Transcript The Standard Normal Distribution

THE STANDARD NORMAL DISTRIBUTION
a.k.a. “bell curve”
THE NORMAL DISTRIBUTION



If a characteristic is normally distributed in
a population, the distribution of scores
measuring that characteristic will form a
bell-shaped curve.
This assumes every member of the
population possesses some of the
characteristic, though in differing degrees.
 examples: height, intelligence, self
esteem, blood pressure, marital
satisfaction, etc.
Researchers presume that scores on most
variables are distributed in a “normal”
fashion, unless shown to be otherwise
 Including communication variables
THE NORMAL DISTRIBUTION


Only interval or ratio level data can be
graphed as a distribution of scores:
 Examples: physiological measures,
ratings on a scale, height, weight,
age, etc.
 Any data that can be plotted on a
histogram
Nominal and ordinal level data cannot
be graphed to show a distribution of
scores
 nominal data is usually shown on a
frequency table, pie chart, or bar
chart
MORE ABOUT THE NORMAL DISTRIBUTION




Lower scores are found toward
the left-hand side of the curve.
Medium scores occupy the
middle portion of the curve
 this is where most scores
congregate, since more
people are average or
typical than not
Higher scores are found
toward the right-hand side of
the curve
In theory, the “tails” of the
curve extend to infinity (e.g.
asymptotic)
lower
scores
medium
scores
higher
scores
MORE ABOUT THE NORMAL DISTRIBUTION


In a normal distribution, the
center point is the exact
middle of the distribution (the
“balance point”)
In a normal, symmetrical
distribution, the mean,
median, and mode all occupy
the same place
mean
median
mode
COMPARING GROUPS BASED ON THEIR
MEANS AND STANDARD DEVIATIONS

Note the height of the curve
does not reflect the size of the
mean, but rather the number
of scores congregated about
the mean
NON-NORMAL DISTRIBUTIONS



Kurtosis refers to how “flat” or
“peaked” a distribution is.
In a “flat” distribution scores
are spread out farther from
the mean
 There is more variability in
scores, and a higher
standard deviation
In a “peaked” distribution
scores are bunched closer to
the mean
 There is less variability in
scores, and a lower
standard deviation
kurtosis
KURTOSIS




Non-normal distributions may
be:
Leptokurtic (or peaked)
 Scores are clustered
closer to the mean
Mesokurtic (normal, bell
shaped)
Platykurtic (flat)
 Scored are spread out
farther from the mean
NON-NORMAL DISTRIBUTIONS



Skewness refers to how
nonsymmetrical or “lopsided” a distribution is.
If the tail extends toward
the right, a distribution is
positively skewed
If the tail extends toward
the left, a distribution is
negatively skewed
skewness
MORE ABUT SKEWNESS



In a positively skewed
distribution, the mean
is larger than the
median
In a negatively skewed
distribution, the mean
is smaller than the
median
Thus, if you know the
mean and median of a
distribution, you can tell
if it is skewed, and
“guesstimate” how
much.
NEGATIVELY SKEWED DISTRIBUTION

Only 2% of Americans
earned more than
$250,000 per year in
2005
STANDARD DEVIATIONS AND THE
NORMAL DISTRIBUTION


Statisticians have calculated
the proportion of the scores
that fall into any specific
region of the curve
For instance, 50% of the
scores are at or below the
mean, and 50% of the scores
are at or above the mean
50%
50%
STANDARD DEVIATIONS AND THE
NORMAL CURVE



Statisticians have
designated different
regions of the curve, based
on the number of standard
deviations from the mean
Each standard deviation
represents a different
proportion of the total area
under the curve
Most scores or
observations (approx. 68%)
fall within +/- one standard
deviation from the mean
68.26%
34.13% 34.13%
-3 SD
-2 SD
-1 SD +1 SD
+2 SD
+3 SD
STANDARD DEVIATIONS AND THE
NORMAL CURVE

Thus, the odds of a
particular score, or set
of scores, falling within
a particular region are
equal to the percentage
of the total area
occupied by that region
34.13%
34.13%
13.59%
13.59%
2.14%
-3 SD
2.14%
-2 SD
-1 SD +1 SD
68.26%
95.44%
99.72%%
+2 SD
+3 SD
THE 68-95-99 % BENCHMARKS



68.2% of all scores should
lie within 1 SD of the mean
95.4% of all scores should
fall within 2 SDs of the
mean
99.7% of all scores should
fall within 3 SDs of the
mean
PROBABILITY THEORY AND STATISTICAL
SIGNIFICANCE


The odds that a score or
measurement taken at
random will fall in a specific
region of the curve are the
same as the percentage of
the area represented by that
region.
Example: The odds that a
score taken at random will
fall in the red area are
roughly 68%.
random
score
-3
-2
+1
+1
68.26%
+2
+3
PROBABILITY THEORY AND STATISTICAL
SIGNIFICANCE


The probability of a random
or chance event happening in
any specific region of the
curve is also equal to the
percentage of the total area
represented by that region
the odds of a chance event
happening two standard
deviations beyond the mean
are approximately 4.28%, or
less than 5%
-3
-2
The odds of a random
or chance event
happening in this
region are 2.14%
+1
+1
+2
+3
The odds of a random
or chance event
happening in this
region are 2.14%
PROBABILITY THEORY AND STATISTICAL
SIGNIFICANCE


When a researcher states
that his/her results are
significant at the p < .05
level, the researcher means
the results depart so much
from what would be expected
by chance that he/she is 95%
confident they could not have
been obtained by chance
alone.
The results are probably due
to the experimental
manipulation, and not due to
chance
-3
-2
-1
+1
+2
By chance
alone, results
should wind up
in either of
these two
regions less
than 5% of the
time
+3
PROBABILITY THEORY AND STATISTICAL
SIGNIFICANCE


When a researcher states
that his/her results are
significant at the p < .01
level, the researcher means
the results depart so much
from what would be expected
by chance alone, that he/she
is 99% confident they could
not have been obtained
merely by chance.
The results are probably due
to the experimental
manipulation and not to
chance
-3
-2
-1
+1
+2
By chance alone, results
should wind up in either
of these two regions less
than 1% of the time.
+3
PROBABILITY THEORY AND STATISTICAL
SIGNIFICANCE


When a researcher employs a
nondirectional hypothesis,
the researcher is expecting a
significant difference at
either “tail” of the curve.
When a researcher employs a
directional hypothesis, the
researcher expects a
significant difference at one
specific “tail” of the curve.
-3
-2
-1
+1
+2
+3
Nondirectional hypothesis
either tail of the curve
Directional hypothesis
one tail or the other
PROBABILITY THEORY AND STATISTICAL
SIGNIFICANCE





The “control” group in an
experiment represents normalcy.
Scores for a “control” group are
expected to be typical, or
“average.”
The “treatment” group in an
experiment is exposed to a
manipulation or stimulus
condition.
Scores for a “treatment” group are
expected to be significantly
different from those of the control
group.
The researcher expects the
“treatment” group to be 2 std. dev.
beyond the mean of the control
group.
-3
-2
-1
+1
+2
+3
The control group should
be in the middle of the
distribution
The treatment group is
expected to be 2 std. dev
beyond the mean