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Standard Scores
Standard scores, or “zscores” measure the
relation between each
score and its distribution.
Equation of z-score of Xi
zX i
Xi  X
Suppose the Mean is 100 and the
Standard Deviation is 15:
(a) Suppose Xi = 70, find z-score
(b) Suppose Xi = 115,
find z-score of this value.
To find a z-score, subtract the mean
and divide by the standard deviation.
In this example, we subtract 100, and
divide the difference by 15:
(a) z = (70 – 100)/15 = –30/15 = –2.
(b) z = (115 – 100)/15 = 15/15 = 1.
More Problems
We might know the z-score and
need to solve for the “raw” score;
That is, we know z and we find X.
If the mean is 100 and sX is 15:
Suppose z = 2; find Xi.
(a) If z = (Xi – Mean)/sX = 2
Then (Xi – 100)/15 = 2
Multiply both sides by 15:
(Xi – 100) = (2)(15) = 30.
Add 100 to both sides:
Xi = 100 + 30 = 130.
Properties of standard
• z- scores always have a mean of
• z-scores always have a variance
and standard deviation of 1.
• If X is above the mean, its zscore is positive; if X is below
its mean, its z-score is negative.
Next Topic: Standard
Normal Distribution
• z-scores are useful to simplify
many problems.
• One use is to convert any
normal distribution to the
standard normal distribution,
which is the next topic.