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

SUMMARY 153
LEARNING CHECK
1. In N = 25 games last season, the college basketball team averaged μ = 74 points
with a standard deviation of σ = 6. In their final game of the season, the team
scored 90 points. Based on this information, the number of points scored in the final
game was _____.
a. a little above average
b. far above average
c. above average, but it is impossible to describe how much above average
d. There is not enough information to compare last year with the average.
2. Under what circumstances would a score that is 15 points above the mean be
considered to be near the center of the distribution?
a. when the population mean is much larger than 15
b. when the population standard deviation is much larger than 15
c. when the population mean is much smaller than 15
d. when the population standard deviation is much smaller than 15
3. Under what circumstances would a score that is 20 points above the mean be
considered to be an extreme, unrepresentative value?
a. when the population mean is much larger than 20
b. when the population standard deviation is much larger than 20
c. when the population mean is much smaller than 20
d. when the population standard deviation is much smaller than 20
ANSWERS
1. B, 2. B, 3. D
SUMMARY
1. Each X value can be transformed into a z-score that
specifies the exact location of X within the distribution.
The sign of the z-score indicates whether the
location is above (positive) or below (negative) the
mean. The numerical value of the z-score specifies the
number of standard deviations between X and μ.
2. The z-score formula is used to transform X values into
z-scores. For a population:
For a sample:
z 5 X 2m
s
z 5 X 2 M
s
3. To transform z-scores back into X values, it usually
is easier to use the z-score definition rather than a
formula. However, the z-score formula can be transformed
into a new equation. For a population:
X = μ + zσ
For a sample: X = M + zs
4. When an entire distribution of X values is transformed
into z-scores, the result is a distribution of z-scores.
The z-score distribution will have the same shape as
the distribution of raw scores, and it always will have
a mean of 0 and a standard deviation of 1.
5. When comparing raw scores from different distributions,
it is necessary to standardize the distributions
with a z-score transformation. The distributions will
then be comparable because they will have the same
parameters (μ = 0, σ = 1). In practice, it is necessary
to transform only those raw scores that are being
compared.