Statistics Tutorials - Z Score
Assume that  has a normal distribution, with mean  and standard deviation  . This is typically
written as
~ 
Then, the Z-score associated to is defined as
Example: Consider the random variable , which as a normal distribution, with mean  and
standard deviation  . Compute the z-score of 
Answer: Using the definition of z-score, we use the following formula:
» What does the z-score represent?
The z-score gives measures how far the random variable is from its mean . This measure is not arbitrary, it
indicates how many standard deviations the value of is away from . In other words, a z-score of 1.75 indicates
that the value of is 1.75 standard deviations away from its mean. Since the z-score is positive, that means that the
value of is 1.75 standard deviations to the right of its mean, to be more precise.
Application Example: Peter took his finance exam last week, and he got 89/100. The mean for his class was 77, with a standard deviation
of 15. Jenna took her math test last week too, and she got 84/100. The mean for her class was 75, with a standard deviation
of 5. Their were arguing on who did better, who do you think did better relative to their class?
Answer: We need to use z-scores. For Peter we have
On the other hand, for Jenna:
The z-score associated with Jenna's score test is higher than the z-score test associated with Peter's score test, which means
that Jenna did better than Peter, relative to her class.
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