Normal Distribution بسم الله الرحمن الرحیم اردیبهشت 1390

Length of Right Foot Number of People with that Shoe Size 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

Length of Right Foot 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. Number of People with that Shoe Size Slide from Del Siegle

Length of Right Foot Number of People with that Shoe Size You will notice that if we smooth the lines, our data almost creates a bell shaped curve. Slide from Del Siegle

Length of Right Foot Number of People with that Shoe Size 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.” Slide from Del Siegle

n=512n=256 n=128 n=64 n=32n=16 n=8n=4

Whenever you see a normal curve, you should imagine the bar graph within it Points on a Quiz Number of Students 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 علامت μ نشان داده می شود. انحراف معیار با حرف б نشان داده می شود. میانگین و انحراف معیار دو پارامتر توزیع می باشند.

توزیع نرمال مقدار μ ، محل 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 مقدارμ وб می باشند.

68.2% توزیع نرمال

68.2% توزیع نرمال

The mean, Points on a Quiz Number of Students = / 51 = 17 Now lets look at quiz scores for 51 students. Slide from Del Siegle

The mean,mode, Points on a Quiz Number of Students Slide from Del Siegle

The mean,mode,and median Points on a Quiz Number of Students , 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. Slide from Del Siegle

خصوصیات توزیع نرمال توزیع نرمال یک توزیع احتمالات است. – سطح زیر منحنی برابر با یک ( یا 100 %) می 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 Points on Math Test Points 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. Slide from Del Siegle

Length of Right Foot Data do not always form a normal distribution. When most of the scores are high, the distributions is not normal, but negatively (left) skewed. Number of People with that Shoe Size 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. Slide from Del Siegle

Length of Right Foot Number of People with that Shoe Size 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. 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. meanmedianmode Negative or Left Skew Distribution meanmedianmode 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 مقدار نمره Z ، تفاوت از میانگین در واحد انحراف معیار را بیان می نماید. مقدار نمره Z ، تا دو عدد اعشار محاسبه می گردد. σ

Tables Areas under the standard normal curve (Appendices of the textbook)

تمرین در نظر بگیرید ضربان قلب طبیعی در افراد سالم دارای توزیع نرمال بوده با میانگین 70 و انحراف معیار 10 ضربه در دقیقه (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 : 1 ) چه بخشی از منحنی بالای 80 ضربه در دقیقه قرار می گیرد؟ Modified from Dawson-Saunders, B & Trapp, RG. Basic and Clinical Biostatistics, 2nd edition, 1994.

منحنی تمرین # % 13.6% The exercises are modified from examples in Dawson-Saunders, B & Trapp, RG. Basic and Clinical Biostatistics, 2nd edition, % 0.13%

2 # تمرین 2 ) چه بخشی از منحنی بالای 90 ضربه در دقیقه قرار می گیرد؟ Modified from Dawson-Saunders, B & Trapp, RG. Basic and Clinical Biostatistics, 2nd edition, 1994.

2.3 % 2.14% The exercises are modified from examples in Dawson-Saunders, B & Trapp, RG. Basic and Clinical Biostatistics, 2nd edition, منحنی تمرین # %

3 # تمرین 3 ) چه بخشی از منحنی بین ضربه در دقیقه قرار می گیرد؟ Modified from Dawson-Saunders, B & Trapp, RG. Basic and Clinical Biostatistics, 2nd edition, 1994.

95.4 % 47.7% The exercises are modified from examples in Dawson-Saunders, B & Trapp, RG. Basic and Clinical Biostatistics, 2nd edition, منحنی تمرین # %

4 # تمرین 4 ) چه بخشی از منحنی بالای 100 ضربه در دقیقه قرار می گیرد؟ Modified from Dawson-Saunders, B & Trapp, RG. Basic and Clinical Biostatistics, 2nd edition, 1994.

0.15 % The exercises are modified from examples in Dawson-Saunders, B & Trapp, RG. Basic and Clinical Biostatistics, 2nd edition, منحنی تمرین # %

5 # تمرین 5 ) چه بخشی از منحنی کمتر از 40 ضربه در دقیقه یا بالای 100 ضربه در دقیقه قرار می گیرد؟ Modified from Dawson-Saunders, B & Trapp, RG. Basic and Clinical Biostatistics, 2nd edition, 1994.

0.3 % The exercises are modified from examples in Dawson-Saunders, B & Trapp, RG. Basic and Clinical Biostatistics, 2nd edition, منحنی تمرین # %

Solution/Answers 1) 15.9% or ) 2.3% or ) 95.4% or ) 0.15 % or ) % or (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.

6 # تمرین 6) اگر فشارخون سیستولیک در افراد سالم نرمال بطور طبیعی با میانگین 120 و انحراف معیار 10 میلی متر جیوه توزیع شده باشد : چه مقداری از فشار خون سیستولیک سطح زیر منحنی نرمال را به دو قسمت پایین 95 % و بالای 5 % تقسیم می کند؟

The exercises are modified from examples in Dawson-Saunders, B & Trapp, RG. Basic and Clinical Biostatistics, 2nd edition, منحنی تمرین # 6 5% 95%

Z = X - μ σ پاسخ تمرین # 6  با استفاده از جدول مقدار Z که مقادیر کمتر از 95% سطح زیر منحنی را از 5% بالایی سطح زیر منحنی جدا می کند، بدست می آوریم =Z  = X  10))(1.645 ) = X  X =

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