Bell-Shaped Curves and Other Shapes
Thought Questions
1. The heights of adult women in the United States follow, at least approximately, a bell-shaped
curve.
What do you think this means?
2. What does it mean to say that a man’s weight is in the 30th percentile for all adult males?
3. A “standardized score” is simply the number of standard deviations an individual falls above
or below the mean for the whole group.
Male heights have a mean of 70 inches and a standard deviation of 3 inches. Female heights
have a mean of 65 inches and a standard deviation of 2 ½ inches. Thus, a man who is 73 inches
tall has a standardized score of 1.
What is the standardized score corresponding to your own height?
Bell-Shaped Curves and Other Shapes
Frequency Curves
Smoothed-out histogram by connecting tops of rectangles with smooth curve.
Frequency curve for population
of British male heights.
The measurements follow a normal
distribution (or a bell-shaped or
Gaussian curve).
Height of curve set so area under entire curve is 1.
Frequency curve for population of dollar
amounts of car insurance damage claims.
The measurements follow a right skewed
distribution. Majority of claims were below
$5,000, but there were occasionally a few
extremely high claims.
Bell-Shaped Curves and Other Shapes
Proportions
Key: Proportion of population of measurements falling in a certain range = area under curve
over that range.
Mean British Height is 68.25
inches. Area to the right of the
mean is 0.50. So about half of
all British men are 68.25 inches
or taller.
Many populations of measurements follow approximately a normal curve:
•
Physical measurements within a homogeneous population – heights of male adults.
•
Standard academic tests given to a large group – SAT scores.
Bell-Shaped Curves and Other Shapes
Percentiles and Standardized Scores
Your percentile = the percentage of the population that falls below you.
Finding percentiles for normal curves requires:
• Your own value.
• The mean for the population of values.
• The standard deviation for the population.
Then any bell curve can be standardized so one table can be used to find percentiles.
Proportions and Percentiles for Standard Normal Scores
Standard
Score, z
-6.00
-5.20
-4.26
-3.00
:
-1.00
:
-0.58
:
0.00
Proportion
Below z
0.000000001
0.0000001
0.00001
0.0013
:
0.16
:
0.28
:
0.50
Percentile
0.0000001
0.00001
0.001
0.13
:
16
:
28
:
50
Standard
Score, z
0.03
0.05
0.08
0.10
:
0.58
:
1.00
:
6.00
Proportion
Below z
0.51
0.52
0.53
0.54
:
0.72
:
0.84
:
0.999999999
Percentile
51
52
53
54
:
72
:
84
:
99.9999999
Bell-Shaped Curves and Other Shapes
Standard Normal Curve
Standard Normal Curve
0.45
0.4
0.35
0.3
0.25
0.2
0.15
0.1
0.05
3.4
3.1
2.8
2.5
2.2
1.9
1.6
1.3
1
0.7
0.4
0.1
-0.2
-0.5
-0.8
-1.1
-1.4
-1.7
-2
-2.3
-2.6
-2.9
-3.2
-3.5
0
The standard normal curve has a mean of 0 and standard deviation of 1.
We convert our data values to the their corresponding values on the standard normal curve.
This enables us to get exact probabilities of certain events.
Standardized Score (or z-score) is : observed value – mean
standard deviation
Bell-Shaped Curves and Other Shapes
Empirical Rule
For any normal curve, approximately …
68% of the values fall within 1 standard deviation of the mean in either direction
95% of the values fall within 2 standard deviations of the mean in either direction
99.7% of the values fall within 3 standard deviations of the mean in either direction
Example
Some IQ tests are standardized to a Normal
Model, with a mean of 100 and a standard
deviation of 16
Approximately 95% of the IQ scores are expected
to be within the interval 68 to 132 IQ points.
Approximately 16% of IQ scores are expected to
be above 116 IQ points.
Approximately 2.5% of the IQ scores are expected
to be above 132.
Bell-Shaped Curves and Other Shapes
Heights of Adult Women in US
Heights of US women in US
0.18
Height of curve
0.16
0.14
0.12
0.1
0.08
0.06
0.04
0.02
55
55.7
56.4
57.1
57.8
58.5
59.2
59.9
60.6
61.3
62
62.7
63.4
64.1
64.8
65.5
66.2
66.9
67.6
68.3
69
69.7
70.4
71.1
71.8
72.5
73.2
73.9
74.6
0
Heights (in inches)
Adult women in U.S. have a mean height of 65 inches with a standard deviation of 2.5 inches
and heights are approximately bell-shaped.
68% of adult women are between 62.5 and 67.5 inches,
95% of adult women are between 60 and 70 inches,
99.7% of adult women are between 57.5 and 72.5 inches.
Bell-Shaped Curves and Other Shapes
Standardized Scores (or z-scores)
Standardized Score (or z-score) is
Heights of US women
observed value – mean
standard deviation
0.18
Height of curve
0.16
0.14
0.12
0.1
Remaining Area
0.08
Area less 61 inches
0.06
0.04
Height of US women - Population
Mean height = 65
Standard deviation = 2.5
What is the probability that a woman selected at
random from this population will be less than 61
inches?
0.02
55
56.4
57.8
59.2
60.6
62
63.4
64.8
66.2
67.6
69
70.4
71.8
73.2
74.6
0
Probability(height is less than 61inches)?
Standard Normal Curve
z-score = (61 – 65)/2.5 = -4/2.5 = -1.6
0.45
0.4
61 is 1.6 standardized scores from the mean of 65.
0.35
0.25
Remaing area
0.2
From Tables 8.1 (pg 157 in text):
Area less than -1.6
0.15
Prob(z-score less than -1.6) = .05
0.1
0.05
0
-3.5
-3
-2.5
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
3
3.5
Axis Title
0.3
The probability that a women selected at random
will have a height less than 61 inches is 0.05.
Bell-Shaped Curves and Other Shapes
Example: Cholesterol
Assume the cholesterol levels of adult American women can be described by a Normal
model with a mean of 188 mg/dL and a standard deviation of 24.
What is the probability that a randomly selected adult women will have a cholesteral level
between 150 and 170 Mg/dL?
Cholesterol Level of 150: Standardized score(z-score) = 150 – 188/24 = -1.583
Cholesterol Level of 170: Standardized score(z-score) = 170 – 188/24 = -0.75
Probability(Cholesterol level between 150 and 170) = Probability(z-score is between -1.583 and -0.75)
From Table 8.1 (rounded to nearest decimal):
Proportion below z=-0.74 minus proportion below z=-1.55 => (0.23 – 0.06) = 0.17
The probability that a randomly selected adult women will have a cholesteral level between 150 and
170 Mg/dL is 0.17.
Bell-Shaped Curves and Other Shapes
Using Excel to Find Normal Curve Proportions
Example: Heights of Adult Women in US
mean height = 65 standard deviation = 2.5. What is the probability that a woman selected at
random from this population will be less than 61 inches?
Excel: In function line type =NORMDIST(height value, mean, standard deviation, 1)
For this example, =NORMDIST(61, 65,2.5,1) equals 0.055
Can also use =NORMSDIST(z-score value). For this example, z-score = (61 – 65)/2.5 = -4/2.5 = -1.6
So entering = NORMSDIST(-1.6) also equals 0.05
Example: Cholesterol
mean = 188 standard deviation =24
Probability(Cholesterol level between 150 and 170)
= NORMDIST(170,188,24,1) – NORMDIST(150,188,24,1) equals 0.169954598
Probability(z-score is between -1.583 and -0.75)
=NORMSDIST(-0.75) - NORMSDIST(-1.583) equals 0.16991662
Bell-Shaped Curves and Other Shapes
Journal’s Paper on ESP Expected to Prompt Outrage – NY Times – January 5th 2011
•One of psychology’s most respected journals has agreed to publish a paper presenting what its
author describes as strong evidence for extrasensory perception, the ability to sense future
events.
•The paper describes nine unusual lab experiments performed over the past decade by its
author, Daryl J. Bem, an emeritus professor at Cornell, testing the ability of college students to
accurately sense random events, like whether a computer program will flash a photograph on
the left or right side of its screen.
•“It’s craziness, pure craziness. I can’t believe a major journal is allowing this work in,” Ray
Hyman, an emeritus professor of psychology at the University Oregon and longtime critic of ESP
research, said. “I think it’s just an embarrassment for the entire field.”
•The editor of the journal, Charles Judd, a psychologist at the University of Colorado, said the
paper went through the journal’s regular review process. “Four reviewers made comments on
the manuscript,” he said, “and these are very trusted people.”
•All four decided that the paper met the journal’s editorial standards, Dr. Judd added, even
though “there was no mechanism by which we could understand the results.”
Bell-Shaped Curves and Other Shapes
Study: Experimental Evidence for Anomalous Retroactive Influences on Cognition and Affect
•The term psi denotes anomalous processes of information or energy transfer that are currently
unexplained in terms of known physical or biological mechanisms.
•This article reports nine experiments designed to test for such retroactive influence by “timereversing” several well-established psychological effects so that the individual’s responses are
obtained before the putatively causal stimulus events occur.
•The trend is exemplified by several recent “presentiment” experiments, pioneered by Radin (1997),
in which physiological indices of participants’ emotional arousal were monitored as participants
viewed a series of pictures on a computer screen.
•Most of the pictures were emotionally neutral, but a highly arousing negative or erotic image was
displayed on randomly selected trials.
•As expected, strong emotional arousal occurred when these images appeared on the screen, but
the remarkable finding is that the increased arousal was observed to occur a few seconds before
the picture appeared, before the computer has even selected the picture to be displayed.
Bell-Shaped Curves and Other Shapes
Study: Experimental Evidence for Anomalous Retroactive Influences on Cognition and Affect
Experiment 1: Precognitive Detection of Erotic Stimuli
One hundred Cornell undergraduates, 50 women and 50 men, were recruited for this experiment
using the Psychology Department’s automated online sign-up system.
Methods: On each trial of the experiment, pictures of two curtains will appear on the screen side
by side. One of them has a picture behind it; the other has a blank wall behind it.
•Your task is to click on the curtain that you feel has the picture behind it. The curtain will then
open, permitting you to see if you selected the correct curtain. There will be 36 trials in all.
•Several of the pictures contain explicit erotic images (e.g., couples engaged in nonviolent but
explicit consensual sexual acts).
•If you object to seeing such images, you should not participate in this experiment.
Results: Across all 100 sessions, participants correctly identified the future position of the erotic
pictures significantly more frequently than the 50% hit rate expected by chance: 53.1%, t(99) =
2.51, p = .01
In contrast, their hit rate on the nonerotic pictures did not differ significantly from chance: 49.8%,
t(99) = -0.15, p = .56
Bell-Shaped Curves and Other Shapes
Text Questions