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Normal Distribution

Normal Distribution. Nojomi, MD, MPH Department of Community Medicine IUMS. Introduction. Parameters and Statistics Value or characteristic associated with population Is called : Parameter Value or characteristic calculated from a sample Is called : Statistics. Introduction.

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Normal Distribution

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  1. Normal Distribution Nojomi, MD, MPH Department of Community Medicine IUMS

  2. Introduction • Parameters and Statistics • Value or characteristic associated with population • Is called : Parameter • Value or characteristic calculated from a sample • Is called : Statistics Normal Disrtibution

  3. Introduction • Parameters are fixed • Parameters such as : µ , σ • Single sample fixed statistic • Statistics such as : x , SD Normal Disrtibution

  4. Frequency Distribution • Is a tabulation , by table or graph , of the frequency at which values occur in a set . Normal Disrtibution

  5. Frequency distribution of cholinestrase in RBC Normal Disrtibution

  6. Frequency Distribution • RBC – Cholinestrase ? • Numeric Continuous variable • Graph ? • Histogram ! Normal Disrtibution

  7. Histogram Normal Disrtibution

  8. Histogram Normal Disrtibution

  9. Normal Disrtibution

  10. The Normal Distribution • The most important frequency distribution in statistic. • It is for numeric variable . • Shape : smooth, bell-shaped, symmetric • It has two parameter : µ , σ Normal Disrtibution

  11. 68% of data : µ ± 1SD 95% of data : µ ± 2SD 99% of data : µ ± 3SD The Normal Distribution Normal Disrtibution

  12. Determining Probabilities Using the Normal Distribution • Example 1: • What is the probability that an individual randomly selected from population will have an RBC- choliestrase value between 11.95 to 13.95? • zi = (xi- µ) / σ xi zi ~ N(0 , 1) • zi= (11.95 – 11) / 2 = 0.48 • zi = (13.95 – 11) / 2 = 1.48 • P (.48 < z <1.48) Normal Disrtibution

  13. Determining Probabilities Using the Normal Distribution Normal Disrtibution

  14. Determining Probabilities Using the Normal Distribution • Area beyond z = .48 = .31 • Area beyond z = 1.48 = .06 • .31 - .06 = .25 Normal Disrtibution

  15. Normal Disrtibution

  16. Example 2 • After prolonged contact with a sample of cholinomimetic compound, a 14-year old boy is brought to Emergency room, with vomiting, headache, bradycardia, ….. Lab test : • RBC-cholinestrase 20 µmol/ml. How unusual is this ? • Solution : • z = (xi - µ) / σ • = (20 – 11) / 2 = 4.5 Normal Disrtibution

  17. Solution • Area beyond z for z = 4.5 ? Normal Disrtibution

  18. Example 3 • Between which two RBC- cholinestrase values, 95% of the values in G.P be expected to fall ? • It means, 95% of the mid of the curve ! • So, the rest is : 2.5% on the right , and 2.5% on the left of curve . • We have to know, area beyond which z is .025? • The curve is symmetric,… Normal Disrtibution

  19. Solution • If the area beyond z = .025 • The z value is +1.96 on the right, and -1.96 on the left . • z = (xi - µ) / σ • +1.96 = (xi – 11) / 2 • xi =3.92 + 11 = 14.92 • - 1.96 = (xi +11) / 2 • xi = -3.92 + 11= 7.08 Normal Disrtibution

  20. Solution • Is there an easier alternative way ? • What is the SD ? • Ok, the SD is equal 2, in normal distribution : • 95% of curve is between : µ± 2SD • So, the mean is : 11 ± 2 × 2 = 11± 4 • 95% of general population have RBC- choliestrase between 7 and 15 . Normal Disrtibution

  21. Problems • In a normal distribution the mean and standard deviation are 40 and 5 respectively. Calculate : • A. Area under curve on the left of xi = 34 ? • xi < 34 ? • z = (xi- µ) /σ • z = (34 – 40) / 5 = -1.2 • Area beyond z = -1.2 ? • = .11 • 11% of the subjects have score less than 34 . Normal Disrtibution

  22. Problems • B. the point on which, 95% of the curve is the left . • It means, .05 is beyond that point . • Z point that .05 is beyond it , is ? • 1.64 • 1.64 = (xi – 40) / 5 • xi = 48.25 Normal Disrtibution

  23. Problems • Distribution of serum glucose is normal with mean and SD 100 and 2.5 respectively. • If select a person randomly, what is the probability of over than 105 for her/his serum glucose? • Z= (105-100) / 2.5 = 2 • Z > 2 ? • .02 = 2% Normal Disrtibution

  24. Problems • نمرات درس آمار دانشجویان پزشکی از توزیع نرمال با میانگین 70 و انحراف معیار 10 برخوردار است. در یک نمونه 100 نفری از دانشجویان مذکور، تقریبأ چند نفر نمره بین 60 و 80 می گیرند؟ • 68 Normal Disrtibution

  25. Problems • اگر Z متغیر نرمال استاندارد باشد، و P(0<Z<Z1)= a باشد آنگاه P(Z<Z1) برابر است با : • 1/2 + a Normal Disrtibution

  26. Problems • کمیت تصادفی X دارای توزیع دو جمله ای با میانگین 20 و انحراف معیار 4 است. پارامتر n چقدر است ؟ • np= 20 • npq=16, np × 0.8= 18, p=0.2 • n × 0.2 = 20 • n=100 Normal Disrtibution

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