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  1. 1. Biostatistics Dr. Priya narayan Post graduate student Department of oral pathology & microbiology Rajarajeswari dental college & hospital.
  2. 2. Contents: Introduction  Measures of central tendency  Measures of dispersion  The normal curve  Tests of significance  References
  3. 3. Introduction  ‘Statistics’ – Italian word ‘statista’ meaning ‘statesman’ or the German word ‘statistik’ which means ‘a political state’.  Originated from 2 main sources:  Government records  Mathematics  Registration of heads of families in ancient Egypt & Roman census on military strength, births & deaths, etc.  John Graunt (1620-1674) – father of health statistics
  4. 4.  STATISTICS : is the science of compiling, classifying and tabulating numerical data & expressing the results in a mathematical or graphical form.  BIOSTATISTICS : is that branch of statistics concerned with mathematical facts & data related to biological events.
  5. 5. Uses of statistics  To assess the state of oral health in the community and to determine the availability and utilization of dental care facilities.  To indicate the basic factors underlying the state of oral health by diagnosing the community and solutions .  To determine success or failure of specific oral health care programs or to evaluate the program action.  To promote health legislation and in creating administrative standards.
  6. 6. MEASURES OF CENTRAL TENDENCY  A single estimate of a series of data that summarizes the data the measure of central tendency.  Objective:  To condense the entire mass of data.  To facilitate comparison.
  7. 7. PROPERTIES  Should be easy to understand and compute.  Should be based on each and every item in the series.  Should not be affected by extreme observations.  Should be capable of further statistical computations.  Should have sampling stability.
  8. 8.  The most common measures of central tendeny that are used in dental sciences are :  Arithmetic mean – mathematical estimate.  Median – positional estimate.  Mode – based on frequency.
  9. 9. Arithmetic mean  Simplest measure of central tendency.  Ungrouped data:  Mean = Sum of all the observations in the data Number of observations in the data  Grouped data:  Mean = Sum of all the variables multiplied by the corresponding frequency in the data Total frequency
  10. 10. MEDIAN: Middle value in a distribution such that one half of the units in the distribution have a value smaller than or equal to the median and one half have a value greater than or equal to the median.  All the observations are arranged in the order of the magnitude.  Middle value is selected as the median.  Odd number of observations : (n+1)/2.  Even number of observations: mean of the middle two values is taken as the mean.
  11. 11. MODE  The mode or the modal value is that value in a series of observations that occurs with the greatest frequency.  When mode is ill defined, it can be calculated using the relation  Mode = 3 median – 2 mean
  12. 12.  Most commonly used: arithmetic mean.  Extreme values in the series : median.  To know the value that has high influence in the series: mode.
  13. 13. Measures of dispersion  Dispersion is the degree of spread or variation of the variable about a central value.  Measures of dispersion used:  To determine the reliability of an average.  To serve as basis for control of variability.  To compare two or more series in relation to their variability.  Facilitate further statistical analysis.
  14. 14. RANGE  It is the simplest method, Defined as the difference between the value of the smallest item and the value of the largest item.  This measure gives no information about the values that lie between the extremes values.  Subject to fluctuations from sample to sample.
  15. 15. MEAN DEVIATION  It is the average of the deviations from the arithmetic mean.  M.D = Ʃ X – Xi , where Ʃ ( sigma ) is the sum of, X is the n arithmetic mean, Xi is the value of each observation in the data, n is the number of observation in the data.
  16. 16. STANDARD DEVIATION(SD)  Most important and widely used.  Also known as root mean square deviation, because it is the square root of the mean of the squared deviations from the arithmetic mean.  Greater the standard deviation, greater will be the magnitude of dispersion from the mean.  A small SD means a higher degree of uniformity of the observations.
  17. 17. CALCULATION  For ungrouped data:  Calculate the mean(X) of the series.  Take the deviations (d) of the items from the mean by : d=Xi – X, where Xi is the value of each observation.  Square the deviations (d2) and obtain the total (∑ d2)  Divide the ∑ d2 by the total number of observations i.e., (n-1) and obtain the square root. This gives the standard deviation.  Symbolically, standard deviation is given by: SD= √ ∑ d2 /(n-1)
  18. 18.  For grouped data with single units for class intervals:  S = √∑(Xi - X) x fi / (N -1)  Where,  Xi is the individual observation in the class interval  fi is the corresponding frequency  X is the mean  N is the total of all frequencies
  19. 19. • For grouped data with a range for the class interval: S =√ ∑(Xi - X) x fi / (N -1) Where, Xi is the midpoint of the class interval fi is the corresponding frequency X is the mean N is the total of all frequencies
  20. 20. COEFFICIENT OF VARIATION(C.V.)  A relative measure of dispersion.  To compare two or more series of data with either different units of measurement or marked difference in mean.  C.V.= (Sx100)/ X  Where, C.V. is the coefficient of variation  S is the standard deviation  X is the mean  Higher the C.V. greater is the variation in the series of data
  21. 21. NORMAL DISTRIBUTION CURVE  Gaussian curve  Half of the observations lie above and half below the mean – Normal or Gaussian distribution
  22. 22. Properties  Bell shaped.  Symmetrical about the midpoint.  Total area of the curve is 1. Its mean zero & standard deviation 1.  Height of curve is maximum at the mean and all three measures of central tendency coincide.  Maximum number of observations is at the value of the variable corresponding to the mean, numbers of observations gradually decreases on either side with few observations at extreme points.
  23. 23.  Area under the curve between any two points can be found out in terms of a relationship between the mean and the standard deviation as follows:  Mean ± 1 SD covers 68.3% of the observations  Mean ± 2 SD covers 95.4% of the observations  Mean ± 3 SD covers 99.7% of the observations  These limits on either side of mean are called confidence limits.  Forms the basis for various tests of significance .
  24. 24. TESTS OF SIGNIFICANCE  Different samples drawn from the same population, estimates differ – sampling variability.  To know if the differences between the estimates of different samples is due to sampling variations or not – tests of significance.  Null hypothesis  Alternative hypothesis
  25. 25. NULL HYPOTHESIS  There is no real difference in the sample(s) and the population in the particular matter under consideration and the difference found is accidental and arises out of sampling variation.
  26. 26. ALTERNATIVE HYPOTHESIS  Alternative when null hypothesis is rejected.  States that there is a difference between the two groups being compared.
  27. 27. LEVEL OF SIGNIFICANCE  After setting up a hypothesis, null hypothesis should be either rejected or accepted.  This is fixed in terms of probability level (p) – called level of significance.  Small p value - small fluctuations in estimates cannot be attributed to sampling variations and the null hypothesis is rejected.
  28. 28. STANDARD ERROR  It is the standard deviation of a statistic like the mean, proportion etc  Calculated by the relation  Standard error of the population = √(p x q)/ n  Where,  p is the proportion of occurrence of an event in the sample  q is (1-p)  n is the sample size
  29. 29. TESTING A HYPOTHESIS  Based on the evidences gathered from the sample  2 types of error are possible while accepting or rejecting a null hypothesis Hypothesis Accepted Rejected True Right Type I error False Type II error Right
  30. 30. STEPS IN TESTING A HYPOTHESIS  State an appropriate null hypothesis for the problem.  Calculate the suitable statistics.  Determine the degrees of freedom for the statistic.  Find the p value.  Null hypothesis is rejected if the p value is less than 0.05, otherwise it is accepted.
  31. 31. TYPES OF TESTS :PARAMETRIC i. student’s ‘t’ test. NON- PARAMETRIC i. Wilcoxan signed rank test. ii. One way ANOVA. ii. Wilcoxan rank sum test. iii. Two way ANOVA. iii. Kruskal-wallis iv. Correlation coefficient. v. Regression analysis. one way ANOVA. iv. Friedman two way ANOVA. v. Spearman’s rank correlation. vi. Chi-square test.
  32. 32. CHI- SQUARE(ᵡ2) TEST  It was developed by Karl Pearson.  It is the alternate method of testing the significance of difference between two proportions.  Data is measured in terms of attributes or qualities.  Advantage : it can also be used when more than two groups are to be compared.
  33. 33. Calculation of ᵡ –statistic :2  ᵡ2 = Ʃ ( O – E )2 E Where, O = observed frequency and E = expected frequency.  Finding the degree of freedom(d.f) : it depends on the number of columns & rows in the original table. d.f = (column -1) (row – 1).  If the degree of freedom is 1, the ᵡ2 value for a probability of 0.05 is 3.84.
  34. 34. CHI-SQUARE WITH YATE’S CORRECTION  It is required for compensation of discrete data in the chi-square distribution for tables with only 1 DF.  It reduces the absolute magnitude of each difference (O- E) by half before squaring.  This reduces chi- square & thus corrects P( i.e., result significance).  Formula used is : ᵡ2 = Ʃ [ ( O – E ) – ½]2 E  It is required when chi-square is in borderline of significance.
  35. 35. LIMITATIONS : It will not give reliable result if the expected frequency in any one cell is less than 5.  In such cases, Yates’ correction is necessary i.e , reduction of the (O-E) by half.  X2 = ∑[(O-E) – 0.5]2 E  The test tells the presence or absence of an association between the two frequencies but does not measure the strength of association.  Does not indicate the cause & effect. It only tells the probability of occurrence of association by chance.
  36. 36. STUDENT ‘T’ TEST : When sample size is small. ‘t’ test is used to test the hypothesis.  This test was designed by W.S Gosset, whose pen name was ‘student’.  It is applied to find the significance of difference between two proportions as,  Unpaired ‘t’ test.  Paired ‘t’ test.  Criterias :  The sample must be randomly selected.  The data must be quantitative.  The variable is assumed to follow a normal distribution in the population.  Sample should be less than 30.
  37. 37. PAIRED ‘T’ TEST  When each individual gives a pair of observations.  To test for the difference in the pair values.  Test procedure is as follows:  Null hypothesis  Difference in each set of paired observation calculated : d=X1 – X2  Mean of differences, D =∑d/n, where n is the number of pairs.  Standard deviation of differences and standard error of difference are calculated.
  38. 38.  Test statistic ‘t’ is calculated from : t=D/SD/√n  Find the degrees of freedom(d.f.) (n-1)  Compare the calculated ‘t’ value with the table value for (n-1) d.f. to find the ‘p’ value.  If the calculated ‘t’ value is higher than the ‘t’ value at 5%, the mean difference is significant and vice-versa.
  39. 39. ANALYSIS OF VARIANCE(ANOVA) TEST : When data of three or more groups is being investigated.  It is a method of partionioning variance into parts( between & within) so as to yield independent estimate of the population variance.  This is tested with F distribution : the distribution followed by the ratio of two independent sample estimates of a population variance.  F = S12/ S22 .The shape depends on DF values associated with S12 & S 22 .
  40. 40.  One way ANOVA : if subgroups to be compared are defined by just one factor.  Two way ANOVA : if subgroups are based on two factors.
  41. 41. Miscellaneous : Fisher’s exact test :  A test for the presence of an association between categorical variables.  Used when the numbers involved are too small to permit the use of a chi- square test.  Friedman’s test :  A non- parametric equivalent of the analysis of variance.  Permits the analysis of an unreplicated randomized block design.
  42. 42.  Kruskal wallis test :  A non-parametric test.  Used to compare the medians of several independent samples.  It is the non-parametric equivalent of the one way ANOVA.  Mann- whitney U test :  A non-parametric test.  Used to compare the medians of two independent samples.  Mc Nemar’s test :  A variant of a chi squared test, used when the data is paired.
  43. 43.  Tukey’s multiple comparison test :  It’s a test used as sequel to a significant analysis of variance test, to determine which of several groups are actually significantly different from one another.  It has built-in protection against an increased risk of a type 1 error.  Type 1 error : being misled by the sample evidence into rejecting the null hypothesis when it is in fact true.  Type 2 error : being misled by the sample evidence into failing to reject the null hypothesis when it is in fact false.
  44. 44. REFERENCES  Park K, Park’s text book of preventive and social medicine, 21st ed, 2011, Bhanot, India; pg- 785-792.  Peter S, essential of preventive and community dentistry, 4th ed; pg- 379- 386.  Mahajan BK, methods in biostatistics. 6th edition.
  45. 45.  John j, textbook of preventive and community dentistry, 2nd ed; pg- 263- 68.  Mahajan BK, methods in biostatistics. 6th edition.  Prabhkara GN, biostatistics; 1st edition.
  46. 46.  Rao K Visweswara, Biostatistics – A manual of statistical methods for use in health, nutrition & anthropology. 2nd edition.2007.  Raveendran R, Gitanjali B, A practical approach to PG dissertation.2005.