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Chapter 5

NORMAL DISTRIBUTION AND Z-SCORE
Understanding Statistics for International Social
Work and Other Behavioral Sciences
Serge Lee, Maria C Silveira Nunes Dinis, Lois Lowe, and
Kelly Anders (2015). Oxford University Press
NORMAL CURVE
2
The sum of the deviations of all observation scores from the mean is
balanced at the center point on the number line, which is equal to or very
near zero
SKEWED DISTRIBUTION
3
 Positively Skewed Distributions
 Negatively Skewed Distributions
Lee. Dinis, Lowe, Anders (2015). Understanding statistics for international social work and other behavioral sciences. Oxford University Press
STANDARD DEVIATION ON PROPORTION
OF THE NORMAL CURVE
4
 The normal curve is divided into six equal units. The six units
collectively reflect the variation that exists within virtually all values of
a normally distributed interval or ratio level scale variable.
Confidence interval
Lee. Dinis, Lowe, Anders (2015). Understanding statistics for international social work and other behavioral sciences. Oxford University Press
DISTANCE BETWEEN THE SAMPLE MEAN AND
STANDARD DEVIATION
5
 The proportion between the mean and their corresponding standard
deviations are divided as below:
Distance from the
mean and standard
deviation
Left side only of the
curve between mean
and SD (-SD)
Right side only of
the curve between
mean and SD (+SD)
Both sides under
areas of the normal
curve
1SD
34.13
34.13
68.26
2SD
13.59
13.59
27.18
3SD
2.15
2.15
4.30
Total areas under
the normal curve
49.87
49.87
99.74
Confidence interval
Lee. Dinis, Lowe, Anders (2015). Understanding statistics for international social work and other behavioral sciences. Oxford University Press
CALCULATE THE Z-SCORE AND CONVERT TO A PERCENTILE
6
Formula:
Z=
X−X
SD
Statistics rules for the conversion of
the Z-score into percentile ranks:
• Always round the calculated Z-score
to two decimal places. For example,
Z = -2.29
• If the calculated Z-score is positive
(+), add 50% or .50 (if use
decimals) to the percentile obtained
from the standard normal curve
table
• If the calculated Z-score is negative
(-), subtract the percentile obtained
from the critical values of the
standard normal curve table from
50% or .50 (if use decimals)