Statistics for the Behavioral Sciences by Frederick J. Gravetter, Larry B. Wallnau ISBN 10: 1305504917 ISBN 13: 9781305504912

DEMONSTRATION 5.2 155
sure that your answer is smaller than the mean, and check that the distance between X and
μ is slightly less than the standard deviation.
2. When comparing scores from distributions that have different standard deviations,
it is important to be sure that you use the correct value for σ in the z-score formula.
Use the σ value for the distribution from which the raw score in question
was taken.
3. Remember that a z-score specifies a relative position within the context of a specific
distribution. A z-score is a relative value, not an absolute value. For example, a z-score
of z = −2.0 does not necessarily suggest a very low raw score—it simply means that
the raw score is among the lowest within that specific group.
DEMONSTRATION 5.1
TRANSFORMING X VALUES INTO Z-SCORES
A distribution of scores has a mean of μ = 60 with σ = 12. Find the z-score for X = 75.
STEP 1
STEP 2
STEP 3
STEP 4
Determine the sign of the z-score. First, determine whether X is above or below the mean.
This will determine the sign of the z-score. For this demonstration, X is larger than (above) μ,
so the z-score will be positive.
Convert the distance between X and μ into standard deviation units. For X = 75 and
μ = 60, the distance between X and μ is 15 points. With σ = 12 points, this distance corresponds
to 15
12 = 1.25 standard deviations.
Combine the sign from step 1 with the numerical value from Step 2. The score is
above the mean (+) by a distance of 1.25 standard deviations. Thus, z = +1.25.
Confirm the answer using the z-score formula. For this example, X = 75, μ = 60,
and σ = 12.
z 5 X 2m
s
5
75 2 60
12
5 115
12 511.25
DEMONSTRATION 5.2
CONVERTING Z-SCORES TO X VALUES
For a population with μ = 60 and σ = 12, what is the X value corresponding to z = −0.50?
Notice that in this situation we know the z-score and must find X.
STEP 1
Locate X in relation to the mean. A z-score of −0.50 indicates a location below the mean
by half of a standard deviation.
STEP 2
Convert the distance from standard deviation units to points.
standard deviation is 6 points.
With σ = 12, half of a
STEP 3
Identify the X value. The value we want is located below the mean by 6 points. The mean
is μ = 60, so the score must be X = 54.