Question

The six numbers below are drawn from N(25, 5). Compute by hand their standard scores. (Just type in your answers; remember, by-hand calculations need not be shown.)

27.5- Standard Score= 0.5     b. 45.0- Standard Score=4     c. 28.4- Standard Score= 0.68    d. 31.2- Standard Score= 1.24    e. 21.1- Standard Score= -0.78    f. 15-Standard Score= -2

What relationship do you see between the number, its z-score, and the mean of the distribution?  

I have already discovered the standard scores but I dont understand the relationships

Answer 1

Solution:

(First we have to confirm that whether   implies the distribution or  . However, here the standard scores or z-scores are already calculated using the notation  .)

Given that six numbers are drawn from .

The six numbers are : (a) 27.5, (b) 45.0, (c) 28.4, (d) 31.2, (e) 21.1, (f) 15.

We know, that the standard scores or the z-scores are given by,

,

where, x is the given observation,

is the population mean,

is the poulation standard deviation.

We got the standard scores or z-scores of the observations which are given in the following table :

Sorting the above data, we can have the following table :

Clearly, we can see that, as the observations (x) increase in magnitude, the standard scores or the z-scores increase as well. The z-score is the lowest (-2.00) for the observation having the lowest magnitude (15). And, the z-score is the highest (+4.00) for the observation having the highest magnitude (45).

Now, since the six numbers or observations (x) are drawn from , if the value of any observation is 25, then we will have,

Thus, standard score or z-score of the population mean is always zero (0).

Thus, these are some relationships between the number, its z-score, and the mean of the distribution.