normal distribution بسم الله الرحمن الرحیم اردیبهشت 1390

Normal Distribution

الرحیم الرحمن الله بسم

1390اردیبهشت

Length of Right Foot

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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.

If our second subject had a 9 inch foot, we would add her to the graph.

As we continued to plot foot lengths, a pattern would begin to emerge.

Slide from Del Siegle


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If we were to connect the top of each bar, we would create a frequency polygon.

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.

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You will notice that if we smooth the lines, our data almost creates a bell shaped curve.


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4 5 6 7 8 9 10 11 12 13 14Length of Right Foot

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You will notice that if we smooth the lines, our data almost creates a bell shaped curve.

This bell shaped curve is known as the “Bell Curve” or the “Normal Curve.”


n=512

n=256

n=128

n=64

n=32

n=16

n=8

n=4

Whenever you see a normal curve, you should imagine the bar graph within it.

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Points on a Quiz

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نرمال توزیع

•. است پیوسته نوع از توزیع این

•. باشد می قرینه و ای زنگوله شکل به آن منحنی

عالمت • با آن در .μمیانگین شود می داده نشان

حرف • با معیار .бانحراف شود می داده نشان•. باشند می توزیع پارامتر دو معیار انحراف و میانگین


میانگین μمقدار • دقیق محل ،می مشخص افقی خط روی را

کند.

• ، معیار شکل бانحراف ،. کند می تعیین را توزیع

مقدار یک دارای هریک و بسیارند نرمال های منحنی μتعداد. бو باشند می

68.2%


The mean,

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Points on a Quiz

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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

867 / 51 = 17

Now lets look at quiz scores for 51 students.


The mean, mode,

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Points on a Quiz

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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


The mean, mode, and median

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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

will all fall on the same value in a normal distribution.


نرمال توزیع خصوصیات

•. است احتماالت توزیع یک نرمال توزیعیا ( – یک با برابر منحنی زیر .100سطح باشد%) می

•. است قرینه توزیع ، نرمال توزیعکه – قائمی خط چپ طرف در منحنی زیر سطح نصف

طرف در دیگر نصف و دارد قرار گذرد می میانگین از. آن راست

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 45 52 59 63Points on Math Test

30 35 40 45 50 55 60Points on a Different Test

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.



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Data do not always form a normal distribution. When most of the scores are high, the distributions is not normal, but negatively (left) skewed.

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Skew refers to the tail of the distribution.

Because the tail is on the negative (left) side of the graph, the distribution has a negative (left) skew.



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When most of the scores are low, the distributions is not normal, but positively (right) skewed.

Because the tail is on the positive (right) side of the graph, the distribution has a positive (right) skew.


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.


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.

mean

median

mode

Negative or Left Skew Distribution

mean

median

mode

Positive or Right Skew Distribution


استاندارد نرمال توزیع

•: که نرمال توزیع– ، باشد μمیانگین صفر با برابر ،معیار، – یک бانحراف با برابر ،

باشد برابر نرمال توزیع یک میانگین که مواقعی در

باشد نمی یک با برابر معیار انحراف و صفر باتبدیل با نرمال Z؛ جدول از براحتی توان می

. نمود استفاده استاندارد

Z Score (Standard Score)

• Z = X - μ

نمره • واحد Zمقدار در میانگین از تفاوت ، . نماید می بیان را معیار انحراف

نمره • می Z مقدار محاسبه اعشار عدد دو تا ،گردد.

σ

Tables

• Areas under the standard normal curve

(Appendices of the textbook)

تمرین

افراد • در طبیعی قلب ضربان بگیرید نظر درمیانگین با بوده نرمال توزیع دارای 70سالم

معیار انحراف دقیقه 10و در ضربه) 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.

تمرین # 1

: اساس این بر

باالی) 1 منحنی از بخشی در 80چه ضربهگیرد؟ می قرار دقیقه

Modified from Dawson-Saunders, B & Trapp, RG. Basic and Clinical Biostatistics, 2nd edition, 1994.

تمرین 1 #منحنی

15.9 %

13.6%


2.14%0.13%

تمرین # 2



2.3 %

2.14%



0.13%

تمرین # 3

بین) 3 منحنی از بخشی ضربه 50 - 90چهگیرد؟ می قرار دقیقه در


95.4 %

47.7%



47.7%

تمرین # 4



0.15 %



0.15%

تمرین # 5

از) 5 کمتر منحنی از بخشی در 40چه ضربهباالی یا قرار 100دقیقه دقیقه در ضربه

گیرد؟ می


0.3 %



0.15%0.15%

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

سالم ) 6 افراد در سیستولیک فشارخون اگرمیانگین با طبیعی بطور و 120نرمال

معیار شده 10انحراف توزیع جیوه متر میلیباشد:

زیر سطح سیستولیک خون فشار از مقداری چهپایین قسمت دو به را نرمال و% 95منحنی

کند؟% 5باالی می تقسیم



5%

95%

• Z = X - μσ

تمرین 6 #پاسخ

مقدار جدول از استفاده از Zبا کمتر مقادیر کهاز% 95 را منحنی زیر زیر% 5سطح سطح باالیی

آوریم . می بدست کند، می جدا منحنی1.645=Z

1.645 = X - 12010

10))(1.645 ) = X - 120

X = 136.45

خسته .نباشید

normal distribution بسم الله الرحمن الرحیم اردیبهشت 1390

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