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ANATOMY OF A STANDARDIZED VARIABLE: Obtaining a z-score
A standardized variable is a variable that always has a mean of 0 and a standard deviation of 1.
In many (most) situations we will be unable to work with a
variable x in its “raw” form. We will first have to
standardize the variable in order to work with it. From a
variable x, we can obtain the standardized variable by
subtracting from it the mean, and then dividing this difference
by the standard deviation. The value obtained is called a zscore. We use the following formula to standardize x:
z
x

Okay, so what is a z-score????? A z-score simply tells us how many standard
deviations the observation x is from the mean. It is a measure of distance from the
mean in terms of the standard deviation. Z-scores can be negative or positive. A
negative z-score indicates that the observation x is less than the mean. A positive
z-score indicates that the observation x is greater than the mean.
Where z is the z-score.
Let’s look at how this is done.
Example:
It is known that the mean height of women ages 18-45 is 65 inches, with a standard deviation of 2.5 inches. If a woman is 72 inches tall, how far
above or below the mean height of 65 inches is she? Here, our variable x is 72 inches. The mean is 65 inches, and the standard deviation is 2.5
inches. To standardize x, we use the formula above, substituting into it the values of x, the mean and the standard deviation, to obtain the following:
z
x


72  65
7

 2.8
2.5
2.5
This particular woman is 2.8 standard deviations above the mean height for women ages 18-45.
  2.8
Pictorially:
57.5
  3
60
  2
62.5
 
65

67.5
 
70
  2
72.5
  3