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CLASS NOTES: The Normal Curve, z-scores & Other Normal Curve Transformations
CONCEPT
CALCULATION/EXAMPLES
APPLICATION
The normal curve:
The “normal curve” is as
theoretical distribution. But, is used
for describing data as so many
distributions of people-related
measurements come so close to this
ideal normal curve.
Features of a normal curve:
* Most of the scores cluster around
the middle of the distribution. As
distance from the middle increases
in either direction, there are fewer
& fewer scores.
* It is symmetrical. Its two halves
are mirror images of each other.
* All three measures of central
tendency (mean, median & mode)
all fall precisely in the center or
midpoint of the distribution.
* There is a constant relationship
w/ the standard deviation.
* When you go out one full
standard deviation from the mean
the curve above the abscissa
reaches its point where the point
changes direction & begins going
out more quickly than it goes
down.
* No matter how far out the tails
are extended, they will never reach
the x-axis.
Probability & the normal
distribution
The percentage of a normal
distribution
b/t the µ of 0 & σ of 1 is 34.13%
The space b/t 1 σ & 2 σ is 13.59%
Beyond the 2nd σ is 2.28%.
This is the same for both above &
below the mean. So, 50% of scores
fall above the mean & 50% of
scores fall below the mean.
There are larger examples of
probability & the normal
distribution at the end of these
notes.
Standardized Distribution:
Composed of scores that have been
transformed to create
predetermined values for a mean
and SD. Standardized distributions
are used to make dissimilar
* Data is often transformed into
standard scores to obtain more
information about a score or set of
scores.
* Two or more different sets of
distributions comparable.
z-score or standard score: The
purpose is to identify & describe
the exact location of every score in
a distribution. A z-score specifies
the precise location of each X value
within a distribution. The sign of
the z-score (+ or -) signifies
whether the score is above the
mean (positive) or below the mean
( - ). The numerical value of the zscore specifies the distance from
the mean by counting the number
of standard deviations b/t X and µ.
The z-score always consists of 2
parts: a sign (+ or -) and a
magnitude.
z-score formula to be used for
transforming back & forth from x
values to z-scores
scores are converted to have the
same mean & SD so that
information & data can be
compared.
* Standard scores make it possible
to compare different scores or
different individuals even though
they may come from completely
different distributions.
z-score formula for a population:
z=x-µ
σ
z-score formula for a sample:
z=X-M
s
Example:
x = 120
µ = 100
σ = 10
z = 120 – 100
10
20 = 2.00
10
The look of a standardized
distribution.
This formula is used when you
want to convert a raw score to a
standard score, or z-score.
X - µ is a deviation score
measuring the distance b/t a
particular score & the standard
mean. σ is used so that the z-score
measures distance in terms of SD
units. For a sample, X represents
the score, while M represents the
sample standard mean & s is the
sample standard deviation score.
In this example available, the raw
score given is 120. You want to
standardize that score to a mean of
100 & SD of 10. Plug the numbers
in the formula & you get an answer
of 2.00 indicating that the raw
score of 120 would fall 2 SD’s
above the mean when the mean is
100 & SD is 10.
Standardized data are all
represented by a bell curve. When
converting to z-scores, the mean
will always be 0 & the SD will
always be + or – 1. Z-scores are not
the only standard scores though.
Others include SAT scores, IQ
scores, etc.. A standard score is any
score in which raw scores have
been converted from its raw form
to a standard that has the same
mean & deviation score.
Converting from standard scores to
raw scores
X = µ + zσ
This formula is used when you
have a standard score, or a z-score
& you wish to convert back to a
raw score.
Example:
In this example, you are given a
standard mean of 50 & a standard
deviation of 10. The z-score is
– 3.333, which means that the
actual score lies 3.333 standard
deviations below the mean.
Following through w/ the formula
for converting standard scores back
into raw, the raw score comes out
to be 17.
µ = 50
z = - 3.333
σ = 10
X = 50 + (- 3.333)(10)
X = 16.67
X = 17
Converting z-scores to standard
deviation
Finding the range
Converting z-score to the mean
More about Z-Scores:
If every X value is transformed into
a z-score, then the distribution of zscores will have the following
properties:
 The shape of the z-score
distribution will be the same
as the original distribution of
raw scores
 The z-score distribution will
always have a mean of 0.
 The z-score distribution will
always have a SD of 1.
SD = X – M
Z
This formula is used when you
have a z-score & you wish to
convert to a standard deviation, or
the standard distance of a score
from the mean. The X value minus
the mean over the z-score
The range is roughly six times the
SD (6)(SD)
M = X – (Z)(SD)
This formula is used when you
have a z-score & you wish to
convert this score to obtain the
mean, or average. The X value
minus the z-score multiplied by the
SD.
The Unit Normal Table: This
table lists proportions of the normal
distribution for a full range of
possible z-score values.
 The body always corresponds to
the larger part of the distribution
whereas the tail always
represents the smaller section
whether it is on the right or the
left.
 B/c the normal distribution is
symmetrical, proportions on the
right side are exactly the same as
proportions on the left.
 Although z-score values can
change depending upon whether
or not they are on the right or the
left of the mean (+ or - ) the
proportions will always be
positive.
The look of the unit normal table as
represented w/in a normal
distribution.
(A)
z
(D)
(C)
(B)
Between
Above
Below z
mean and
z
z
0.00 0.5000
0.5000 0.0000
0.01 0.5040
0.4960 0.0040
0.02 0.5080
0.4920 0.0080
0.03 0.5120
0.4880 0.0120
0.04 0.5160
0.4840 0.0160
0.05 0.5199
0.4801 0.0199
0.06 0.5239
0.4761 0.0239
0.07 0.5279
0.4721 0.0279
0.08 0.5319
0.4681 0.0319
0.09 0.5359
0.4641 0.0359
To the left in an example or a
portion of the z unit normal table.
Column A lists z-scores.
Column B lists the area that falls
below that particular z-score.
Column C represents the area that
falls above the z-score, &
column D represents the area that
falls between the mean & the zscore.
If a question asks you to determine
the distance under the normal curve
“greater” than the given score, you
obtain the z-score, then you will
mark that score on your normal
distribution & “color in” the space
to the right of that score. If it asks
you to determine the distance under
the normal curve “less” than the
given score, you obtain the z-score,
mark it on your normal distribution
& “color in” the space to the left of
the given score. Locate your zscore on the UNT. If more than
50% of your distribution is
“colored in,” then you will use the
proportion in the body. If less than
50% of your distribution is
“colored in,” then you will use the
proportion in the tail.
This picture corresponds with
column B (the area below the zscore)
The area C corresponds with
column C (the area that falls above
the z-score)
The area D corresponds with
column D (the area that
Other Standardized
Distributions Based on Z-Scores:
Before standardizing a set of raw
scores into scores w/ a specific
mean & SD, raw scores are
transformed into z-scores & then
into scores w/ a specific mean &
SD
SAT scores (Scholastic Aptitude
Test)
200 300 400
IQ scores (Intelligence Tests)
65 70
T-scores
20
Calculating T-scores
85
30 40
500 600 700 800
SAT scores have a µ of 500 & σ of
100
100 115 130 145
IQ tests are typically converted to a
µ of 100 & σ of 15
50
T-score values range from 20 to 80
& are all positive scores as are the
SAT’s & IQ tests.
60
70
80
X = (z)(SD) + M
Or
T = (z)(10) + 50
Scroll down to view the percentages covered under the normal curve…..
These tables below represent the distance covered under the normal curve. The first curve indicates the
percentages between each half standard deviation (or z-score) above (+) & below (-) the normal curve.
The second curve indicates the same, but shows the percentages covered under the normal curve between
each whole standard deviation (or z-score) & their negative counterpart (ex: +1.00 to -1.00 = 68.27%).
The total percentage covered under the normal curve is 100%. If none of the space under the normal
curve is indicated, that would then be represented by 0%.