‫بسم هللا الرحمن الرحیم‬
‫‪Normal‬‬
‫‪Distribution‬‬
‫اردیبهشت ‪1390‬‬
Suppose we measured the right foot length of 30 persons and
graphed the results.
Assume the first person had a 10 inch foot. We could create
a bar graph and plot that person on the graph.
Number of People with
that Shoe Size
If our second subject had a 9 inch foot, we would add her to
the graph.
As we continued to plot foot lengths, a
8
pattern would begin to emerge.
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2
1
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8
9 10 11 12 13 14
Length of Right Foot
Slide from Del Siegle
Number of People with
that Shoe Size
Notice how there are more people (n=6) with a 10 inch right foot
than any other length. Notice also how as the length becomes
larger or smaller, there are fewer and fewer people with that
measurement. This is a characteristics of many variables that
we measure. There is a tendency to have most measurements
in the middle, and fewer as we approach the high and low
extremes.
If we were to connect the top of each bar, we
8
would create a frequency polygon.
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2
1
4
5
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8
9 10 11 12 13 14
Length of Right Foot
Slide from Del Siegle
Number of People with
that Shoe Size
You will notice that if we smooth the lines, our data almost
creates a bell shaped curve.
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5
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2
1
4
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8
9 10 11 12 13 14
Length of Right Foot
Slide from Del Siegle
You will notice that if we smooth the lines, our data almost
creates a bell shaped curve.
Number of People with
that Shoe Size
This bell shaped curve is known as the “Bell Curve” or the
“Normal Curve.”
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Length of Right Foot
Slide from Del Siegle
n=4
n=32
n=8
n=64
n=16
n=128
n=256
n=512
Number of Students
Whenever you see a normal curve, you should imagine the
bar graph within it.
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15 16 17 18 19 20 21 22
Points on a Quiz
Slide from Del Siegle
‫توزیع نرمال‬
‫• این توزیع از نوع پیوسته است‪.‬‬
‫• منحنی آن به شکل زنگوله ای و قرینه می باشد‪.‬‬
‫• میانگین در آن با عالمت ‪ μ‬نشان داده می شود‪.‬‬
‫• انحراف معیار با حرف ‪ б‬نشان داده می شود‪.‬‬
‫• میانگین و انحراف معیار دو پارامتر توزیع می باشند‪.‬‬
‫توزیع نرمال‬
‫• مقدار ‪ ، μ‬محل دقیق میانگین را‬
‫روی خط افقی مشخص می کند‪.‬‬
‫• انحراف معیار ‪ ، б ،‬شکل توزیع‬
‫را تعیین می کند‪.‬‬
‫‪ ‬تعداد منحنی های نرمال بسیارند و هریک دارای یک مقدار‪ μ‬و‪ б‬می باشند‪.‬‬
‫توزیع نرمال‬
‫‪68.2%‬‬
‫توزیع نرمال‬
‫‪68.2%‬‬
Now lets look at quiz scores for 51 students.
The mean,
12+13+13+14+14+14+14+15+15+15+15+15+15+16+16+16+16+16+16+16+16+
17+17+17+17+17+17+17+17+17+18+18+18+18+18+18+18+18+19+19+19+19+
19+ 19+20+20+20+20+ 21+21+22 = 867
Number of Students
867 / 51 = 17
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1
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Points on a Quiz
Slide from Del Siegle
The mean, mode,
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13 13
14 14 14 14
15 15 15 15 15 15
16 16 16 16 16 16 16 16
17 17 17 17 17 17 17 17 17
18 18 18 18 18 18 18 18
19 19 19 19 19 19
Number of Students
20 20 20 20
21 21
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22
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15 16 17 18 19 20 21 22
Points on a Quiz
Slide from Del Siegle
The mean, mode, and median will all fall on the same
value in a normal distribution.
Number of Students
12, 13, 13, 14, 14, 14, 14, 15, 15, 15, 15, 15, 15, 16, 16, 16, 16, 16, 16, 16, 16, 17, 17, 17, 17, 17, 17, 17, 17, 17, 18, 18, 18, 18, 18, 18, 18, 18, 19, 19, 19, 19, 19, 19, 20, 20, 20, 20, 21, 21, 22
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1
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Points on a Quiz
Slide from Del Siegle
‫خصوصیات توزیع نرمال‬
‫• توزیع نرمال یک توزیع احتماالت است‪.‬‬
‫– سطح زیر منحنی برابر با یک ( یا ‪ )%100‬می باشد‪.‬‬
‫• توزیع نرمال ‪ ،‬توزیع قرینه است‪.‬‬
‫– نصف سطح زیر منحنی در طرف چپ خط قائمی که از میانگین‬
‫می گذرد قرار دارد و نصف دیگر در طرف راست آن‪.‬‬
If your data fits a normal distribution, approximately 68% of
your subjects will fall within one standard deviation of the
mean.
Slide from Del Siegle
If your data fits a normal distribution, approximately 68% of
your subjects will fall within one standard deviation of the
mean.
Approximately 95% of your subjects will fall within two
standard deviations of the mean.
Slide from Del Siegle
If your data fits a normal distribution, approximately 68% of
your subjects will fall within one standard deviation of the
mean.
Approximately 95% of your subjects will fall within two
standard deviations of the mean.
Over 99% of your subjects will fall within three standard
deviations of the mean.
Slide from Del Siegle
If your data fits a normal distribution, approximately 68% of
your subjects will fall within one standard deviation of the
mean.
Approximately 95% of your subjects will fall within two
standard deviations of the mean.
Over 99% of your subjects will fall within three standard
deviations of the mean.
The number of points that one standard deviations equals
varies from distribution to distribution. On one math test, a
standard deviation may be 7 points. If the mean were 45, then
we would know that 68% of the students scored from 38 to 52.
24
31
38
On another test, a
standard deviation may
equal 5 points. If the mean
were 45, then 68% of the
students would score from
40 to 50 points.
Slide from Del Siegle
45
52
59
Points on Math Test
30
35
63
40
45
50
55
Points on a Different Test
60
Data do not always form a normal distribution. When most of
the scores are high, the distributions is not normal, but
negatively (left) skewed.
Skew refers to the tail of the distribution.
Number of People with
that Shoe Size
Because the tail is on the negative (left) side of the graph, the
distribution has a negative (left) skew.
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Length of Right Foot
Slide from Del Siegle
When most of the scores are low, the distributions is not
normal, but positively (right) skewed.
Number of People with
that Shoe Size
Because the tail is on the positive (right) side of the graph,
the distribution has a positive (right) skew.
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Length of Right Foot
Slide from Del Siegle
 When data are skewed, they do not possess the
characteristics of the normal curve (distribution).
 For example, 68% of the subjects do not fall
within one standard deviation above or below the
mean.
 The mean, mode, and median do not fall on the
same score.
 The mode will still be represented by the highest
point of the distribution, but the mean will be toward
the side with the tail and the median will fall between
the mode and mean.
Slide from Del Siegle
When data are skewed, they do not possess the
characteristics of the normal curve (distribution). For
example, 68% of the subjects do not fall within one
standard deviation above or below the mean. The
mean, mode, and median do not fall on the same score.
The mode will still be represented by the highest point
of the distribution, but the mean will be toward the side
with the tail and the median will fall between the mode
and mean.
Negative or Left Skew Distribution
Positive or Right Skew Distribution
Slide from Del Siegle
‫توزیع نرمال استاندارد‬
‫• توزیع نرمال که‪:‬‬
‫– میانگین ‪ ، μ ،‬برابر با صفر باشد‬
‫– انحراف معیار‪ ، б ،‬برابر با یک باشد‬
‫‪ ‬در مواقعی که میانگین یک توزیع نرمال برابر با صفر‬
‫و انحراف معیار برابر با یک نمی باشد ؛ با تبدیل ‪Z‬‬
‫می توان براحتی از جدول نرمال استاندارد استفاده نمود‪.‬‬
‫)‪Z Score (Standard Score‬‬
‫• ‪Z = X-μ‬‬
‫‪σ‬‬
‫• مقدار نمره ‪ ،Z‬تفاوت از میانگین در واحد انحراف معیار‬
‫را بیان می نماید‪.‬‬
‫• مقدار نمره ‪ ، Z‬تا دو عدد اعشار محاسبه می گردد‪.‬‬
Tables
• Areas under the standard normal curve
(Appendices of the textbook)
‫تمرین‬
‫• در نظر بگیرید ضربان قلب طبیعی در افراد سالم‬
‫ و انحراف‬70 ‫دارای توزیع نرمال بوده با میانگین‬
‫ ضربه در دقیقه‬10 ‫معیار‬
) Mean = 70 and Standard Deviation =10 beats/min(
The exercises are modified from examples in Dawson-Saunders, B &
Trapp, RG. Basic and Clinical Biostatistics, 2nd edition, 1994.
1 # ‫تمرین‬
:‫بر این اساس‬
‫ ضربه در دقیقه‬80 ‫) چه بخشی از منحنی باالی‬1
‫قرار می گیرد؟‬
Modified from Dawson-Saunders, B & Trapp, RG. Basic and Clinical
Biostatistics, 2nd edition, 1994.
1 # ‫منحنی تمرین‬
13.6%
2.14%
15.9 %
The exercises are modified from examples in Dawson-Saunders, B &
Trapp, RG. Basic and Clinical Biostatistics, 2nd edition, 1994.
0.13%
2 # ‫تمرین‬
‫ ضربه در دقیقه‬90 ‫) چه بخشی از منحنی باالی‬2
‫قرار می گیرد؟‬
Modified from Dawson-Saunders, B & Trapp, RG. Basic and Clinical
Biostatistics, 2nd edition, 1994.
2 # ‫منحنی تمرین‬
2.14%
0.13%
2.3 %
The exercises are modified from examples in Dawson-Saunders, B &
Trapp, RG. Basic and Clinical Biostatistics, 2nd edition, 1994.
3 # ‫تمرین‬
‫ ضربه در‬50 - 90 ‫) چه بخشی از منحنی بین‬3
‫دقیقه قرار می گیرد؟‬
Modified from Dawson-Saunders, B & Trapp, RG. Basic and Clinical
Biostatistics, 2nd edition, 1994.
3 # ‫منحنی تمرین‬
47.7%
47.7%
95.4 %
The exercises are modified from examples in Dawson-Saunders, B &
Trapp, RG. Basic and Clinical Biostatistics, 2nd edition, 1994.
4 # ‫تمرین‬
‫ ضربه در دقیقه‬100 ‫) چه بخشی از منحنی باالی‬4
‫قرار می گیرد؟‬
Modified from Dawson-Saunders, B & Trapp, RG. Basic and Clinical
Biostatistics, 2nd edition, 1994.
4 # ‫منحنی تمرین‬
0.15%
0.15 %
The exercises are modified from examples in Dawson-Saunders, B &
Trapp, RG. Basic and Clinical Biostatistics, 2nd edition, 1994.
5 # ‫تمرین‬
‫ ضربه در دقیقه‬40 ‫) چه بخشی از منحنی کمتر از‬5
‫ ضربه در دقیقه قرار می گیرد؟‬100 ‫یا باالی‬
Modified from Dawson-Saunders, B & Trapp, RG. Basic and Clinical
Biostatistics, 2nd edition, 1994.
5 # ‫منحنی تمرین‬
0.15%
0.15%
0.3 %
The exercises are modified from examples in Dawson-Saunders, B &
Trapp, RG. Basic and Clinical Biostatistics, 2nd edition, 1994.
Solution/Answers
1) 15.9% or 0.159
2) 2.3% or 0.023
3) 95.4% or 0.954
4) 0.15 % or 0.0015
5) 0. 15 % or 0.0015 (for each tail)
The exercises are modified from examples in Dawson-Saunders, B &
Trapp, RG. Basic and Clinical Biostatistics, 2nd edition, 1994.
Application/Uses of Normal Distribution
•
It’s application goes beyond describing distributions
• It is used by researchers and modelers.
• The major use of normal distribution is the role it plays in
statistical inference.
• The z score along with the t –score, chi-square and F-statistics is
important in hypothesis testing.
• It helps managers/management make decisions.
‫تمرین ‪6 #‬‬
‫‪ )6‬اگر فشارخون سیستولیک در افراد سالم نرمال بطور‬
‫طبیعی با میانگین ‪ 120‬و انحراف معیار ‪ 10‬میلی متر‬
‫جیوه توزیع شده باشد‪:‬‬
‫چه مقداری از فشار خون سیستولیک سطح زیر منحنی‬
‫نرمال را به دو قسمت پایین ‪ %95‬و باالی ‪ %5‬تقسیم‬
‫می کند؟‬
6 # ‫منحنی تمرین‬
95%
5%
The exercises are modified from examples in Dawson-Saunders, B &
Trapp, RG. Basic and Clinical Biostatistics, 2nd edition, 1994.
‫پاسخ تمرین ‪6 #‬‬
‫• ‪Z = X-μ‬‬
‫‪σ‬‬
‫‪ ‬با استفاده از جدول مقدار ‪ Z‬که مقادیر کمتر از ‪ %95‬سطح‬
‫زیر منحنی را از ‪ %5‬باالیی سطح زیر منحنی جدا می کند‪،‬‬
‫‪Z=1.645‬‬
‫بدست می آوریم ‪.‬‬
‫‪120‬‬
‫ ‪ )10)(1.645 ) = X‬‬‫‪ X =136.45‬‬
‫‪120‬‬
‫‪10‬‬
‫‪ 1.645 = X -‬‬
‫خسته نباشید‪.‬‬