Transcript The Normal Curve and Z-scores

The Normal Probability
Distribution and Z-scores
Using the Normal Curve to Find
Probabilities
Carl
Gauss
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The normal probability distribution or 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.
From http://www.willamette.edu/~mjaneba/help/normalcurve.html
Properties of the Normal
Distribution:
 Theoretical construction
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Also called Bell Curve or Gaussian Curve
Perfectly symmetrical normal distribution
The mean (µ) 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 (σ) from the mean (also called Z).
Total area under the curve is an area of 1.00
The Theoretical Normal Curve
(from http://www.music.miami.edu/research/statistics/normalcurve/images/normalCurve1.gif
Properties (cont.)
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Has a mean µ = 0 and standard deviation σ = 1.
General relationships:
±1 σ = about 68.26%
±2 σ = about 95.44%
±3 σ = about 99.72%*
*Also, when z=±1 then p=.68, when z=±2, p=.95, and when z=±3, p=.997
68.26%
95.44%
99.72%
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Using Table A to Find the Area Beyond Z
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Table A (p. 668) can be used to find the area beyond
the part of the curve cut off by the z-score.
This is the probability associated with the area of the
curve beyond ±z.
The probability of falling within z standard deviations
of the mean is found by subtracting the area beyond
from .5000 for both the positive and negative sides
of the curve or by doubling the area beyond z and
subtracting it from a total area of 1.00.
Using Table A (cont.)
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( p) = area beyond the Z score
This area is the probability associated with
your ±z-score
p
p
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 which is known as a probability.
Formula:
z
y

yi  y

s
The formula changes a “raw” score (yi) to a standardized
score (Z). Table A can then be used to determine the area
(or the probability) beyond z, the area between the mean
and z, or the area below z.
Using Table A (cont.)
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For instance, if you calculate Z to be +1.67, the
area beyond Z = 1.67 would be .0475. The
probability of a score lying beyond z = 1.67
would be p = .0475
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The area between the mean and Z would be
.5000 - .0475 = .4525. Therefore the total
probability associated with finding a score
between the mean (µ) and Z is p = .4575.
Probability of any score less than Z=1.67
- This can be found by subtracting the area beyond Z =
1.67 from the total curve area of 1.00 (p=.9525)
.9525
Using Table A (cont.)
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The area below Z = - 1.67 is .0475.
Probabilities can be expressed as %: 4.75%.
Frequency
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Area = .0475
Scores
z= -1.67
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 a mean of 13 and a
standard deviation of 4, what is the
probability of randomly selecting a score of
19 or more?
Find the Z score.
For yi = 19, Z = 1.50.
Find area above in Table A.
Probability is 0.0668 or 0.07.
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?
What percentile is your mark at?
Your Z-score of +1.0 is exactly 1 σ above the mean. The
probability beyond z is .1587 and below z is .3413+.5000 = .8413.
This also means 16% of scores are above yours and 84% are below.
Your mark is at the 84th percentile.
< Mean = 60
Area .3413>
<Area .3413
< Z = +1.0
.6826
Area .5000------->
.9544
<-------Area .5000
<---Area .1587
.9972
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Another Question…
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Suppose you have 72% and your
classmate has 55%.
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What is the probability of someone having
a mark between your score and your
classmate’s?
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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The probability of having a mark between 72 and 55
for this distribution is p = .5764 or 58%
Probability:
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What is the probability of having a mark
between 60% and 70%?
The probability of having a mark between 60 and 70% is
.3413
:
< Mean = 60%
Area .3413>
<Area .3413
< Z = +1.0 (70%)
.6826
Area .5000------>
<------Area .5000
.9544
.9972
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Other points:
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Note that when:
z = 1.64, p = .05
z = 2.33, p = .01
z = 3.10, p = .001
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When the Normal Probability Distribution is used
in inferential statistics, it is usually referred to as
the Standard Normal Distribution. It has a mean
µ = 0 and an s.d. σ = 1
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Try P. 112, #7(a-d) and #9(a-f).