个体变异与变量分布.ppt

1、Statistical Description (1),Xiaojin Yu Introductory biostatistics http:/ biostatistics for the health sciencehttp:/faculty.ksu.edu.sa/hisham/Documents/eBooks/Introductory_Biostatistics_for_the_Health.pdf,Review,What is Medical statistics about? key terms in Statistics,2,3,1.2 key words in Statistics

2、,Population(individual) & sampleVariation & random variableRandom Variable & dataStatistic & parameterSampling errorProbability,4,Framework of statistical analysis,populationindividual, variation,parameterunknown,samplerepresentative,sampling error,Randomly sampling,Statisticsknown,Statistical infer

3、ence based on probabililty,Statistical Description,5,Statistical Description CONTENTS,For quantitative(numerical) dataFrequency distributionMeasures of central tendencyMeasures of dispersionFor qualitative(categorical) data,6,Raw Data (quantitative),Example: 120 values of height (cm) for 12-year-old

4、 boys in 1997: 142.3 156.6 142.7 145.7 138.2 141.6 142.5 130.5 134.5 148.8134.4 148.8 137.9 151.3 140.8 149.8 145.2 141.8 146.8 135.1150.3 133.1 142.7 143.9 151.1 144.0 145.4 146.2 143.3 156.3141.9 140.7 141.2 141.5 148.8 140.1 150.6 139.5 146.4 143.8143.5 139.2 144.7 139.3 141.9 147.8 140.5 138.9 1

5、34.7 147.3138.1 140.2 137.4 145.1 145.8 147.9 150.8 144.5 137.1 147.1142.9 134.9 143.6 142.3 125.9 132.7 152.9 147.9 141.8 141.4140.9 141.4 160.9 154.2 137.9 139.9 149.7 147.5 136.9 148.1134.7 138.5 138.9 137.7 138.5 139.6 143.5 142.9 129.4 142.5141.2 148.9 154.0 147.7 152.3 146.6 132.1 145.9 146.7

6、144.0135.5 144.4 143.4 137.4 143.6 150.0 143.3 146.5 149.0 142.1140.2 145.4 142.4 148.9 146.7 139.2 139.6 142.4 138.7 139.9,7,Data Summary For continuous variable data,Numerical methods Description of tendency of central Description of dispersionTabular and graphical methods,8,Tabular & Graphical Me

7、thods,Frequency table Histogram,9,FREQUENCY TABLE,10,SOLUTION TO EXAMPLE,1.number of intervals k=102 calculate the width R=Xmax-Xmin= 160.9- 125.9=35 w=R/k W=35/10=3.53.form the intervals4.counting frequencyA recommended step is to present the proportion or relative frequency.,11,Class intervals,12,

8、12,Tally and Counting,13,13,Final Frequency Table,A recommended step is to present the proportion or relative frequency.,14,Basic Steps to Form Frequency Table,step1: determining the number of intervals 5-15step2: calculating the width of intervalsStep3: forming intervals- certain range of valuesSte

9、p4: count the number of observation with certain interval the final table consists of the intervals and the frequencies.,15,Figure 2.1 Distribution of heights of 120 boys from China,1997,Frequency,16,Present data graphically,presenting data visuallyintuitivelyeasy to read and understandself-explanat

10、ory stand alone from text Statistical table and graph are intended to communicate information, so it should be easy to read and understand.The shape of the distribution is the characteristic of the variable.,17,Application,One lead to a research questionconcerns unimodal and symmetry of the distribu

11、tion,18,18,Shape of frequency distribution,DistributionUnimodal/bimodal Symmetry /skew,19,Unimodal/bimodal,Homogeneous /heterogeneousThe definition of population or the classification is approapriate.,20,SYMMETRY & SKEWNESS,Symmetric means the distribution has the same shape on both side of the peak

12、 location.Skewness means the lack of symmetry in a probability distribution. (The Cambridge Dictionary of Statistics in the Medical Sciences.)An asymmetric distribution is called skew. (Armitage: Statistical Methods in Medical Research.),21,Figure 2.2 Symmetric And Asymmetric Distribution,positive s

13、kewness,negative skewness,22,Positive & Negative Skewness,A distribution is said to have positive skewness when it has a long thin tail at the right, and to have negative skewness when it has a long thin tail to the left.A distribution which the upper tail is longer than the low, would be called pos

14、itively skew.,23,Frequency,24,Fig. The distribution of scores of QOL (quality of life ) of 892 senior citizen,0 10 20 30 40 50 60 70 80 90 100,QOL,400300200100 0,Frequency,25,Frequency,26,Fig. The distribution of ages at death of males in 19901992,0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85,

15、Age at death (year),2500200015001000 500 0,Frequency,27,Numerical methods,Central tendencyTendency of dispersion,arithmetic mean, Median, geometric mean range, interquartile range, standard deviation, variance, coefficient of variation,28,Mean,Concept and notationCalculation Application,29,CONCEPT O

16、F MEAN,Arithmetic mean, meanPopulation mean The Sample mean will be denoted by x (x-bar).,30,CALCULATION OF MEAN,given a data set of size n x1,x2,，xn,The mean is computed by summing all the xs and divided the sum by n. symbolically,31,GROUPED DATA,The mean can be approximated using the formulaWhere

17、f denotes the frequency ,m the interval midpoint ,and the summation is across the intervals.,32,Midpoint,The midpoint for an interval is obtained by calculating the average of the interval lower true boundary and the upper true boundary.The midpoint for the first interval is The midpoint for the sec

18、ond interval is,33,Example 1,34,Average: Limitation in describing data,It has been said that a fellow with one leg frozen in ice and the other leg in boiling water is comfortable ON AVERAGE !,35,Geometric Mean-notation,The geometric mean is defined as thenth rootof the productof n numbers, i.e., for

19、 a set of numbers.G /GM,36,Geometric Mean-calculation,As the definition, the expression is,Example like, the G for 2, 4, 8(n=3) should be like:,37,Geometric mean:,37,38,38,Geometric Mean-calculation,Example1_geo given a data set consisting of survival times to relapse in weeks of 21 acute leukemia p

20、atients that received some drug.1,1,2,2,3,4,4,5,5,8,8,8,8,11,11,12,12,15,17,22,23(n=21)The mean is 8.67 weeks,Ex. Serum HI antibody dilution from 107 testees after measles vaccination,39,hemagglutination inhibition(HI),application,positive skew data_if log transformation creates symmetric, unimodalG

21、eometric series.,40,41,Median,Concept of medianCalculation Application-disadvantage $ advantage,42,Concept of Median,If the data are arranged in increasing or decreasing order, the median is the middle value, which divided the set into equal halves.,M sample median,43,Calculation-how do we get it?,M

22、=56,Example1 n=11,a. When n is odd,44,Calculation-how do we get it?,Example2 n=12,M=(56+58)/2=57,b. When n is even,45,Application-Advantage,It is robust to the extreme value.,Mean=58.42 mean=149.9Median=57,46,Application-when is it used?,Fig.A skew distribution.,47,Data described by Median,Skew data

23、Normal distribution dataOrdinal data!,48,For normal distribution,49,Figure 3 the average of height of basketball players.,50,Disadvantage of median,the precise magnitude of most of the observations are not taken. if two groups of observations are pooled, the median of the combined group cannot be ex

24、pressed in terms of the medians of the two component.,51,Summary: Choosing the most appropriate measure,symmetric, unimodal-mean if log transformation creates symmetric, unimodal-geometric meandistribution free, uncertain data-median Outlier or skewed data-median Ordinal data-median,52,Measure of Di

25、spersion,range, interquartile range, Variance& standard deviation, coefficient of variation,53,Percentile(quantile),X% PX (100-X)%Quartiles:Lower (First) quartile: 25% (QL) p25Second quartile:medianUpper (Third) quartile:75% (QU)p75,54,Measures Of Dispersion,55,Range & Inter-quartile Range,R = xmaxx

26、min QU QL P75 P 25Obviously, range and inter-quartile are simple and easy to explain. However, there are a few difficulties about use of the range. 1.The first is that the value of the range is determined by only two of the original observations. 2.Second, the interpretation of the range depends on

27、the number of observations in a complicated way, which is a undesirable feature.,56,variance s2,An alternative approach is to make use of deviations from the mean, x-xbar; the greater the variation in the data set, the larger the magnitude of these deviations will tend to be.From this deviation, the

28、 variance s2 is computed by squaring each deviation, adding them and dividing their sum by one less than n.,n-1: degree of freedom, df,57,Variance,A population variance is denoted by 2,A sample variance is denoted by s2,57,58,The following should be noted,It would be no use to take the mean of devia

29、tions becauseTaking the mean of the absolute values, for example, is possibility. However, this measure has the drawback of being difficult to handle mathematically.,59,standard deviation, SD,The variance s2 have the units that are the square of the original units. For example , if x is the time in

30、seconds, the variance is measured in seconds squared(sec2). So it is convenient to have a measure of variation expressed in the same units as the original data, and this can be done by taking the square root of the variance. This quantity is the standard deviation,60,Formula for Calculation,In gener

31、al the calculation using mean is likely to cause some trouble. If the mean is not a round number, say mean is 10/3, it will need to be rounded off, and errors arise in the subtraction of this figure from each x. this difficulty can be overcome by using the following shortcut formula for the variance

32、 or SD.,Solution to calculation of s,61,62,Example:,range variance sd meanGroup A: 8 10.03.16 30Group B: 1222.54.74 30Group C: 8 8.52.92 30,63,Coefficient Of Variation, CV,nonzero mean.Make comparison between different distributions.for variables with different scale or unit;for variables with more

33、different means.,64,Example:Comparing The Dispersion Of Two Variables,mean sdHeight: 166.06(cm)4.95(cm)Weight:53.72(kg)4.96(kg),65,What do the variance and SD tell us?,Large variance (or SD) means:more variable, wider range,lower degree of representativeness of mean.small variance (or SD) means:less

34、 variable, narrower range,higher degree of representativeness of mean.,66,Which measure should be used?,sd, variancefor unimodal, symmetric, CVfor different units; for more different means.Rangefor any distribution, Wasteful of information.Interquartilefor any distribution, robust, Wasteful of infor

35、mation.The subjects should be homogeneity!,67,Summary of Average and dispersion,Meansd(min,max)Medianinterquartile range(min,max)Using both average and dispersion.,68,SUMMARY,Each variable has its own distribution;Descriptive Using graphsUsing statisticsaverage：Mean, G, M Dispersion: sd, variance, Q

36、, CV, RChoosing appropriate measurement；Using average with dispersion.,69,DATA SUMMARIZATION,Tabular and graphical methodsFrequency tablehistogramNumerical methods -Using statistics measures of location: arithmetic mean, Median geometric mean, measures of dispersion: range, inter-Quartile range(IQR), standard deviation, variance, coefficient of variation,70,Thanks,