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Normal distribution Zhang Guozhen School of Public Health Xinjiang Medical University Version 1-Q2 XJMU EPI&MedSTAT content Introduction to continuous probability distributions Normal distribution and it’s general features The standard normal distribution The applications of normal distributions 2 Section 1--Introduction to continuous probability distributions 3 For a defined population, every random variable has an associated distribution that defines the probability of occurrence of each possible value of that variable (if there are a finitely countable number of unique values) or all possible sets of possible values (if the variable is defined on the real line). 4 Frequency Frequency 30 25 20 15 10 5 0 4 4.2 4.4 4.6 4.8125 5.2 5.4 5.6 5.8 RBC(10 /L) 5 probability distribution curve When the sample size increases and the width of the class intervals decreases infinitively, the peak line of histogram will turn into a smooth curve (like a bell shape). This is normal distribution. 6 Distribution Curve Because the cumulative frequency is 1, so the area under normal curve is also 1. The characteristics of this family of curves were developed by Abraham de Moivreand karlFriedrich Gauss. In fact, this distribution is sometimes called the Gauss distribution. 7 Section 2--Normal distribution and it’s general features 8 The Normal Distribution The most widely used continuous distribution many other distributions that are not themselves normal can be made approximately normal by transforming the data onto a different scale Generally speaking ,any random variable that can be expressed as a sum of many other random variables can be well approximated by a normal distribution. 9 Many medical phenomenon meets with normal distribution. RBC WBC Hb Blood pressure Height weight 10 The probability density function for a normal random variable is the form 11 If X has a normal distribution with mean μ and variance σ2 Then we denote this X~N（μ，σ2） 12 σ μ The normal distributions are symmetric, single- peaked, bell-shaped density curves. 13 General features of the Normal Distribution Symmetric about x=μ Peak at x=μ Total area under the curve equals 1 The normal distribution is completely determined by the parameters μ (the location parameter) and σ(the shape parameter) In other words, a different normal distribution is specified for each different value of μ and σ. 14 1 2 15 16 The areas distribution under the normal curve has certain rules. 68% of total area is between - and +. -3 -2 -1 0 1 2 3 μ-2.58σ μ-1.96σ μ-σ μ μ+σ μ+1.96σ μ+2.58σ 68.27% 95.00% 99.00% 17 The above results show that although any normal variable may take values anywhere in (-∞,+∞),the chance that its value falls in ( 1.96 , 1.96 ) is always 95% and the chance that its value falls in ( 2.58 , 2.58 ) is always 99%. 18 Section 3 --The standard normal distribution 19 The Standard Normal Distribution Suppose X has a normal distribution with mean and standard deviation , denoted X ~ N(, 2). Then a new random variable defined as Z=(X- )/ , has the standard normal distribution, denoted Z ~ N(0,1). 20 All normal random variables can be converted to the standard normal random variable. any value Xi in any normal distribution is corresponding 3 2 2 3 to the value Zi in the standard 3 2 2 3 -3 -2 -1 0 +1 +2 +3 normal distribution. -3 -2 -1 0 +1 +2 +3 21 General Features of the Standard Normal Distribution The distribution is centered at 0 The distribution is bell shaped and symmetric. The curve extends to infinity in both directions The areas distribution under the normal curve has certain rules 22 23 It can be shown that about 68% of the area under the stand normal density lies between +1 and -1,about 95% of the area lies between +1.96 and -1.96,and about 99% lies between +2.58 and -2.58. These relationships can be more precisely by saying that p(1 x 1) 0.68 p(1.96 x 1.96) 0.95 p(2.58 x 2.58) 0.99 24 normal standard normal area distribution distribution ~ -1~+1 68.27% 1.96 ~ 1.96 -1.96~+1.96 95.00% 2.58 ~ 2.58 -2.58~+2.58 99.00% 25 The area under the curve between any two points of the normal distribution is equal to the probability of observing a value between those two points. How to get the probability under the standard normal distribution curve between any two value Z1 and Z2 ? the probability of any event on a normal random variable and standard normal random variable can be computed from tables of the standard normal distribution. 26 Normal Table 0.0 1.0 Z Table 1 gives areas left of z. This table from a previous edition gives areas right of z. 27 The areas (probability) in the table denote the shaded area from negative infinitive to z Φ(z) σ =1 μ =0 Z Φ(-z)=1- Φ(z) 28 P( Z1 Z Z 2 ) P( Z Z 2 ) P( Z Z1 ) =Φ(z2)- Φ(z1) σ=1 Z1 0 Z2 29 Probability Using symmetry and the fact that the Problems area under the density curve is 1. P(Z > 1.83) = 0.0336 P(Z < 1.83)= 1-P( Z> 1.83) =1-0.0336 = 0.9664 -1.83 1.83 P(Z < -1.83) = P( Z> 1.83) =0.0336 By Symmetry 30 P( -0.6 < Z < 1.83 )= P( Z < 1.83 ) - P( Z < -0.6 ) = 0.7257 - 0.0336 = 0.6921 -0.6 1.83 31 Calculate P(x<1.96) and P(x<1), P(x<-1.96), P(-1<x<1.5) if x~N(0,1) 32 To obtain the probability (or area) under the normal distribution curve between any two specified values X1 and X2 on X-axis that we are interested in. X1 X2 X 1 Z1 X 2 Z2 P( X 1 X X 2 ) P( Z1 Z Z 2 ) =Φ(z2)- Φ(z1) 33 Find P(2 < X < 4) when X ~ N(5,2). The standardization equation for X is: Z = (X-)/ = (X-5)/2 when X=2, Z= -3/2 = -1.5 when X=4, Z= -1/2 = -0.5 P(2<X<4) = P(X<4) - P(X<2) P(X<2) = P( Z< -1.5 ) = P( Z > 1.5 ) (by symmetry) P(X<4) = P(Z < -0.5) = P(Z > 0.5) (by symmetry) P(2 < x < 4) = P(X<4)-P(X<2) = P(Z>0.5) - P( Z > 1.5) = 0.3085 - 0.0668 = 0.2417 34 Example 1 Suppose that the scores on an aptitude test are normally distributed with a mean of 100 and standard deviation of 10.( some of the original IQ tests were purported to have these parameters) what is the probability that a randomly selected score is below 90? 35 36 37 The probability from this intersection is 0.1587, so the probability of a score less than 90 is 15.87% 38 Example 2 What is the probability of a score between 90 and 115? 39 40 41 Exercise 1 Suppose that diastolic blood pressure X in hypertensive women centers about 100mmHg and has a standard deviation of 16mmHg and is normally distributed. Find P(X<90) and P(X>124). 42 43 44 Section 4--The Application of Normal Distributions 45 Estimate the medical reference range Estimate the frequency distribution Many statistic methods are based on normal distribution, or their limit are normal distribution 46 Estimate the Frequency Distribution Example 3 Suppose that the scores on CET-4 are normally distributed with a mean of 70 and standard deviation of 6. what is the probability that a randomly selected score is larger than 80? 47 48 Estimate the Medical Reference Range In medical field, towards a useful index (a variable in statistics) people frequently try to measure a large group of ‘ normal’ people to determine the reference range or normal range of such an index. Reference range, meaning ”normal range”, is the value range of most normal individuals. It is used to define a normal physiological status. In addition, when making reference range, you must randomly select enough sample size of “normal people” from population. 49 If someone’ s value is outside this range, then he or she become suspect and need to pay intensive attention. the 95% reference range of heights of seven-year old: (110cm, 130cm) the 95% reference range of hemoglobin of healthy female adults: (10g/L, 14g/L) 50 How to estimate the reference range The normal distribution method The percentile method 51 Determination of the Reference Range When the distribution of the variable is the normal distribution, we use the normal distribution method to determine the reference range μ and σ are always unknown, so when the sample size is large enough we can use and s to replace them x The 95% reference range: X 1.96S The 99% reference range: X 2.58S 52 When the distribution of the variable is the skew distribution, we use the percentile method to determine the reference range If the value above certain value is abnormal, the 95% P95 reference range: for example: hair mercury If the value below certain value is abnormal, the 95% reference range: P5 for example: respiratory capacity If the value below certain value and above certain value are both abnormal, the 95% reference range: P2.5 ~ P97.5 53 Example 4 Suppose the concentration of hemoglobin in 120 health women is normally distributed with a mean of 117.4g/L and standard deviation 10.2g/L. what is the 95% medical reference range of hemoglobin? 54 55 Exercise 2 Suppose the concentration of RBC in 144 health men is normally distributed with a mean of 5.38 *1012g/L and standard deviation of 0.44*10.2g/L. what is the 95% medical reference range of RBC? 56 57 Summary The normal distribution—the most important continuous distribution Two parameters Standard normal distribution Reference range 58 Review questions What is a standard normal distribution? What is the area to the left of -0.2 under a standard normal distribution? What symbol is used to represent this area? What is the area to the right of 0.3 under a standard normal distribution? What symbol is used to represent this area? What is Z0.30?what does it mean? What is Z0.75?what does it mean? 59 Suppose tree diameters of a certain species of tree from some defined forest area are assumed to be normally distributed with mean 8 in. and standard deviation 2 in. Find the probability of a tree having an unusually large diameter, which is defined as >12 in. 60 Solution We have x~N(8,4)and require P( x>12)=1-P(x<12)=1-P{z<(12-8)/2} =1-P(z<2.0)=1-0.977=0.023 Thus 2.3% of trees from this area have an unusually large diameter. 61 Exercise Assume the diastolic pressure of healthy high school students follows a normal distribution with mean 9.3kPa and variance 1.3kPa.What is the percentage of the students whose diastolic levels are in between of 8kPa and 10.6kPa,higher than 12.7kPa and lower than 6.7kPa respectively? 62

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