}.

Now the mean of X will be, [tex]\mu[/tex] = [tex]\frac{\sum X}{N}[/tex] where N= 2

so mean = [tex]\frac{4+8}{2}[/tex] =6

Standard deviation formula = [tex]\sqrt{\frac{\sum (X-\mu )^{2}}{N}}[/tex] =[tex]\sqrt{\frac{-2^{2}+ 2^{2}}{2}}[/tex] = 2

The z -score has a formula z = [tex]\frac{X-\mu }{\sigma }[/tex]

so z 1 = [tex]\frac{4-6 }{2}[/tex] = -2

z 2 = [tex]\frac{8-6 }{2}[/tex] =2

Now to calculate mean of z score = [tex]\frac{-2 + 2 }{2}[/tex] = 0

Standard deviation of z score = [tex]\sqrt{\frac{-1^{2}+1^{2}}{2}}[/tex] = 1

Hence our option A and B are also correct.