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Fundamentals of biostatistics

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Fundamentals of biostatistics

1. 1. K i n g s u k S a r k a r , M D A s s t . P r o f . D e p t . o f C o m m u n i t y M e d i c i n e , D S M C H FUNDAMENTALS OF BIOSTATISTICS
2. 2. statistics: - It refers to the subject of scientific activity dealing with the theories and methods of collection, compilation, analysis and interpretation of data. Bio-statistics: - An art & science of collection, compilation, analysis and interpretation of data. Data(sing. Datum): - A set of observations, usually obtained by
3. 3. Classification of data- Qualitative/Attribute Quantitative/Variable: Continuous & Discreet Qualitative Data: - Can not be expressed in number - Not measurable - Can only be categorized under different categories & frequencies - E.g., Religion is an attribute; can be categorized into Hindu, Muslim, Christian - Human Blood Group: A,B,AB or O - Sex: M/F
4. 4. Quantitative Data/variable: - In statistical language, any character, characteristic or quality that varies is called variable - It has got magnitude Continuous variable: - It is expressed in numbers & can be measured - Can take up infinite no. of values in a certain range - E.g., weight, height, blood sugar
5. 5. Discreet variable: - Countable only - Takes only some isolated values - E.g., numbers of a family members, no. of workers in a factory, no. of persons suffering from a particular disease According to source- Primary Data Secondary Data
6. 6. Primary Data: - Collected directly from the field of enquiry - original in nature - E.g., measurement of BP, weight, height, blood sugar Secondary Data: - Collected previously by some other agency/organization - Used afterwards by another - E.g., hospital records, census data
7. 7. Nominal scales Ordinal Scales Interval Scales Ratio  Nominal Scales: - Used when data are classified by major categories or subgroups of population - Religion can be assigned to following categories- Muslim, Hindu, Christian - Outcome of treatment: cured or not cured; died or survived
8. 8.  Ordinal Scales: - Assign rank order to categories placed in an order - E.g., students rank in a class; Grades A,B,C,D; - Literacy status: illiterate, just literate, primary, secondary, higher secondary, graduate, post graduate - Disease condition: mild, moderate, severe  Interval Scale: - Distance between two measurement is defined, not their ratio - E.g., intelligence score in IQ tests, temperature in Centigrade
9. 9.  Ratio Scale: - Both the distance & ratio between two measurements are defined - E.g., length, weight, incidence of disease, no. of children in a family  Dichotomy/ Binary Scale: - A scale with only two categories - E.g., disease→ present/absent; sex→male /female  Population: - An aggregate of objects, animate or inanimate, under study - A group of units defined according to aims & objective of the study  Sample: - a finite subset of or part of population - Every member of population should have equal chance to be included in sample
10. 10.  Parameter: - constant, describes the characteristics of population  Statistic: - Function of observation, which describes a sample Statistic Parameter Mean x (x bar) µ(Mu) Standard Deviation s s (sigma) No. of Subject n N Proportion P P
11. 11. • Main sources for collection of medical statistics are: 1. Experiments: - Performed in the laboratories of physiology, biochemistry, pharmacology,, clinical pathology - Hospital words→ for investigations & fundamental research - Used in preparation of thesis/dissertation, scientific paper for publication in scientific journals & books 2. Surveys: - Carried out for epidemiological studies in the field by trained teams to find out incidence or prevalence of health or disease situations in a community - Used in OR→ assessment of existing condition, how to follow a program, to study merits of different methods adopted to control of a disease - Provide trends in health status, morbidity, mortality, nutritional status, health practices, environmental hazards - Provide feedback needed to modify policy - Provide timely earning of public health hazards
12. 12. 3. Records: - Maintained as a routine in registers or books over a long period of time - Used for keeping vital statistics: births, deaths, marriage, hospitalization following illness, - Used in demography & public health practices - Collected data are qualitative
13. 13.  DATA INFORMATION  Statistical data is presented usually in tabular forms through different types of tables and in pictorial forms; diagrams, charts  Method of presentation: A. Tabulation B. Drawing
14. 14.  Tabular presentation: - A form of presenting data from a mass of statistical data - at first frequency distribution table is prepared - Table can be simple or complex • Frequency distribution table or frequency table: - All frequencies considered together form “frequency distribution” - No of person in each group is called the frequency of that group - Frequency distribution table of most biological variables develop normal, binomial or Poisson distribution.
15. 15. • Presentation of quantitative data is more cumbersome as - Characteristic has a measured magnitude as well as frequency - Table x: presentation of quantitative data of height in markingsHeight of groups in Cm Markings Frequency of each group 160-162 //// //// 10 162-164 //// //// //// 15 164-166 //// //// //// // 17 166-168 //// //// //// //// 19 168-170 //// //// //// //// 20 170-172 //// //// //// //// //// / 26 172-174 //// //// //// //// //// //// 29 174-176 //// //// //// //// //// //// 30 176-178 //// //// //// //// // 22 178-180 //// //// // 12 Total 200
16. 16. - Data needs consolidation by way of tabulation to express some meaning - Tabulation → a process of summarizing raw data & displaying it in a compact form for further analysis - Orderly management of data in columns & rows
17. 17. •General Principle in designing Table: - Table should be numbered - Brief & self-explanatory title should be there mentioning time, place, person - Headings of columns & rows should be clear & concise - Data to be presented according to size of importance chronologically, alphabetically, geographically - Data must be presented meaningfully - Table should not be too large - Foot notes given, if necessary - Total no of observations ; the denominator should be written - Information obtained should be summarized in the table
18. 18. • Frequency distribution drawings: - After classwise or groupwise tabulation, the frequencies of a charecteristics can be presented by two kinds of drawings - Graphs & Diagrams - May be shown by either lines, dots, figures o Presentation of quantitative data is through graphs o Presentation of qualitative, discreet, counted data is through diagrams
19. 19. 1. Histogram - Graphical presentation of frequency distribution - Variable characters of different groups are indicated in the horizontal line (x-axis) is called abscissa - No. of observations marked on the vertical line (y-axis) is called ordinate - Frequency of each group forms a triangle
20. 20. 2. Frequency Polygon: - An area diagram of frequency distribution developed over a histogram - Mid points of the class intervals at the height of frequency are joined by straight lines - It gives a polygon, figure with many angles
21. 21. 3. Frequency Curve: - If no. of observation are very large & group interval reduced - Frequency polygon tends to loose its angulation - Gives rise to a smooth curve → frequency curve
22. 22. 4. Line Chart or Graph: - A frequency polygon presenting variation by lin - Shows trend of event occurring over a period of time - Shows rise, fall or periodic fluctuations vertical axis may not start from zero, but some point above frequency
23. 23. 5. Cumulative Frequency Diagram or “Ogive” - Graph of the cumulative frequency distribution - An ordinary frequency distribution table→ relative frequency table - Cumulative frequency: total no. of persons in each particular range from lowest value of the characteristic up to & including any higher group value
24. 24. 6. Scatter or Dot Diagram: - Prepared after tabulation in which frequencies of at least two variables have been cross classified - Shows nature of correlation between two variable character in same person(s)( e.g., height & weight) - Also called correlation diagram
25. 25. 1. Bar Diagram: - Graphically present frequencies of different categories of qualitative data - Vertical/ horizontal - May be descending/ascending order - Widths should be equal - Spacing between bars should also be equal i. Simple Bar Diagram: - Each bar represents frequency of a single category with a distinct gap from one another
26. 26. ii. Multiple bar diagram:- - Used to show comparison of two or more sets of related statistical data iii. Component/ proportional bar diagram: - Used to compare sizes of different component parts among themselves - Also shows relation between each part & the whole
27. 27. 2. Pie/ sector Diagram: - A circle whose area is divided into different segments by different straight lines from cenre to circumference - Each segment express proportional components of the attributes - Angle ( ) of a sector is calculated by Class frequency X 3.6 or (Class frequency/total frequency)X 360
28. 28. 3. Pictogram/ Picture Diagram: - A popular method to denote the frequency of the occurrence of events to common man such as attacks, deaths, number operated, admitted, discharged, accidents, etc. in a population.
29. 29. • 4. Map diagram/ spot Map: - These diagrams are prepared to visualize the geographic distribution of frequency of characteristics - One point denotes occurrence of one more events
30. 30. • When a series of observations have been tabulated in the form of frequency distribution →→it is felt necessary to convert a series of observation in a single value, that describes the characteristics of that distribution,→ called Measure Of Central Tendency • All data or values are clustered round it • These values enable comparisons to be made between one series of observations and another • Individual values may overlap, two distributions have different central tendency • E.g., average incubation period of measles is 10 days and that of chicken pox is 15 days.
31. 31. Measures of Central tendency Mean Mode Median Arithmetic Geometric Harmonic Mean(AM) Mean(GM) Mean(HM)
32. 32. • Arithmetic mean: - Sum of all observations divided by number of observations - Mean(x)=Sx/n; x is a variable taking different observational values & n= no. of observations - Exmp. • ESR of 7 subjects are 8,7,9,10,7,7, & 6 mm for 1st hr. Calculate mean ESR. - Mean(x)= (8+7+9+10+7+7+6)/7=54/7=7.7 mm
33. 33. • Median : when observations are arranged in ascending or descending order of magnitude, the middle most value is known as Median. • Problem: - From same example of ESR, observations are arranged first in ascending order: 6,7,7,7,8,9,10. - Median= {7+1}/2=8/2=4th observation I,e., 7 - When n is Odd no., Median={n+1}2 th observation - When n is Even no., Median={n/2th + (n/2+1)th}/2 th observation • Problem: suppose, there are 8 observations of ESR like 5,6,7,7,7,8,9,10 • Median={8/2th +(8/2+1)th}/2={4th+5th obs}/2=(7+7)/2=7
34. 34. • Mode: - The observation, which occurs most frquently in series • Problem: ESR of 7 subjects are 8,7,9,10,7,7, & 6 mm for 1st hr. Calculate the Mode. - Mode is 7.
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37. 37. • Geometric mean: - Used when data contain a few extremely large or small values - It’s the nth root product of n observastions • GM=ⁿ√(x₁.x₂.x₃….xn) • Harmonic Mean: - Reciprocal of the arithmetic mean of reciprocals of observations arithmetic mean of reciprocals of observations=S(⅟x) - HM=n/S⅟x - got limited use - A.M>GM>HM
38. 38. • Measures of central tendency do not provide information about spread or scatter values around them • Measures of dispersion helps us to find how individual observations are dispersed or scattered around the mean of a large series of data • Different measures of Dispersion are: i. Range ii. Mean deviation iii. Standard deviation iv. Variance v. Coefficient of variation
39. 39. • Range: - Difference between highest & lowest value - Defines normal value of a biological characteristic • Problem: Systolic blood pressure (mm of Hg) of 10 medical students as follows: 140/70, 120/88, 160/90, 140/80, 110/70, 90/60, 124/64, 100/62, 110/70 & 154/90 • Range of Systolic BP of medical students = highest value- lowest value=160-90=70mm of Hg • Range of Diastolic BP= 90-60=30 mm of Hg
40. 40. • Mean deviation: - Average deviations of observations from mean value - Mean Deviation(S) =(x-x)/n, where x=observation, x=Mean
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42. 42. • To estimate variability in population from values of a sample, degree of freedom is used in placed of no. of observations • Standard deviation is calculated by following stages: - Calculate the mean - Calculate the difference between each observation & mean - Square the difference - Sum the squared values - Divide the sum of squares by the no. of observations(n) to get mean square deviation or variances(s) - Find the square root of variance to get “Root-Mean- Square-Deviation” • Use: sample size calculation of any study - Summarizes deviation of a large series of observation around mean in a single value
43. 43. • Coefficient of Variation: - Used to denote the comparability of variances of two or more different sets of observations - Coefficient of Variation=(Sd/Mean)X100 - Coefficient of Variation indicates relative variability
44. 44. NORMAL DISTRIBUTION • Most important useful distribution in theoretical statistics • Quantitative data can be represented by a histogram & by joining midpoints of each rectangle in the histogram we can get a frequency polygon • when no. of observations become very large & class intervals get very much reduced→ frequency polygon loses its angulation →gives rise to a smooth curve known as frequency curve, • Most biological variables , e.g., height, weight, blood cholesterol etc, follows normal distribution can be graphically represented by “normal curve”
45. 45. • If a large no. of observations of any variables such as height, weight, blood pressure, pulse rate etc. are taken at random to make a representative sample of the world and if a frequency distribution table is made, it will show following characteristics: - Exactly half the observations will lie above & half below the mean and all observations are symmetrically distributed on either side of mean - Maximum no. of frequencies will be seen in the middle around the mean and fewer at extremities, decreasing smoothly on both sides
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47. 47. • Normal Curve: - Observations of a variable, which are normally distributed in a population, when plotted as a frequency curve will give rise to Normal Curve • Characteristics of a Normal Curve: - Smooth - Bell shaped - Bilaterally symmetrical - Mean, Median, Mode coincide - Distribution of observation under normal curve follows the same pattern of normal distribution as already mentioned
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50. 50. SAMPLING TECHNIQUE  Universe/population: - Aggregate of units of observation about which certain information is required - Population is a set of persons (or objects) having a common observable characteristics - E.g., while recording pulse rate of boys in a school, all boys in the school constitute the population/universe  Sample: - A portion or part of total population selected in some manner  Sapling Frame: - A complete, non-overlapping list of all the sampling units (persons or objects) of the population from which the sample is to be drawn - E.g., telephone directory acts as a frame for conducting opinion
51. 51. • Statistic: - A characteristic of a sample, whereas a • parameter - a character of a population Types of sampling: non-probability & probability/random sampling • Non-probability sampling: - Easier, less expensive o perform - Sampling is done by choice & not by chance - Information collected cannot be presumed to be representative of the whole universe - E.g, Quota Sampling, convenience sampling, Purposive sampling, Snowball Sampling, Case Study
52. 52. • Probability/Random Sampling: - Sample are selected from universe by proper sampling technique - Each member of the universe has equal opportunity to get selected - Composition of sample from universe occurs only by chance Types: oSimple Random Sampling:
53. 53. oStratified Random Sampling: oSystemic Random Sampling: oCluster Sampling: oMultistage sampling: oMultiphase Sampling:
54. 54. • Exercise no. 1 Following are the diastolic blood pressure values (in mmHg) of 10 male adults. 80, 60, 70, 80,65, 74, 66, 80, 70, 55 Solution: Mode= 80 Arranging in ascending order: 55,60,65,66,70,70,74,80,80,80 Median={10/2th+(10/2+1)th}/2={5th + 6th}/2={70+70}/2=70 Mean=700/10=70
55. 55. Exercise No. 5. The following table shows the number of children per family in a village Calculate the measure of central tendency: No of children per family No of families 0 30 1 40 2 70 3 30 4 20 5 10
56. 56. Solution: Table 1.1 showing number of children in families • Average (x)no. of children=400/200=2 No. of children in a family(x) No. of families(f) Total no. of children(fx) 0 30 0x30=0 1 40 1x40=40 2 70 2x70=140 3 30 3x30=90 4 20 4x20=80 5 10 5x10=50 Total 200 400
57. 57. Exercise no. 8 Marks obtained by 50 students in community medicine in final MBBS Part-I Exam as follows: Calculate central tendency. Marks No. of students 41-50 5 51-60 18 61-70 15 71-80 7 81-90 5
58. 58. • Solution: Average marks obtained by students=3165/50=63.3 Marks obtained No. of students(f) Mid value of marks group(x) of students Total marks obtained by each group(fx) 41-50 5 45.5 227.5 51-60 18 55.5 999 61-70 15 65.5 982.5 71-80 7 75.5 528.5 81-90 5 85.5 427.5 Total 50 3165
59. 59. Calculation of Median: N/2=3165/2=1582.5 Median class=60.5-70.5 Median=L+{(N/2 –cf) xh}/f • where: • L = lower boundary of the median class h= class width N = total frequency cf = cumulative frequency of the class previous to the median class f = frequency in the median class Class boundary frequency Cumulative frequency 40.5-50.5 227.5 227.5 <N/2 50.5-60.5 999 Cf=1226.5 <N/2 60.5-70.5 f=982.5 2209 >N/2 70.5-80.5 528.5 2737.5 80.5-90.5 427.5 3165 Total 3165
60. 60. • Median= 60.5+ (1582.5 - 1226.5)x10/982.5 = 60.5 + 3560/982.5 = 60.5 + 3.62 = 64.12 *Modal class: the class having maximum frequency Class boundary frequency 40.5-50.5 f1=227.5 50.5-60.5 fm=999 Modal Class 60.5-70.5 f2=982.5 70.5-80.5 528.5 80.5-90.5 427.5 Total 3165
61. 61. • Mode=L + (fm –f1)/(2fm- f1 – f2)x h Where, L= lower boundary of modal class fm =Frequency of modal class f1= frequency of pre-modal class f2= Frequency of post-modal class h= width of modal class Median= 60.5 +(999 –227.5 )/(2x 999- 227.5- 982.5 )x10 =60.5 -771.5/(1998-1210)x10 =60.5 – 771.5/788x10 =60.5 – 9.79 =50.71
62. 62. • Exercise no. 11 Calculate measures of dispersion from following data: 15,17,19,25,30,35,48 Solution: Range=48- 15= 33 Mean deviation= Σ(x- x)/n Observation(x) Mean(x) (x-x) 15 X=Σx/n=189/7=27 -12 17 -10 19 -8 25 -2 30 3 35 8 48 11 Σx=189 Σ(x-x)=54, ignoring- or + signs
63. 63. X • Standard deviation: SD=√(506/10)=√50.6= Observatio n(x) Mean(x) Deviation (x-x) (x-x)2 15 X=Σx/n=189 /7=27 -12 144 17 -10 100 19 -8 64 25 -2 4 30 3 9 35 8 64 48 11 121 Σx=189 Σ(x-x)=54, Σ(x-x)=506
64. 64. • Coefficient of variation=(SD/Mean)x 100 =√50.6/27 x 100 =
65. 65. • Exercise no. 20 In the following data A & B are given below: Calculate mean deviation & standard deviation. A-item B-frequency 10-20 4 20-30 8 30-40 8 40-50 16 50-60 12 60-70 6 70-80 4
66. 66. • Solution: a=assumed mean SD=√{(sumfd1)2 – (sum fd1)/N}2/√(N-1) x h • x= sumfd1 x h + a Data A - Class interval Data B- frequency (f) Mid value (x) d1=(x-a)/h fd1 fd1 2 10-20 4 15 (15-35)/10=- 2 -8 64 20-30 8 25 -1 -8 64 30-40 8 a=35 0 0 0 40-50 16 45 1 16 256 50-60 12 55 2 24 576 60-70 6 65 3 18 324 total 54 Σfd1=74 Σfd1 2=1284
67. 67. • SD=√{1284- 74/54}/√(54-1) x 10 = √{1284- 1.37}/√53 x 10 = √( 1282.63/53) x 10 = √24.2 x 10