個(gè)體變異與變量分布

1、Statistical Description (1),Xiaojin Yu Introductory biostatistics http://www.hstathome.com/tjziyuan/Introductory%20Biostatistics%20Le%20C.T.%20%20(Wiley,%202003)(T)(551s).pdfIntroductory biostatistics for the health sc

2、iencehttp://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,Population(individua

3、l) & sampleVariation & random variableRandom Variable & dataStatistic & parameterSampling errorProbability,4,Framework of statistical analysis,populationindividual, variation,parameterunknown,sampl

4、erepresentative,sampling error,,Randomly sampling,Statisticsknown,,,,Statistical inference based on probabililty,,Statistical Description,5,Statistical Description CONTENTS,For quantitative(numerical) dataFrequency

5、 distributionMeasures of central tendencyMeasures of dispersionFor qualitative(categorical) data,6,Raw Data (quantitative),Example: 120 values of height (cm) for 12-year-old boys in 1997: 142.3 156.6 142.7 145.

6、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

7、.1 150.6 139.5 146.4 143.8143.5 139.2 144.7 139.3 141.9 147.8 140.5 138.9 134.7 147.3138.1 140.2 137.4 145.1 145.8 147.9 150.8 144.5 137.1 147.1142.9 134.9 143.6 142.3 125.9 132.7 152.9 14

8、7.9 141.8 141.4140.9 141.4 160.9 154.2 137.9 139.9 149.7 147.5 136.9 148.1134.7 138.5 138.9 137.7 138.5 139.6 143.5 142.9 129.4 142.5141.2 148.9 154.0 147.7 152.3 146.6 132.1 145.9 146.7 1

9、44.0135.5 144.4 143.4 137.4 143.6 150.0 143.3 146.5 149.0 142.1140.2 145.4 142.4 148.9 146.7 139.2 139.6 142.4 138.7 139.9,7,Data Summary For continuous variable data,Numerical methods Description

10、 of tendency of central Description of dispersionTabular and graphical methods,8,Tabular & Graphical Methods,Frequency table Histogram,9,FREQUENCY TABLE,10,SOLUTION TO EXAMPLE,1.number of intervals k=10

11、2 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 f

12、requency.,11,Class intervals,,12,12,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 numb

13、er of intervals 5-15step2: calculating the width of intervalsStep3: forming intervals- certain range of valuesStep4: count the number of observation with certain interval the final table consists of the interva

14、ls 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-explanatory stand a

15、lone 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

16、 lead to a research questionconcerns unimodal and symmetry of the distribution,18,18,Shape of frequency distribution,DistributionUnimodal/bimodal Symmetry /skew,,19,Unimodal/bimodal,Homogeneous /heterogeneousThe d

17、efinition of population or the classification is approapriate.,20,SYMMETRY & SKEWNESS,Symmetric means the distribution has the same shape on both side of the peak location.Skewness means the lack of symmetry in a pr

18、obability 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

19、And Asymmetric Distribution,positive skewness,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 skewne

20、ss when it has a long thin tail to the left.A distribution which the upper tail is longer than the low, would be called positively skew.,23,Frequency,24,,,,,,,,,Fig. The distribution of scores of QOL (quality of life

21、) 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 1990~1992,0 5 10

22、 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85,Age at death (year),2500200015001000 500 0,Frequency,27,Numerical methods,Central tendencyTendency of dispersion,arithmetic mean, Median, geometric mean r

23、ange, interquartile range, standard deviation, variance, coefficient of variation,,28,Mean,Concept and notationCalculation Application,29,CONCEPT OF MEAN,Arithmetic mean, meanPopulation mean μThe Sample mean will

24、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 x’s and divided the sum by n. symbolically,31,GROUPED DATA

25、,The mean can be approximated using the formulaWhere 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

26、 the average of the interval lower true boundary and the upper true boundary.The midpoint for the first interval is The midpoint for the second interval is,33,Example 1,34,Average: Limitation in describing data,It has

27、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 the nth root of the produc

28、t of n numbers, i.e., for 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

29、-calculation,Example1_geo given a data set consisting of survival times to relapse in weeks of 21 acute leukemia patients 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

30、 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, unimodalGeometric series.,40,4

31、1,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 equ

32、al halves.,M sample median,,43,Calculation-how do we get it?,M=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

33、 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 dataNormal distribution dataOrdinal data!!,48,For

34、 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 me

35、dian of the combined group cannot be expressed in terms of the medians of the two component.,,51,Summary: Choosing the most appropriate measure,symmetric, unimodal-mean if log transformation creates symmetric, unimod

36、al-geometric meandistribution free, uncertain data-median Outlier or skewed data-median Ordinal data-median,52,Measure of Dispersion,range, interquartile range, Variance& standard deviation

37、, coefficient of variation,53,Percentile(quantile),X% PX (100-X)%Quartiles:Lower (First) quartile: 25% (QL) p25Second quartile:medianUpper (Third) quartile:75%

38、(QU)p75,,54,Measures Of Dispersion,,,,,,,,,,,,,,,,,,,,55,Range & Inter-quartile Range,R = xmax－xmin QU － QL ＝ P75 － P 25Obviously, range and inter-quartile are simple and easy to explain. However,

39、 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 the number o

40、f 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 ma

41、gnitude of these deviations will tend to be.From this deviation, the 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

42、 population variance is denoted by σ2,A sample variance is denoted by s2,,57,58,The following should be noted,It would be no use to take the mean of deviations becauseTaking the mean of the absolute values, for exam

43、ple, 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 , i

44、f x is the time in 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 squar

45、e root of the variance. This quantity is the standard deviation,,60,Formula for Calculation,In general the calculation using mean is likely to cause some trouble. If the mean is not a round number, say mean is 10/3, it

46、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 or SD.,Solution to calculation of s,61

47、,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,

48、CV,nonzero mean.Make comparison between different distributions.for variables with different scale or unit;for variables with more different means.,64,Example:Comparing The Dispersion Of Two Variables,,mean sd

49、Height: 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 varianc

50、e (or SD) means:less 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.Ra

51、ngefor any distribution, Wasteful of information.Interquartilefor any distribution, robust, Wasteful of information.The subjects should be homogeneity!,67,Summary of Average and dispersion,Mean±sd(min,max)Medi

52、an±interquartile 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,

53、 CV, RChoosing appropriate measurement；Using average with dispersion.,69,DATA SUMMARIZATION,Tabular and graphical methodsFrequency tablehistogramNumerical methods -Using statistics measures of location: arithme

54、tic mean, Median geometric mean, measures of dispersion: range, inter-Quartile range(IQR), standard deviation