is a field of study concerned with collection, organization, summarization and analysis of, Summaries of Medicine

Download is a field of study concerned with collection, organization, summarization and analysis of and more Summaries Medicine in PDF only on Docsity! 7 Normal Distribution Objectives At the end of this lecture students will be able to: • Explain what is normal distribution and standard normal distribution • Explain properties of normal distribution and test normality • Find the z-scores of a data set • Explain the distribution of sample mean and Standard error • Calculate standard error 2  Has a Bell Shape Curve and is Symmetric  It is Symmetric around the mean: Two halves of the curve are the same (mirror images) Characteristics of Normal Distribution 5  Hence Mean = Median = mode  The total area under the curve is 1 (or 100%)  Normal Distribution has the same shape as Standard Normal Distribution. Characteristics of Normal Distribution Cont’d 6  The mean ± 1 standard deviation covers 68% of the area under the curve  The mean ± 2 standard deviation covers 95% of the area under the curve  The mean ± 3 standard deviation covers 99.7% of the area under the curve Distinguishing Features 7 80 90 100 110 120 130 140 150 160 0 5 10 15 20 25 P e r c e n t POUNDS 127.8 143.3112.3 68% of 120 = .68x120 = ~ 82 runners In fact, 79 runners fall within 1-SD (15.5 lbs) of the mean. 10 8 0 9 0 1 0 0 1 1 0 1 2 0 1 3 0 1 4 0 1 5 0 1 6 0 0 5 1 0 1 5 2 0 2 5 P e r c e n t P O U N D S 127.896.8 95% of 120 = .95 x 120 = ~ 114 runners In fact, 115 runners fall within 2-SD’s of the mean. 158.8 11 8 0 9 0 1 0 0 1 1 0 1 2 0 1 3 0 1 4 0 1 5 0 1 6 0 0 5 1 0 1 5 2 0 2 5 P e r c e n t P O U N D S 127.881.3 99.7% of 120 = .997 x 120 = 119.6 runners In fact, all 120 runners fall within 3-SD’s of the mean. 174.3 12 z-score formula x z     Where x represents an element of the data set, the mean is represented by and standard deviation by .   15 Example  Example: Ahmad gets a 50 on his Statistics midterm and an 50 on his epidemiology midterm. Did he do equally well on these two exams? 16 GRADE 0 20406080100 0 5 10 15 •In one case, Ahmad’s exam score is 10 points above the mean •In the other case, Ahmad’s exam score is 10 points below the mean •In an important sense, we must interpret Ahmad’s grade relative to the average performance of the class Statistics Epidemiology Mean Statistics = 40 Mean Epidemiology = 60 Example 1 17 An example where the means are identical, but the two sets of scores have different spreads Ahmad’s Stats Z-score (50-40)/5 = 2 Ahmad’s Epide Z-score (50-40)/20 = .5 GRADE 0 2040608010 0 5 10 15 20 25 30 Statistics Epidemiology Example 2 20 Three Properties of Standard Scores  1. The mean of a set of z-scores is always zero  2. The SD of a set of standardized scores is always 1 21 Three Properties of Standard Scores cont.  3. The distribution of a set of standardized scores has the same shape as the unstandardized scores  beware of the “normalization” misinterpretation 22 SCORE -4 -2 0 2 4 0.0 0.1 0.2 0.3 0.4 34% 34% 14%14% 2%2% 50% The area under a normal curve 25 Advantages of Standard Scores cont. 2. Standard scores provides a way to standardize or equate different metrics. We can now interpret Ahmad’s scores in Statistics and Epidemiology on the same metric (the z-score metric). (Each score comes from a distribution with the same mean [zero] and the same standard deviation [1].) 26 Disadvantages of Standard Scores 1. Because a person’s score is expressed relative to the group (X - M), the same person can have different z-scores when assessed in different samples Example: If Ahmad had taken his statistic exam in a class in which everyone knew statistic well his z-score would be well below the mean. If the class didn’t know statistic very well, however, Ahmad would be above the mean. Ahmad’s score depends on everyone else’s scores. 27 Answer Suppose biology scores among college students are normally distributed with a mean of 50 and a standard deviation of 10. If a student scores a 70, what would be her z-score? 70-50 Z = =2 10 Her z-score would be 2 which means her score is two standard deviations above the mean. 30 Question? • A set of math test scores has a mean of 70 and a standard deviation of 8. • A set of English test scores has a mean of 74 and a standard deviation of 16. For which test would a score of 78 have a higher standing? Answer Now 31 Answer 78-70 math -score = 1 8 z  To solve: Find the z-score for each test. 78-74 English -score= .25 16 z The math score would have the highest standing since it is 1 standard deviation above the mean while the English score is only .25 standard deviation above the mean. 32 Question? A group of data with normal distribution has a mean of 45. If one element of the data is 60, will the z-score be positive or negative? Answer Now 35 Answer A group of data with normal distribution has a mean of 45. If one element of the data is 60, will the z-score be positive or negative? The z-score must be positive since the element of the data set is above the mean. 36 T Score  T score have a mean of 50 and a standard deviation of 10.  A T score is computed by multiplying the Z score by 10 and adding 50.  T =10(Z) + 50  It often used for personality inventories 37 Srandard Devstions Frocn The Peter Cuarvedar hea 2 Scores The Normal Distributian Fred I “4 30 11S 2.935 1 som ans crea t ' t 1 a a “2 a a sin +40 +h m1 ao un oo 7 an Many statistics books have z-score tables, giving us this information: z (a) Area between mean and z (b) Area beyond z 0.00 0.0000 0.5000 0.01 0.0040 0.4960 0.02 0.0080 0.4920 : : : 1.00 0.3413 0.1587 : : : 2.00 0.4772 0.0228 : : : 3.00 0.4987 0.0013 (a) (b) 10/27/2013 41 Surdad Carden in 1s a oie ° ie ae aler “dn 1 ! ! I i Percernos + T iad oh q oe TT oP 1 ria i Vaccankhecs q i Moma apenas BS tr . Fame ab a6 at | 3 +16 aie a44 ad T aco. sj 4 it a Et “a ‘Stared Mone 1 z]3]4|5 17) al 7 [Starwren Parcenkege ro Pe Pixie em | el | ee 7” % 13.07% 2.28% 0.13% -3 -2 -1 μ 1 2 3 Diagram of Exercise # 2 0.028 34.13% 45 Then: 3) What area of the curve is between 50-90 beats/min? Third Exercise 46 13.07% 2.28% 0.13% -3 -2 -1 μ 1 2 3 Diagram of Exercise # 3 0.954 34.13% 47 5) What area of the curve is below 40 beats per min or above 100 beats per min? Fifth Exercise 50 13.07% 2.28% 0.13% -3 -2 -1 μ 1 2 3 Diagram of 5th exercise 0.00130.0013 34.13% 51 1) 15.9% or 0.159 2) 2.8% or 0.028 3) 95.4% or 0.954 4) 0.13 % or 0.0013 5) 0.26 % or 0.0013 (for each tail) Solution/Answers 52 Testing Normal Distribution cont. 3. Histogram  Frequency and shape  Gaps in the data and outlying values 55 Testing Normal Distribution cont. Histogram 80 604 Frequency 204 Std. dev = 35.78 Mean = 38.1 0 N = 132.00 0.0 40.0 80.0 120.0 160.0 200.0 240.0 20.0 60.0 100.0 140.0 180.0 220.0 Length of stay 56 Testing Normal Distribution cont. 4. Normal Q–Q plot  Data value plotted against the value that would be expected if the data came from a normal distribution  If the variable was normally distributed, the points would fall directly on the straight line  Any deviations from the straight line indicate some degree of non-normality 57 4 Testing Normal Distribution cont. Box plot of birth weight 5000 4000 3000 2000 1000 139 Birth weight 60 4 Testing Normal Distribution cont. 300 4 2004 = 100 4 —100 Box plot of length of stay 432 Length of stay 61 Outlier Cutoffs  Outlier Cutoffs  “Large” outliers: values > upper hinge +1.5*IQR i.e. > 75th percentile +1.5*(75th percentile – 25th percentile)  “Small” outliers: values < lower hinge - 1.5*IQR i.e. < 25th percentile -1.5*(75th percentile – 25th percentile) 62 A distribution of sample means is: the collection of sample means for all the possible random samples of a particular size (n) that can be obtained from a population. All possible sample = Distribution of Sample Means 65 Population 1 2 3 4 5 6 1 2 3 4 5 6 7 8 9 66 Distribution of Sample Means from Samples of Size n = 2 1 2, 2 2 2 2,4 3 3 2,6 4 4 2,8 5 5 4,2 3 6 4,4 4 7 4,6 5 8 4,8 6 9 6,2 4 10 6,4 5 11 6,6 6 12 6,8 7 13 8,2 5 14 8,4 6 15 8.6 7 16 8.8 8 Sample # Scores Mean ( )  X 67 A key distinction Population Distribution – distribution of all individual scores in the population Sample Distribution – distribution of all the scores in your sample Sampling Distribution – distribution of all the possible sample means when taking samples of size n from the population. Also called “the distribution of sample means”. 70 1 2 3 4 5 6 1 2 3 4 5 6 7 8 9 sample mean Distribution of Sample Means Things to Notice 1. The sample means tend to pile up around the population mean 2. The distribution of sample means is approximately normal in shape, even though the population distribution was not. 3. The distribution of sample means has less variability than does the population distribution. 71 * What if we took a larger sample? 72 Central Limit Theorem For any population with mean  and standard deviation , the distribution of sample means for sample size n 1. will have a mean of  2. will have a standard deviation of 3. will approach a normal distribution as n approaches infinity (∞)   n 75 Notation The mean of the sampling distribution The standard deviation of sampling distribution (“standard error of the mean”)   X n X    76 The “standard error” of the mean is: The standard deviation of the distribution of sample means. The standard error measures the standard amount of difference between x-bar and  that is reasonable to expect simply by chance. Standard Error SE =   n 77 1 2 3 4 5 6 1 2 3 4 5 6 7 8 9 Distribution of Individuals in Population Distribution of Sample Means 1 2 3 4 5 6 1 2 3 4 5 6 7 8 9 sample mean  = 5,  = 2.24 X = 5, X = 1.58 58.1 2 24.2  X  80 1 2 3 4 5 6 2 4 6 8 10 12 7 8 9 sample mean 14 16 18 20 22 24 Sampling Distribution (n=3) X = 5 X = 1.29 29.1 3 24.2  X  81 Population Sample Distribution of Sample Means Clarifying Formulas N X  n X X     X N x   2)(   1 )( 2     n xx s n X    nX 2 2    notice 82