Introduction of biostatistics
- 1. Statistics and Biostatistics Mrs. Khushbu K. Patel Assistant professor Shri Sarvajanik Pharmacy College
- 2. What is Statistics? Different authors have defined statistics differently. The best definition of statistics is given by Croxton and Cowden according to whom statistics may be defined as the science, which deals with collection, presentation, analysis and interpretation of numerical data. The science and art of dealing with variation in data through collection, classification, and analysis in such a way as to obtain reliable results. —(John M. Last, A Dictionary of Epidemiology ) Branch of mathematics that deals with the collection, organization, and analysis of numerical data and with such problems as experiment design and decision making. —(Microsoft Encarta Premium 2009)
- 3. A branch of mathematic staking and transforming numbers into useful information for decision makers. Methods for processing & analyzing numbers Methods for helping reduce the uncertainty inherent indecision making
- 4. What is biostatistics? It is the science which deals with development and application of the most appropriate methods for the: Collection of data. Presentation of the collected data. Analysis and interpretation of the results. Making decisions on the basis of such analysis The methods used in dealing with statistics in the fields of medicine, biology and public health.
- 5. Why study statistics? Decision Makers Use Statistics To: Present and describe data and information properly Draw conclusions about large groups of individuals or information collected from subsets of the individuals or items. Improve processes.
- 6. Statistics Descriptive Statistics Experimental Statistics Inferential Statistics Methods for processing, summarizing, presenting and describing data Drawing conclusions and / or making decisions concerning a population based only on sample data Techniques for planning and conducting experiments
- 7. DATA Definition:- A set of values recorded on one or more observational units. Data are raw materials of statistics. Data set : A collection of data is data set Data point : A single observation Raw data : Information before it arranged and analysed Sources of data:- Experiments Surveys Records
- 8. Example of Raw data: Blood Pressure Systolic BP Diastolic BP 120 80 135 90 125 85 140 95 138 86
- 9. Elements, Variables, and Observations The elements are the entities on which data are collected. A variable is a characteristic of interest for the elements. The set of measurements collected for a particular element is called an observation. The total number of data values in a data set is the number of elements multiplied by the number of variables.
- 10. Data, Data Sets, Elements, Variables, and Observations Stock Exchange Annual Sales($M) Earn/ Share($) Company V ariables Element Names Data Set
- 11. Descriptive statistics ◦ Summarizing and describing the data ◦ Uses numerical and graphical summaries to characterize sample data
- 12. Descriptive Statistics n • Collect data – e.g., Survey • Present data – e.g., Tables and graphs • Characterize data – e.g., Sample mean = Xi
- 13. Inferential Statistics • Estimation – e.g., Estimate the population mean weight using the sample mean weight • Hypothesis testing – e.g., Test the claim that the population mean weight is 120 pounds Drawing conclusions about a large group of individuals based on a subset of the large group.
- 14. Inferential statistics It refers to the process of selecting and using a sample to draw inference about population from which sample is drawn. Two forms of statistical inference ◦ Hypothesis testing ◦ Estimation
- 15. Basic Vocabulary of Statistics POPULATION : A population consists of all the items or individuals about which you want to draw a conclusion. Ex: People who live within 25 kms of radius from centre of the city. SAMPLE : A sample is the portion of a population selected for analysis. It has to be representative. PARAMETER : A parameter is a numerical measure that describes a characteristic of a population. STATISTIC : A statistic is a numerical measure that describes a characteristic of a sample.
- 16. Population vs. Sample Population Sample Measures used to describe the population are called parameters Measures computed from sample data are called statistics
- 18. Types of data Quantitative data(numerical) Qualitative data(categorical) continuous Discrete Nominal Ordinal take forever to count Ex: time countable in a finite amount of time Ex: count change of money in your pocket
- 19. Type of variables Categorical (qualitative) variables have values that can only be placed into categories, such as “yes” and “no.” Numerical (quantitative) variables have values that represent quantities.
- 20. Qualitative Data Non Numerical Categorical No numbers are use to describe it Word, picture, image Ex. Do you smoke? Yes No
- 21. Quantitative Data Numerical Non Metric Binary Nominal Ordinal Metric Discrete Continuous
- 22. REASONS FOR ASSIGNING NUMBERS Numbers are usually assigned for two reasons: numbers permit statistical analysis of the resulting data numbers facilitate the communication of measurement rules and results
- 24. TYPES OF MEASUREMENT SCALES Non Metric Scales Nominal: (Description) Ordinal: (Order) Metric Scales Interval: (Distance) Ratio: (Origin) Nominal Ordinal Interval Ratio
- 25. Nominal Notes Lowest Level of measurement Discrete Categories No natural order Categorical or dichotomous May be referred to a qualitative or categorical Examples Gender 0 = Male 1 = Female Group Membership 1= Experimental 2 = Placebo 3 = Routine Marital Status, Colour, religion, type of car etc.
- 26. Nominal Nominal sounds like name Notes Lowest Level Classification of data Order is arbitrary Gender Marital Status Religion Types of Car Driven Possible Measures Mode Model Percentage Range Frequency Distribution
- 27. Ordinal Notes Ordered Categories Relative rankings Unknown distance between rankings Zero arbitrary Examples Likert Scales Socioeconomic status Size Size, ranking of favorite sports, class rankings, wellness rankings
- 28. Ordinal The values in an ordinal scale simply express an order Customers Satisfaction Are you Very Satisfied Satisfied Neither satisfied nor dissatisfied Dissatisfied Very dissatisfied Movie Ratings
- 29. Ordinal Notes Order matters But not the difference between values Unknown distance between rankings Relative rankings Likert scales Socioeconomic status Pain intensity Non numeric concepts Possible Measures All Nominal level tests Median Percentile Semi quartile range Rank order coefficients of correlation
- 30. Interval Notes Ordered categories Equal distance Between values An accepted unit of measurement Zero is arbitrary Examples
- 31. Interval Notes Ordered categories Equal distance Can measure differences Zero is arbitrary Temperature Celsius or Fahrenheit Elevation Time Possible Measures All Ordinal tests Mean Standard deviation Addition and subtraction Can not multiply or divide
- 32. Ratio Notes Most Precise Ordered Exact Value Equal Intervals • Natural Zero When variable equals zero it means there is none of that variable Not Arbitrary zero Examples Weight Height Pulse Blood Pressure Time Degrees Kelvin
- 33. Ratio Note Precise, Ordered, Exact Equal intervals Natural Zero Weight Time Degree Kelvin Possible Measures All operations are possible Descriptive and inferential statistics Can make comparisons An 8 kg baby is twice as heavy as a 4 kg baby Can add, subtract, multiply, divide
- 34. CHARACTERISTICS OF LEVEL OF MEASUREMENT Nominal Ordinal Interval Ratio Labeled Yes Yes Yes Yes Ordered No Yes Yes Yes Known difference No No Yes Yes Zero is arbitrary N/A Yes Yes No Zero Means None N/A No No Yes
- 35. LEVEL OF MEASUREMENT DECISION TREE Ordered? Yes, Equally Spaced Yes, Zero means none? Yes, Ratio No, Interval No Ordinal No Nominal
- 36. Scale Number system Example Permissible statistics Nominal : Unique definition of numbers ( 0,1,2,……..9) Roll number of students, Numbers assign to basket ball players. Percentages, Mode, Binomial test, Chi- Square test Ordinal: Order Numbers (0<1<2……….<9) Student’s Rank Percentiles, Median, Rank-order co- relation, Two-way ANOVA Interval : Equality of differences (2-1 = 7-6) Temperature Range, Mean, Standard deviation, Product Movement Correlation t- test and f -test Ratio: Equality of Ratio (5/10 = 3/6) Weight, height, distance Geometric Mean, Harmonic Mean, Coefficient of variation
- 37. SOME STATISTICAL TESTS Nominal Ordinal Interval Ratio Mode Yes Yes Yes Yes Median No Yes Yes Yes Mean No No Yes Yes Frequency Distribution Yes Yes Yes Yes Range No Yes Yes Yes Add and Subtract No No Yes Yes Multiply and Divide No No No Yes Standard Deviation No No Yes Yes
- 38. NOIR Remember Example Central Tendency Notes Nominal Named classifications; Mutually exclusive categories Gender Mode No order; Limited in descriptive ability Ordinal Ordered or Relative rankings; Numbers are not equidistant; Zero is arbitrary Pain scale Mode, median Not necessarily equal intervals Interval Rank ordering; Approximately equal intervals; Can have negative numbers Exam marks Mode, median, mean Exact difference between numbers is known; Zero is arbitrary Ratio Rank ordering; Equal intervals; absolute Zero Length Weight Mode, Median, Mean Zero means none
- 39. Methods of presentation of data 1 Tabular presentation 2 Graphical presentation Purpose: To display data so that they can be readily understood. Principle: Tables and graphs should contain enough information to be self- sufficient without reliance on material within the text of the document of which they are a part. •Tables and graphs share some common features, but for any specific situation, one is likely to be more suitable than the other.
- 40. Tabular Presentation Types of tables:- 1.list table:- for qualitative data, count the number of observations ( frequencies) in each category. A table consisting of two columns, the first giving an identification of the observational unit and the second giving the value of variable for that unit. Example : number of patients in each hospital department are Department Number of patients Medicine 100 Surgery 88 ENT 54 Opthalmology 30
- 41. Tabular Presentation 2. Frequency distribution table:- for qualitative and quantitative data Simple frequency distribution table:-
- 42. Tabular Presentation complex frequency distribution table Smoking Lung cancer Total positive negative No. % No. % No. % Smoker 15 65.2 8 34.8 23 100 Non smoker 5 13.5 32 86.5 37 100 Total 20 33.3 40 66.7 60 100
- 43. Graphical presentation For quantitative, continuous or measured data Histogram Frequency polygon Frequency curve Line chart Scattered or dot diagram For qualitative, discrete or counted data Bar diagram Pie or sector diagram Spot map
- 44. Bar diagram It represent the measured value (or %) by separated rectangles of constant width and its lengths proportional to the frequency Use:- discrete qualitative data Types:- simple multiple component Conditions for Which Patients were referred for treatment 0 20 40 80 100 120 B ac k and Neck A rthritis A nxiety Sk in D igestive Headache Gynecologic Respiratory Circulatory General Blood Endocrine Condition 60 N u m b e r of Patients
- 45. Bar diagram Multiple bar chart:- Each observation has more than one value represented, by a group of bars. Component bar chart:-subdivision of a single bar to indicate the composition of the total divided into sections according to their relative proportion.
- 46. Pie diagram Consist of a circle whose area represents the total frequency (100%) which is divided into segments. Each segment represents a proportional composition of the total frequency
- 47. Histogram it is very similar to the bar chart with the difference that the rectangles or bars are adherent (without gaps). It is used for presenting continuous quantitative data. Each bar represents a class and its height represents the frequency (number of cases), its width represent the class interval.
- 48. Frequency polygon Derived from a histogram by connecting the mid points of the tops of the rectangles in the histogram. The line connecting the centers of histogram rectangles is called frequency polygon. We can draw polygon without rectangles so we will get simpler form of line graph
- 49. Scattered diagram It is useful to represent the relationship between two numeric measurements. Each observation being represented by a point corresponding to its value on each axis
- 50. Organizing Numerical Data: Frequency Distribution The frequency distribution is a summary table in which the data are arranged in to numerically ordered classes. You must give attention to selecting the appropriate number of class groupings for the table, determining a suitable width of a class grouping, and establishing the boundaries of each class grouping to avoid overlapping. The number of classes depends on the number of values in the data. With a larger number of values, typically there are more classes. In general, a frequency distribution should have at least 5 but no more than 15 classes. To determine the width of a class interval, you divide the range (Highest value–Lowest value) of the data by the number of class groupings desired.
- 51. Example: A manufacturer of insulation randomly selects 20 winter days and records the daily high temperature 24, 35, 17, 21, 24, 37, 26, 46, 58, 30, 32, 13, 12, 38, 41, 43, 44, 27, 53, 27
- 52. Sort raw data in ascending order:12, 13, 17, 21, 24, 24, 26, 27, 27, 30, 32, 35, 37, 38, 41, 43, 44, 46, 53, 58 Find range: 58 -12 = 46 Select number of classes: 5 (usually between 5 and 15) Compute class interval (width): 10 (46/5 then round up) Determine class boundaries (limits): Class 1: 10 to less than 20 Class 2: 20 to less than 30 Class 3: 30 to less than 40 Class 4: 40 to less than 50 Class 5: 50 to less than 60 Compute class midpoints: 15, 25, 35, 45, 55 Count observations & assign to classes
- 53. Data in ordered array: 12, 13, 17, 21, 24, 24, 26, 27, 27, 30, 32, 35, 37, 38, 41, 43, 44, 46, 53, 58 1 2 3 4 5
- 54. Tabulating Numerical Data: Cumulative Frequency Data in ordered array: 12, 13, 17, 21, 24, 24, 26, 27, 27, 30, 32, 35, 37, 38, 41, 43, 44, 46, 53, 58
- 55. Why Use a Frequency Distribution? • It condenses the raw data into a more useful form • It allows for a quick visual interpretation of the data • It enables the determination of the major characteristics of the data set including where the data are concentrated / clustered
- 56. Frequency Distributions: Some Tips Different class boundaries may provide different pictures for the same data (especially for smaller data sets) Shifts in data concentration may show up when different class boundaries are chosen As the size of the data set increases, the impact of alterations in the selection of class boundaries is greatly reduced When comparing two or more groups with different sample sizes, you must use either a relative frequency or a percentage distribution
- 57. How to make distribution table ? https://www.statisticshowto.com/probability-and- statistics/descriptive-statistics/frequency-distribution-table/ Online generate frequency distribution https://www.socscistatistics.com/descriptive/frequencydistribution/de fault.aspx Practice work https://www.mathsisfun.com/data/frequency-distribution.html
- 58. Measures of central tendacy • The central tendency is the extent to which all the data values group around a typical or central value. . The three most commonly used averages are: • The arithmetic mean • The Median • The Mode
- 59. Measures of central tendacy 1. Mean:- ◦ The arithmetic average of the variable x. ◦ It is the preferred measure for interval or ratio variables with relatively symmetric observations. ◦ It has good sampling stability (e.g., it varies the least from sample to sample), implying that it is better suited for making inferences about population parameters. ◦ It is affected by extreme values
- 60. Measures of Central Tendency: The Median Median:- The middle value (Q2, the 50th percentile) of thevariable. In an ordered array, the median is the “middle” number (50% above, 50% below) It is appropriate for ordinal measures and for interval or ratio measures. • Not affected by extreme values 0 1 2 3 4 5 6 7 8 9 10 Median = 3 0 1 2 3 4 5 6 7 8 9 10 Median = 3
- 61. Measures of Central Tendency: The Median The rank of median for is (n + 1)/2 if the number of observation is odd and n/2 if the number is even If the number of values is odd, the median is the middle number If the number of values is even, the median is the average of the two middle numbers Note that is not the value of the median, only the position of the median in the ranked data.
- 62. Median for Grouped Data Formula for Median is given by Median = Where L =Lower limit of the median class n = Total number of observations = m = Cumulative frequency preceding the median class f = Frequency of the median class c = Class interval of the median class L (n/2) m c f f (x)
- 63. Median for Grouped Data Example Find the median for the following continuous frequency distribution: Class 0-1 1-2 2-3 3-4 4-5 5-6 Frequency 1 4 8 7 3 2
- 64. Solution for the Example Class Frequency Cumulative Frequency 0-1 1 1 1-2 4 5 2-3 8 13 3-4 7 20 4-5 3 23 5-6 2 25 Total 25 Substituting in the formula the relevant values, Median = = , we have Median = = 2.9375 L (n/2) m c f 2 (25/ 2) 5 1 8 L =Lower limit of the median class n = Total number of observations m = Cumulative frequency preceding the median class f = Frequency of the median class c = Class interval of the median class
- 65. Measures of Central Tendency: The Mode 3 Mode:- ◦ The most frequently occurring value in the data set. ◦ May not exist or may not be uniquely defined. ◦ It is the only measure of central tendency that can be used with nominal variables, but it is also meaningful for quantitative variables that are inherently discrete. 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 Mode = 9 0 1 2 3 4 5 6 No Mode
- 66. Mode for Grouped Data Mode = Where L =Lower limit of the modal class = Frequency of the modal class = Frequency preceding the modal class = Frequency succeeding the modal class. C = Class Interval of the modal class c d1 d1 d2 L d1f1f0 d2 f1f2 f1 f0 f2
- 67. Mode for Grouped Data Example Example: Find the mode for the following continuous frequency distribution: Class 0-1 1-2 2-3 3-4 4-5 5-6 Frequency 1 4 8 7 3 2
- 68. Solution for the Example Class Frequency 0-1 1 1-2 4 2-3 8 3-4 7 4-5 3 5-6 2 Total 25 Mode = L = 2 = 8 - 4 = 4 = 8 - 7 = 1 C = 1 Hence Mode = = 2.8 c d1 d1 d2 L d1 f1f0 d2 f1 f2 2 4 1 5
- 69. Measure of dispersion Measures of variability depict how similar observations of a variable tend to be. Variability of a nominal or ordinal variable is rarely summarized numerically. The measure of dispersion describes the degree of variations or dispersion of the data around its central values: (dispersion = variation = spread = scatter). Range - R Standard Deviation - SD Coefficient of Variation -COV
- 70. Measures of Variation Same center, different variation Measures of variation give information on the spread or variability or dispersion of the data values. Variation Standard Deviation Coefficient of Variation Range Variance
- 71. Measures of Variation: The Range Example: 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 Range = 14 - 1 = 13 Simplest measure of variation Difference between the largest and the smallest values: Range = X largest – X smallest
- 72. Measure of dispersion Range:- It is the difference between the largest and smallest values. It is the simplest measure of variation. Disadvantage:- it is based only on two of the observations and gives no idea of how the other observations are arranged between these two.
- 73. Measures of Variation: Why The Range Can Be Misleading Ignores the way in which data are distributed 7 8 9 10 11 12 Range = 12 - 7 = 5 7 8 9 10 11 12 Range = 12 - 7 = 5 Sensitive to outliers 1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,5 1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,120 Range = 5 - 1 = 4 Range = 120 - 1 = 119
- 74. Measures of Variation: The Variance • Average (approximately) of squared deviations of values from the mean – Sample variance: n -1 n 2 i (X X) S2 i1 Where X = arithmetic mean n = sample size Xi = ith value of the variable X
- 75. Measures of Variation: The Standard Deviation • Most commonly used measure of variation • Shows variation about the mean • Is the square root of the variance • Has the same units as the original data – Sample standard deviation: n 2 i i1 n -1 (X X) S
- 76. Measures of Variation: The Standard Deviation Steps for Computing Standard Deviation 1. Compute the difference between each value and the mean. 2. Square each difference. 3. Add the squared differences. 4. Divide this total by n-1 to get the sample variance. 5. Take the square root of the sample variance to get the sample standard deviation.
- 77. Measure of Standard Deviation Uses:- 1. It summarizes the deviations of a large distribution from mean in one figure used as a unit of variation. 2. Indicates whether the variation of difference of an individual from the mean is by chance, i.e. natural or real due to some special reasons. 3. It also helps in finding the suitable size of sample for valid conclusions. https://www.mathsisfun.com/data/standard-deviation.html
- 78. Measures of Variation: Sample Standard Deviation Sample Data (Xi) : 10 12 14 15 17 18 18 24 n = 8 Mean = X = 16 Example 7 130 4.3095 81 n 1 S (1016)2 (1216)2 (1416)2 (2416)2 (10 X)2 (12 X)2 (14 X)2 (24 X)2
- 79. Standard Deviation (Sample) for Grouped Data Frequency Distribution of Return on Investment of Mutual Funds Return on Investment Number of Mutual Funds 5-10 10-15 15-20 20-25 25-30 Total 10 12 16 14 8 60
- 80. Solution for the Example From the spreadsheet of Microsoft Excel in the previous slide, it is easy to see Mean = = 1040/60=17.333 = = 6.44 Standard Deviation = S X f X n f(X X)2 n 1 2 4 4 8 . 3 3 5 9
- 81. Assignment Class Frequency 700-799 4 800-899 7 900 8 1000 10 1100 12 1200 17 1300 13 1400 10 1500 9 1600 7 1700 2 1800-1899 1 Find sample standard deviation S.D.
- 82. Measures of Variation: Comparing Standard Deviations The coefficient of variation (CV) is a measure of relative variability. It is the ratio of the standard deviation to the mean (average). Always in percentage (%) Shows variation relative to mean Can be used to compare the variability of two or more sets of data measured in different units S CV 100% X
- 83. Measure of dispersion Coefficient of variation:- The coefficient of variation expresses the standard deviation as a percentage of the sample mean. C. V = SD / mean * 100 C.V is useful when, we are interested in the relative size of the variability in the data.
- 84. Measures of Variation: Comparing Standard Deviations A B Which curve has higher SD?
- 85. Measures of Variation: Comparing Standard Deviations The coefficient of variation (CV) is a measure of relative variability. It is the ratio of the standard deviation to the mean (average). Mean = 15.5 S = 3.338 11 12 13 14 15 16 17 18 19 20 21 11 12 13 14 15 16 17 18 19 20 21 Data B DataA Mean = 15.5 S = 0.926 11 12 13 14 15 16 17 18 19 20 21 Mean = 15.5 S = 4.570 Data C CV =21.53 CV =5.97 CV =29.48
- 86. Measures of Variation: Comparing Coefficients of Variation • Drug A sale – Average price last year = $50 – Standard deviation = $5 • Drug B sale: – Average price last year = $100 – Standard deviation = $5 $5 X $50 A 100% 10% 100% S CV $100 $5 X B 100% 5% 100% S CV Both stocks have the same standard deviation, but stock B is less variable relative to its price