The Normal Curve
and Z-scores
Using the Normal Curve to Find
Probabilities
Outline
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1. Properties of the normal curve.
2. Mean and standard deviation of the
normal curve.
3. Area under the curve.
4. Calculating z-scores.
5. Probability.
Carl
Gauss
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The normal curve is often called the
Gaussian distribution, after Carl Friedrich
Gauss, who discovered many of its
properties. Gauss, commonly viewed as one
of the greatest mathematicians of all time (if
not the greatest), is honoured by Germany on
their 10 Deutschmark bill.
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From http://www.willamette.edu/~mjaneba/help/normalcurve.html
The Histogram and the Normal Curve
The Theoretical Normal Curve
(from http://www.music.miami.edu/research/statistics/normalcurve/images/normalCurve1.gif
Properties of the Normal Curve:
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Theoretical construction
Also called Bell Curve or Gaussian Curve
Perfectly symmetrical normal distribution
The mean of a distribution is the midpoint of
the curve
The tails of the curve are infinite
Mean of the curve = median = mode
The “area under the curve” is measured in
standard deviations from the mean
Properties (cont.)
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Has a mean = 0 and standard deviation = 1.
General relationships: ±1 s = about 68.26%
±2 s = about 95.44%
±3 s = about 99.72%
68.26%
95.44%
99.72%
-5
-4
-3
-2
-1
0
1
2
3
4
5
Z-Scores
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Are a way of determining the position of a
single score under the normal curve.
Measured in standard deviations relative to
the mean of the curve.
The Z-score can be used to determine an
area under the curve known as a probability.
Formula:
Z = (Xi –
) /S
Using the Normal Curve: Z Scores
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Procedure:
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To find areas, first compute Z scores.
Substitute score of interest for Xi
Use sample mean for
and sample
standard deviation for S.
The formula changes a “raw” score (Xi) to a
standardized score (Z).
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Using Appendix A to Find Areas Below a
Score
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Appendix A can be used to find the areas above and
below a score.
First compute the Z score, taking careful note of the
sign of the score.
Make a rough sketch of the normal curve and shade
in the area in which you are interested.
Using Appendix A (cont.)
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Appendix A has three columns (a), (b), and (c)
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(a) = Z scores.
(b) = areas between the score and the mean
b
b
Using Appendix A (cont.)
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( c) = areas beyond the Z score
c
c
Using Appendix A
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Suppose you calculate
a Z-score = +1.67
Find the Z-score in
Column A.
To find area below a
positive score:
(a)
(b)
(c)
.
.
.
1.66
0.4515
0.0485
Add column b area to
.50.
1.67
0.4525
0.0475
To find area above a
positive score
1.68
0.4535
0.0465
.
.
.
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Look in column c.
Using Appendix A (cont.)
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The area below Z = 1.67 is
0.4525 + 0.5000 = 0.9525.
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Areas can be expressed as percentages:
 0.9525 = 95.25%
Normal curve with z=1.67
95.25%
Using Appendix A
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What if the Z score is
negative (–1.67)?
(a)
(b)
(c)
To find area below a
negative score:
.
.
.
Look in column c.
1.66
0.4515
0.0485
1.67
0.4525
0.0475
1.68
0.4535
0.0465
.
.
.
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To find area above a
negative score
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Add column b .50
Using Appendix A (cont.)
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The area below Z = - 1.67 is .0475.
Areas can be expressed as %: 4.75%.
Frequency
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Scores
Area = .0475
z= -1.67
Finding Probabilities
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Areas under the curve can also be expressed
as probabilities.
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Probabilities are proportions and range from
0.00 to 1.00.
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The higher the value, the greater the
probability (the more likely the event). For
instance, a .95 probability of rain is higher
than a .05 probability that it will rain!
Finding Probabilities
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If a distribution has:
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X
= 13
s =4
What is the probability of randomly selecting
a score of 19 or more?
Finding Probabilities
(a)
(b)
(c)
.
.
.
1.49
0.4319
0.0681
1.50
0.4332
0.0668
1.51
0.4345
0.0655
.
.
.
1.
2.
3.
4.
Find the Z score.
For Xi = 19, Z = 1.50.
Find area above in
column c.
Probability is 0.0668
or 0.07.
In Class Example
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After an exam, you learn that the mean for
the class is 60, with a standard deviation of
10. Suppose your exam score is 70. What is
your Z-score?
Where, relative to the mean, does your score
lie?
What is the probability associated with your
score (use Z table Appendix A)?
To solve:
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Available information:
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Formula:
Z = (Xi –
Xi = 70
= 60
S = 10
) /S
= (70 – 60) /10
= +1.0
Your Z-score of +1.0 is exactly 1 s.d. above the mean (an
area of 34.13% + 50%
) You are at the 84.13 percentile.
< Mean = 60
Area 34.13%>
<Area 34.13%
< Z = +1.0
68.26%
Area 50%------->
<-------Area 50%
95.44%
99.72%
-5
-4
-3
-2
-1
0
1
2
3
4
5
What if your score is 72?
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Calculate your Z-score.
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What percentage of students have a score
below your score? Above?
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What percentile are you at?
Answer:
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Z = 1.2
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The area beyond Z = .1151
(11.51% of marks are above yours)
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Area between mean and Z = .3849 + .50 = .8849
(% of marks below = 88.49%)
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Your mark is at the 88th percentile!
What if your mark is 55%?
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Calculate your Z-score.
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What percentage of students have a score
below your score? Above?
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What percentile are you at?
Answer:
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Z = -.5
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Area between the mean and Z = .1915 + .50 = .6915
(% of marks above = 69.15%)
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The area beyond Z = .3085
(30.85% of the marks are below yours)
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Your mark is only at the 31st percentile!
Another Question…
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What if you want to know how much better
or worse you did than someone else?
Suppose you have 72% and your
classmate has 55%?
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How much better is your score?
Answer:
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Z for 72% = 1.2 or .3849 of area above mean
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Z for 55% = -.5 or .1915 of area below mean
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Area between Z = 1.2 and Z = -.5 would be .3849 +
.1915 = .5764
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Your mark is 57.64% better than your classmate’s
mark with respect to the rest of the class.
Probability:
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Let’s say your classmate won’t show you the
mark….
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How can you make an informed guess about
what your neighbour’s mark might be?
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What is the probability that your classmate
has a mark between 60% (the mean) and
70% (1 s.d. above the mean)?
Answer:
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Calculate Z for 70%......Z = 1.0
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In looking at Appendix A, you see that the
area between the mean and Z is .3413
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There is a .34 probability (or 34% chance)
that your classmate has a mark between 60%
and 70%.
The probability of your classmate having a mark between
60 and 70% is .34
:
< Mean = 60
Area 34.13%>
<Area 34.13%
< Z = +1.0 (70%)
68.26%
Area 50%------->
<------Area 50%
95.44%
99.72%
-5
-4
-3
-2
-1
0
1
2
3
4
5
Homework Questions:
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Healey 1st Cdn: #5.1, 5.3, 5.5*
Healey 2/3 Cdn: #4.1, 4.3, 4.5*
*Answers can be found at back of text
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Read SPSS section at end of chapter
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