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Edit Comment Close Premium member Presentation Transcript PowerPoint Presentation: DEPT OF PHARMACOLOGY WELCOMESTATISTICAL ANALYSIS: DEPT OF PHARMACOLOGY STATISTICAL ANALYSIS BY GOPINATH.V.E. DEPARTMENT OF PHARMACOLOGY SHREE DEVI COLLEGE OF PHARMACYPowerPoint Presentation: DEPT OF PHARMACOLOGY Definition of Statistics: DEPT OF PHARMACOLOGY Definition of Statistics Statistics is the science of dealing with numbers . It is used for c ollection , s ummarization , p resentation and a nalysis of data. Statistics provides a way of organizing data to get information on a wider and more formal (objective) basis than relying on personal experience (subjective).Uses of medical statistics : DEPT OF PHARMACOLOGY Uses of medical statistics Medical statistics are used in 1 - Planning , monitoring and evaluating community health care programs. 2- Epidemiological research studies. 3- Diagnosis of community health problems. 4- Comparison of health status and diseases in different countries and in one country over years. 5- To form standards for the different biological measurements as weight, height. 6- To differentiate between diseased and normal groups.Types of data: DEPT OF PHARMACOLOGY Types of data Any aspect of an individual that is measured, is called variable. Variables are either 1-Quantitative or 2-Qualitative . 1- Quantitative data : it is numerical data. Discrete data : are usually whole numbers , such as number of cases of certain disease, number of hospital beds (no decimal fraction). Continuous data: it implies the measurement on a continuous scale e.g. height, weight, age (a decimal fraction can be present ).2- Qualitative data: DEPT OF PHARMACOLOGY 2- Qualitative data Qualitative data : It is non numerical data and is subdivided into Two Types: A- Categorical : data are purely descriptive and imply no ordering of any kind such as sex, area of residence. B- Ordinal data : are those which imply some kind of ordering like - Level of education: - Socio-economic status: - Degree of severity of disease:Normal Distribution curve: DEPT OF PHARMACOLOGY Normal Distribution curve NDC is a Graphical Presentation < Frequency Polygon> of any Quantitative Biologic Variables The Normal Distribution Curve is the frequency polygon of a quantitative variable measured in large number. It is a form of presentation of frequency distribution of biologic variables such as weights, heights, hemoglobin level and blood pressure or any continuous data. It occupies a major role in the techniques of statistical analysis.PowerPoint Presentation: DEPT OF PHARMACOLOGYCharacteristics of Normal Distribution curve: DEPT OF PHARMACOLOGY Characteristics of Normal Distribution curve 1- It is bell shaped, continuous curve. 2- It is symmetrical i.e. can be divided into two equal halves vertically. 3- The tails never touch the base line but extended to infinity in either direction. 4- T he mean , median and mode values coincide 5- I t is described by two parameters: arithmetic mean determine the location of the center of the curve and standard deviation represents the scatter around the mean.Areas under the normal curve: DEPT OF PHARMACOLOGY Areas under the normal curve X ± 1 SD = 68% of the area on each side of the mean. X ± 2 SD = 95% of area on each side of the mean. X ± 3 SD = 99% of area on each side of the mean.Skewed data: DEPT OF PHARMACOLOGY Skewed data If we represent a collected data by a frequency polygon graph and the resulted curve does not simulate the normal distribution curve (with all its characteristics) then these data are not normally distributedCauses of Skewed Curve Not Normally Distributed Data: DEPT OF PHARMACOLOGY Causes of Skewed Curve Not Normally Distributed Data The curve may be skewed to the right or to the left side This is because The data collected are from: certain heterogeneous group or from diseased or abnormal population therefore the results obtained from these data can not be applied or generalized on the whole population.PowerPoint Presentation: DEPT OF PHARMACOLOGY NDC can be used in distinguishing between normal from abnormal measurements . Example: If we have NDC for hemoglobin levels for a population of normal adult males with mean ± SD = 11 ±1.5 If we obtain a hemoglobin reading for an individual = 8.1 and we want to know if he/she is normal or anemic. If this reading lies within the area under the curve at 95% of normal (i.e. mean ± 2 SD) he /she will be considered normal. If his reading is less then he is anemic.PowerPoint Presentation: DEPT OF PHARMACOLOGY The normal range for hemoglobin in this example will be: the higher level of hemoglobin : 11 + 2 ( 1.5 ) =14. The lower hemoglobin level 11 – 2 ( 1.5 ) = 8. i.e the normal range of hemoglobin of adult males is from 8 to 14. our sample (8.1 ) lies within the 95% of his population. therefore this individual is normal because his reading lies within the 95% of his population.Data Summarization: DEPT OF PHARMACOLOGY Data Summarization To summarize data, we need to use one or two parameters that can describe the data. Measures of Central tendency which describes the center of the data and the Measures of Dispersion , which show how the data are scattered around its center.Measures of central tendency: DEPT OF PHARMACOLOGY Measures of central tendency Variable usually has a point (center) around which the observed values lie. These averages are also called measures of central tendency. The three most commonly used averages are: The arithmetic mean: The Median The Mode1- The arithmetic mean:: DEPT OF PHARMACOLOGY 1- The arithmetic mean: the sum of observation divided by the number of observations: x = ∑ x n Where : x = mean ∑ denotes the (sum of) x the values of observation n the number of observation1- The arithmetic mean:: DEPT OF PHARMACOLOGY Example: In a study the age of 5 students were: 12 , 15, 10, 17, 13 Mean = sum of observations / number of observations Then the mean X = (12 + 15 + 10 + 17 + 13) / 5 =13.4 years 1- The arithmetic mean: Calculation of Mean For frequency Distribution Data: DEPT OF PHARMACOLOGY Calculation of Mean For frequency Distribution Data In case of frequency distribution data we calculate the mean by this equation: x = ∑ fx n where f = frequency for example : we want to calculate the mean incubation period of this group. Calculation of Mean For frequency Distribution Data: DEPT OF PHARMACOLOGY Calculation of Mean For frequency Distribution Data Calculation of Mean For frequency Distribution Data with class intervals : DEPT OF PHARMACOLOGY If data is presented in frequency table with class intervals we calculate mean by the same equation summation of f x1 /n , x1 denotes the midpoint of class interval. Example : calculate the mean of blood pressure of the following group : Calculation of Mean For frequency Distribution Data with class intervalsPowerPoint Presentation: DEPT OF PHARMACOLOGYPowerPoint Presentation: DEPT OF PHARMACOLOGY2- Median: DEPT OF PHARMACOLOGY 2- Median It is the middle observation in a series of observation after arranging them in an ascending or descending manner. The rank of median for is (n + 1)/2 if the number of observation is odd and n/2 if the number is even2- Median: DEPT OF PHARMACOLOGY Calculate the median of the following data 5, 6, 8, 9, 11 n = 5~ Odd!! -The rank of the median = n + 1 / 2 i.e. (5+ 1)/ 2 = 3 The median is the third value in these groups when data are arranged in ascending (or descending) manner. - So the median is 8 (the third value) 2- Median2- Median: DEPT OF PHARMACOLOGY - If the number of observation is even , the median will be calculated as follows: e.g. 5, 6, 8, 9 n = 4 - The rank of median = n / 2 i.e. 4 / 2 = 2 .The median is the second value of that group. If data are arranged ascendingly then the median will be 6 and if arranged descendingly the median will be 8 therefore the median will be the mean of both observations i.e. (6 + 8)/2 =7. 2- Median2- Median: DEPT OF PHARMACOLOGY For simplicity we can apply the same equation used for odd numbers i.e. n + 1 / 2. The median rank will be 4 + 1 /2 = 2 ½ i.e. the median will be the second and the third values i.e. 6 and 8, take their mean = 7. 2- Median3- Mode: DEPT OF PHARMACOLOGY The most frequent occurring value in the data is the mode and is calculated as follows: Example: 5, 6, 7, 5, 10. The mode in this data is 5 since number 5 is repeated twice. Sometimes, there is more than one mode and sometimes there is no mode especially in small set of observations. 3- Mode3- Mode: DEPT OF PHARMACOLOGY Example : 20 , 18 , 14, 20, 13, 14, 30, 19. There are two modes 14 and 20. Example : 300, 280 , 130, 125 , 240 , 270 . Has no mode . Unimodal Bimodal Nomodal 3- ModeAdvantages and disadvantages of the measures of central Tendency:: DEPT OF PHARMACOLOGY Advantages and disadvantages of the measures of central Tendency: - Mean: is the preferred CTM since it takes into account each individual observation but its main disadvantage is that it is affected by the extreme valus of observations.Advantages and disadvantages of the measures of central Tendency:: DEPT OF PHARMACOLOGY Median: it is a useful descriptive measure if there are one or two extremely high or low values. -Mode : is seldom used. Advantages and disadvantages of the measures of central Tendency:Measures of Dispersion: DEPT OF PHARMACOLOGY Measures of Dispersion The measure of dispersion describes the degree of variations or scatter or dispersion of the data around its central values: (dispersion = variation = spread = scatter). Range - R Variance - V Standard Deviation - SD Coefficient of Variation - COV 1- Range:: DEPT OF PHARMACOLOGY 1- Range: is the difference between the largest and smallest values. is the simplest measure of variation. disadvantages , it is based only on two of the observations and gives no idea of how the other observations are arranged between these two. Also, it tends to be large when the size of the sample increases 2- Variance: DEPT OF PHARMACOLOGY If we want to get the average of differences between the mean and each observation in the data, we have to reduce each value from the mean and then sum these differences and divide it by the number of observation. V = ∑ (mean – x i ) / n 2- Variance 2- Variance: DEPT OF PHARMACOLOGY To overcome this zero we square the difference between the mean and each value so the sign will be always positive . Thus we get: V = ∑ (mean – x) 2 / n - 1 2- Variance3- Standard Deviation SD: DEPT OF PHARMACOLOGY 3- Standard Deviation SD The main disadvantage of the variance is that it is the square of the units used. So, it is more convenient to express the variation in the original units by taking the square root of the variance. This is called the standard deviation (SD). Therefore SD = √ V i.e. SD = √ ∑ (mean – x) 2 / n - 14- Coefficient of variation CoV: DEPT OF PHARMACOLOGY 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. Example : if we have observations 5, 7, 10, 12 and 16. Their mean will be 50/5=10. SD = √ (25+9 +0 + 4 + 36 ) / (5-1) = √ 74 / 4 = 4.3 C.V. = 4.3 / 10 x 100 = 43% 4- Coefficient of variation CoVInferential statistics: DEPT OF PHARMACOLOGY Inferential statistics Inference involves making a Generalization about a larger group of individuals on the basis of a subset or sample.Inferential statistics Hypothesis Testing : DEPT OF PHARMACOLOGY Inferential statistics Hypothesis Testing In hypothesis testing we want to find out whether the observed variation among sampling is explained by chance alone ???? (i.e., the chance of random sampling variations ) , or due to a real difference ???? between groups.Hypothesis Testing: DEPT OF PHARMACOLOGY Hypothesis Testing It involves conducting a test of statistical significance quantifying the chance of random sampling variations that may account for observed results. In hypotheses testing, we are asking whether the sample mean for example is consistent with a certain hypothesis value for the population mean .Hypothesis Testing: DEPT OF PHARMACOLOGY Hypothesis Testing The method of assessing the hypotheses testing is known as significance test . The significance testing is a method for assessing whether a result is likely to be due to chance or due to a real effect .Hypothesis Testing –Steps : DEPT OF PHARMACOLOGY Hypothesis Testing –Steps >>> Formulate Hypothesis >>> Collect the Data >>>> Test Your Hypothesis >>> Accept of Reject Your HypothesisNull and alternative hypotheses: DEPT OF PHARMACOLOGY Null and alternative hypotheses In hypotheses testing, a specific hypothesis ( Null and alternative Hypothesis ) are formulated and tested. The null hypotheses H0 means : X1=X 2 Or X1-X 2=0 this means that there is no difference between x1 and x2 The alternative hypotheses H1 means X1>X2 or X1< X2Null and alternative hypotheses: DEPT OF PHARMACOLOGY Null and alternative hypotheses The alternative hypotheses H1 means X1>X2 or X1< X2 If we reject the null hypothesis, i.e there is a difference between the two readings, it is either H1 : x1 < x2 or H2 : x1> x2 in other words the null hypothesis is rejected because x1 is different from x2.General principles of significance tests: DEPT OF PHARMACOLOGY General principles of significance tests set up a null hypothesis and its alternative. find the value of the test statistic. refer the value of the test statistic to a known distribution which it would follow if the null hypothesis was true.General principles of significance tests: DEPT OF PHARMACOLOGY General principles of significance tests 4-conclude that the data are consistent or inconsistent with the null hypothesis. If the data are not consistent with the null hypotheses, the difference is said to be statistically significant. If the data are consistent with the null hypotheses it is said that we accept it i.e. statistically insignificant.General principles of significance tests P<0.05: DEPT OF PHARMACOLOGY General principles of significance tests P<0.05 In medicine, we usually consider that differences are significant if the probability is less than 0.05. This means that if the null hypothesis is true, we shall make a wrong decision less than 5 in a hundred timesPowerPoint Presentation: DEPT OF PHARMACOLOGY Probability is chance of an outcome. What is meant by chance is the relative frequency of the outcome. Probability = Of finding out error. If probability is low then finding out error will be rare and error will become uncommon.. P = 0.01 = 1 ÷ 100 REAL = 99 NOT REAL / ERROR = 1 P = 0.0001 = 0.1 ÷ 100 REAL = 99.9 NOT REAL / ERROR = 0.1Tests of significance: DEPT OF PHARMACOLOGY Tests of significance The selection of test of significance depends essentially on the type of data that we have. 1-Quantitative Data ( Means & SD): t test , paired t test and , ANOVA 2-Qualitative Data>>> Chi , and Z test .Tests of significance: DEPT OF PHARMACOLOGY Tests of significance Comparison of means : 1-comparing two means of large samples using the normal distribution: (z test or SND standard normal deviate) If we have a large sample size i.e. 60 or more and it follows a normal distribution then we have to use the z-test. z = (population mean — sample mean) / SD. If the result of z >2 then there is significant difference.Tests of significance: DEPT OF PHARMACOLOGY Tests of significance Since the normal range for any biological reading lies between the mean value of the population reading ± 2 SD. (this range includes 95% of the area under the normal distribution curve). Student’s t-test: DEPT OF PHARMACOLOGY Student’s t-test 2-Comparing two means of small samples using t-test: If we have a small sample size (less than 60), we can use the t distribution instead of the normal distribution. T = mean1 — mean2 / (SD 1 2 / n1) + (SD 2 2 / n2)PowerPoint Presentation: DEPT OF PHARMACOLOGY T-Test Case 1: The variable has a normal distribution and is known. In this case the test statistic is which has a standard normal distribution if . Case 2: The variable has a normal distribution and is unknown. The test statistic is which has a distribution if is true.PowerPoint Presentation: DEPT OF PHARMACOLOGY T-Test Case 3: The variable is not normal but n is large (which n>30), may be known or unknown. The test statistic is By central limit theorem it has approximately standard normal distribution (0,1) if is true.t-test: DEPT OF PHARMACOLOGY The value of t will be compared to values in the specific table of "t distribution test" at the value of the degree of freedom. If the value of t is less than that in the table , then the difference between samples is insignificant. If the t value is larger than that in the table so the difference is significant i.e. the null hypothesis is rejected. t-testt-test: DEPT OF PHARMACOLOGY 2-Comparing two means of small samples using t-test: If we have a small sample size (less than 60), we can use the t distribution instead of the normal distribution. T = mean1 — mean2 / (SD 1 2 / n1) + (SD 2 2 / n2) t-testPaired t-test: DEPT OF PHARMACOLOGY 3-paired t-test: If we are comparing repeated observation in the same individual or difference between paired data, we have to use paired t-test where the analysis is carried out using the mean and standard deviation of the difference between each pair. Paired t-testANOVA: DEPT OF PHARMACOLOGY 4-comparing several means: Sometimes we need to compare more than two means, this can be done by the use of several t-test which is not only tedious but can lead to spurious significant results. Therefore we have to use what we call analysis of variance or ANOVA. ANOVAANOVA: DEPT OF PHARMACOLOGY 4-comparing several means: There are two main types: one-way analysis of variance and two-way analysis of variance. One-way analysis of variance is appropriate when the subgroups to be compared are defined by just one factor, for example comparison between means of different socio-economic classes. The two-way analysis of variables is used when the subdivision is based upon more than one factor ANOVAANOVA: DEPT OF PHARMACOLOGY The main idea in the analysis of variance is that we have to take into account the variability within the groups and between the groups and value of F is equal to the ratio between the means sum square of between the groups and within the groups. F = between-groups MS / within-groups MS ANOVAChi-Squared Test: DEPT OF PHARMACOLOGY b-Qualitative variables: 1)Chi -squared test : Qualitative data are arranged in table formed by rows and columns. One variable define the rows and the categories of the other variable define the column. Chi-Squared TestChi-Squared Test: DEPT OF PHARMACOLOGY A chi-squared test is used to test whether there is an association between the row variable and the column variable or, in other words whether the distribution of individuals among the categories of one variable is independent of their distribution among the categories of the other. X 2 = (O-E) 2 / E Chi-Squared TestChi-Squared Test: DEPT OF PHARMACOLOGY 1)Chi -squared test : degree of freedom = (row - 1) (column - 1) O = observed value in the table E = expected value calculated as follows: E = Rt x Ct / GT total of row x total of column / grand total Chi-Squared TestSelected nonparametric tests Chi-Square test. Example: DEPT OF PHARMACOLOGY Selected nonparametric tests Chi-Square test. Example Question: whether men are treated more aggressively for cardiovascular problems than women? Sample: people have similar results on initial testing Response: whether or not a cardiac catheterization was recommended Independent: sex of the patientSelected nonparametric tests Chi-Square test. Example: DEPT OF PHARMACOLOGY Selected nonparametric tests Chi-Square test. Example Result : observed frequenciesSelected nonparametric tests Chi-Square test. Example: DEPT OF PHARMACOLOGY Selected nonparametric tests Chi-Square test. Example Result : expected frequenciesSelected nonparametric tests Chi-Square test. Example: DEPT OF PHARMACOLOGY Selected nonparametric tests Chi-Square test. Example Result : 2 = 2.52, df=1 (2-1) (2-1) p > 0.05 Null hypothesis is accepted at 5% level Conclusion: Recommendation for cardiac catheterization is not related to the sex of the patientTHANK YOU: DEPT OF PHARMACOLOGY THANK YOU You do not have the permission to view this presentation. In order to view it, please contact the author of the presentation.
statiatical analysis gopinathve Download Post to : URL : Related Presentations : Share Add to Flag Embed Email Send to Blogs and Networks Add to Channel Uploaded from authorPOINT lite Insert YouTube videos in PowerPont slides with aS Desktop Copy embed code: (To copy code, click on the text box) Embed: URL: Thumbnail: WordPress Embed Customize Embed The presentation is successfully added In Your Favorites. Views: 32 Category: Education License: All Rights Reserved Like it (0) Dislike it (0) Added: February 13, 2012 This Presentation is Public Favorites: 0 Presentation Description PHARMACEUTICAL STATISTICS Comments Posting comment... By: nrsbb (3 month(s) ago) sir i am studying m.pharm please send your ppts to my mail it will be usefull for me please this ppt to my mail nareshgd6@gmail.com Saving..... Post Reply Close Saving..... Edit Comment Close Premium member Presentation Transcript PowerPoint Presentation: DEPT OF PHARMACOLOGY WELCOMESTATISTICAL ANALYSIS: DEPT OF PHARMACOLOGY STATISTICAL ANALYSIS BY GOPINATH.V.E. DEPARTMENT OF PHARMACOLOGY SHREE DEVI COLLEGE OF PHARMACYPowerPoint Presentation: DEPT OF PHARMACOLOGY Definition of Statistics: DEPT OF PHARMACOLOGY Definition of Statistics Statistics is the science of dealing with numbers . It is used for c ollection , s ummarization , p resentation and a nalysis of data. Statistics provides a way of organizing data to get information on a wider and more formal (objective) basis than relying on personal experience (subjective).Uses of medical statistics : DEPT OF PHARMACOLOGY Uses of medical statistics Medical statistics are used in 1 - Planning , monitoring and evaluating community health care programs. 2- Epidemiological research studies. 3- Diagnosis of community health problems. 4- Comparison of health status and diseases in different countries and in one country over years. 5- To form standards for the different biological measurements as weight, height. 6- To differentiate between diseased and normal groups.Types of data: DEPT OF PHARMACOLOGY Types of data Any aspect of an individual that is measured, is called variable. Variables are either 1-Quantitative or 2-Qualitative . 1- Quantitative data : it is numerical data. Discrete data : are usually whole numbers , such as number of cases of certain disease, number of hospital beds (no decimal fraction). Continuous data: it implies the measurement on a continuous scale e.g. height, weight, age (a decimal fraction can be present ).2- Qualitative data: DEPT OF PHARMACOLOGY 2- Qualitative data Qualitative data : It is non numerical data and is subdivided into Two Types: A- Categorical : data are purely descriptive and imply no ordering of any kind such as sex, area of residence. B- Ordinal data : are those which imply some kind of ordering like - Level of education: - Socio-economic status: - Degree of severity of disease:Normal Distribution curve: DEPT OF PHARMACOLOGY Normal Distribution curve NDC is a Graphical Presentation < Frequency Polygon> of any Quantitative Biologic Variables The Normal Distribution Curve is the frequency polygon of a quantitative variable measured in large number. It is a form of presentation of frequency distribution of biologic variables such as weights, heights, hemoglobin level and blood pressure or any continuous data. It occupies a major role in the techniques of statistical analysis.PowerPoint Presentation: DEPT OF PHARMACOLOGYCharacteristics of Normal Distribution curve: DEPT OF PHARMACOLOGY Characteristics of Normal Distribution curve 1- It is bell shaped, continuous curve. 2- It is symmetrical i.e. can be divided into two equal halves vertically. 3- The tails never touch the base line but extended to infinity in either direction. 4- T he mean , median and mode values coincide 5- I t is described by two parameters: arithmetic mean determine the location of the center of the curve and standard deviation represents the scatter around the mean.Areas under the normal curve: DEPT OF PHARMACOLOGY Areas under the normal curve X ± 1 SD = 68% of the area on each side of the mean. X ± 2 SD = 95% of area on each side of the mean. X ± 3 SD = 99% of area on each side of the mean.Skewed data: DEPT OF PHARMACOLOGY Skewed data If we represent a collected data by a frequency polygon graph and the resulted curve does not simulate the normal distribution curve (with all its characteristics) then these data are not normally distributedCauses of Skewed Curve Not Normally Distributed Data: DEPT OF PHARMACOLOGY Causes of Skewed Curve Not Normally Distributed Data The curve may be skewed to the right or to the left side This is because The data collected are from: certain heterogeneous group or from diseased or abnormal population therefore the results obtained from these data can not be applied or generalized on the whole population.PowerPoint Presentation: DEPT OF PHARMACOLOGY NDC can be used in distinguishing between normal from abnormal measurements . Example: If we have NDC for hemoglobin levels for a population of normal adult males with mean ± SD = 11 ±1.5 If we obtain a hemoglobin reading for an individual = 8.1 and we want to know if he/she is normal or anemic. If this reading lies within the area under the curve at 95% of normal (i.e. mean ± 2 SD) he /she will be considered normal. If his reading is less then he is anemic.PowerPoint Presentation: DEPT OF PHARMACOLOGY The normal range for hemoglobin in this example will be: the higher level of hemoglobin : 11 + 2 ( 1.5 ) =14. The lower hemoglobin level 11 – 2 ( 1.5 ) = 8. i.e the normal range of hemoglobin of adult males is from 8 to 14. our sample (8.1 ) lies within the 95% of his population. therefore this individual is normal because his reading lies within the 95% of his population.Data Summarization: DEPT OF PHARMACOLOGY Data Summarization To summarize data, we need to use one or two parameters that can describe the data. Measures of Central tendency which describes the center of the data and the Measures of Dispersion , which show how the data are scattered around its center.Measures of central tendency: DEPT OF PHARMACOLOGY Measures of central tendency Variable usually has a point (center) around which the observed values lie. These averages are also called measures of central tendency. The three most commonly used averages are: The arithmetic mean: The Median The Mode1- The arithmetic mean:: DEPT OF PHARMACOLOGY 1- The arithmetic mean: the sum of observation divided by the number of observations: x = ∑ x n Where : x = mean ∑ denotes the (sum of) x the values of observation n the number of observation1- The arithmetic mean:: DEPT OF PHARMACOLOGY Example: In a study the age of 5 students were: 12 , 15, 10, 17, 13 Mean = sum of observations / number of observations Then the mean X = (12 + 15 + 10 + 17 + 13) / 5 =13.4 years 1- The arithmetic mean: Calculation of Mean For frequency Distribution Data: DEPT OF PHARMACOLOGY Calculation of Mean For frequency Distribution Data In case of frequency distribution data we calculate the mean by this equation: x = ∑ fx n where f = frequency for example : we want to calculate the mean incubation period of this group. Calculation of Mean For frequency Distribution Data: DEPT OF PHARMACOLOGY Calculation of Mean For frequency Distribution Data Calculation of Mean For frequency Distribution Data with class intervals : DEPT OF PHARMACOLOGY If data is presented in frequency table with class intervals we calculate mean by the same equation summation of f x1 /n , x1 denotes the midpoint of class interval. Example : calculate the mean of blood pressure of the following group : Calculation of Mean For frequency Distribution Data with class intervalsPowerPoint Presentation: DEPT OF PHARMACOLOGYPowerPoint Presentation: DEPT OF PHARMACOLOGY2- Median: DEPT OF PHARMACOLOGY 2- Median It is the middle observation in a series of observation after arranging them in an ascending or descending manner. The rank of median for is (n + 1)/2 if the number of observation is odd and n/2 if the number is even2- Median: DEPT OF PHARMACOLOGY Calculate the median of the following data 5, 6, 8, 9, 11 n = 5~ Odd!! -The rank of the median = n + 1 / 2 i.e. (5+ 1)/ 2 = 3 The median is the third value in these groups when data are arranged in ascending (or descending) manner. - So the median is 8 (the third value) 2- Median2- Median: DEPT OF PHARMACOLOGY - If the number of observation is even , the median will be calculated as follows: e.g. 5, 6, 8, 9 n = 4 - The rank of median = n / 2 i.e. 4 / 2 = 2 .The median is the second value of that group. If data are arranged ascendingly then the median will be 6 and if arranged descendingly the median will be 8 therefore the median will be the mean of both observations i.e. (6 + 8)/2 =7. 2- Median2- Median: DEPT OF PHARMACOLOGY For simplicity we can apply the same equation used for odd numbers i.e. n + 1 / 2. The median rank will be 4 + 1 /2 = 2 ½ i.e. the median will be the second and the third values i.e. 6 and 8, take their mean = 7. 2- Median3- Mode: DEPT OF PHARMACOLOGY The most frequent occurring value in the data is the mode and is calculated as follows: Example: 5, 6, 7, 5, 10. The mode in this data is 5 since number 5 is repeated twice. Sometimes, there is more than one mode and sometimes there is no mode especially in small set of observations. 3- Mode3- Mode: DEPT OF PHARMACOLOGY Example : 20 , 18 , 14, 20, 13, 14, 30, 19. There are two modes 14 and 20. Example : 300, 280 , 130, 125 , 240 , 270 . Has no mode . Unimodal Bimodal Nomodal 3- ModeAdvantages and disadvantages of the measures of central Tendency:: DEPT OF PHARMACOLOGY Advantages and disadvantages of the measures of central Tendency: - Mean: is the preferred CTM since it takes into account each individual observation but its main disadvantage is that it is affected by the extreme valus of observations.Advantages and disadvantages of the measures of central Tendency:: DEPT OF PHARMACOLOGY Median: it is a useful descriptive measure if there are one or two extremely high or low values. -Mode : is seldom used. Advantages and disadvantages of the measures of central Tendency:Measures of Dispersion: DEPT OF PHARMACOLOGY Measures of Dispersion The measure of dispersion describes the degree of variations or scatter or dispersion of the data around its central values: (dispersion = variation = spread = scatter). Range - R Variance - V Standard Deviation - SD Coefficient of Variation - COV 1- Range:: DEPT OF PHARMACOLOGY 1- Range: is the difference between the largest and smallest values. is the simplest measure of variation. disadvantages , it is based only on two of the observations and gives no idea of how the other observations are arranged between these two. Also, it tends to be large when the size of the sample increases 2- Variance: DEPT OF PHARMACOLOGY If we want to get the average of differences between the mean and each observation in the data, we have to reduce each value from the mean and then sum these differences and divide it by the number of observation. V = ∑ (mean – x i ) / n 2- Variance 2- Variance: DEPT OF PHARMACOLOGY To overcome this zero we square the difference between the mean and each value so the sign will be always positive . Thus we get: V = ∑ (mean – x) 2 / n - 1 2- Variance3- Standard Deviation SD: DEPT OF PHARMACOLOGY 3- Standard Deviation SD The main disadvantage of the variance is that it is the square of the units used. So, it is more convenient to express the variation in the original units by taking the square root of the variance. This is called the standard deviation (SD). Therefore SD = √ V i.e. SD = √ ∑ (mean – x) 2 / n - 14- Coefficient of variation CoV: DEPT OF PHARMACOLOGY 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. Example : if we have observations 5, 7, 10, 12 and 16. Their mean will be 50/5=10. SD = √ (25+9 +0 + 4 + 36 ) / (5-1) = √ 74 / 4 = 4.3 C.V. = 4.3 / 10 x 100 = 43% 4- Coefficient of variation CoVInferential statistics: DEPT OF PHARMACOLOGY Inferential statistics Inference involves making a Generalization about a larger group of individuals on the basis of a subset or sample.Inferential statistics Hypothesis Testing : DEPT OF PHARMACOLOGY Inferential statistics Hypothesis Testing In hypothesis testing we want to find out whether the observed variation among sampling is explained by chance alone ???? (i.e., the chance of random sampling variations ) , or due to a real difference ???? between groups.Hypothesis Testing: DEPT OF PHARMACOLOGY Hypothesis Testing It involves conducting a test of statistical significance quantifying the chance of random sampling variations that may account for observed results. In hypotheses testing, we are asking whether the sample mean for example is consistent with a certain hypothesis value for the population mean .Hypothesis Testing: DEPT OF PHARMACOLOGY Hypothesis Testing The method of assessing the hypotheses testing is known as significance test . The significance testing is a method for assessing whether a result is likely to be due to chance or due to a real effect .Hypothesis Testing –Steps : DEPT OF PHARMACOLOGY Hypothesis Testing –Steps >>> Formulate Hypothesis >>> Collect the Data >>>> Test Your Hypothesis >>> Accept of Reject Your HypothesisNull and alternative hypotheses: DEPT OF PHARMACOLOGY Null and alternative hypotheses In hypotheses testing, a specific hypothesis ( Null and alternative Hypothesis ) are formulated and tested. The null hypotheses H0 means : X1=X 2 Or X1-X 2=0 this means that there is no difference between x1 and x2 The alternative hypotheses H1 means X1>X2 or X1< X2Null and alternative hypotheses: DEPT OF PHARMACOLOGY Null and alternative hypotheses The alternative hypotheses H1 means X1>X2 or X1< X2 If we reject the null hypothesis, i.e there is a difference between the two readings, it is either H1 : x1 < x2 or H2 : x1> x2 in other words the null hypothesis is rejected because x1 is different from x2.General principles of significance tests: DEPT OF PHARMACOLOGY General principles of significance tests set up a null hypothesis and its alternative. find the value of the test statistic. refer the value of the test statistic to a known distribution which it would follow if the null hypothesis was true.General principles of significance tests: DEPT OF PHARMACOLOGY General principles of significance tests 4-conclude that the data are consistent or inconsistent with the null hypothesis. If the data are not consistent with the null hypotheses, the difference is said to be statistically significant. If the data are consistent with the null hypotheses it is said that we accept it i.e. statistically insignificant.General principles of significance tests P<0.05: DEPT OF PHARMACOLOGY General principles of significance tests P<0.05 In medicine, we usually consider that differences are significant if the probability is less than 0.05. This means that if the null hypothesis is true, we shall make a wrong decision less than 5 in a hundred timesPowerPoint Presentation: DEPT OF PHARMACOLOGY Probability is chance of an outcome. What is meant by chance is the relative frequency of the outcome. Probability = Of finding out error. If probability is low then finding out error will be rare and error will become uncommon.. P = 0.01 = 1 ÷ 100 REAL = 99 NOT REAL / ERROR = 1 P = 0.0001 = 0.1 ÷ 100 REAL = 99.9 NOT REAL / ERROR = 0.1Tests of significance: DEPT OF PHARMACOLOGY Tests of significance The selection of test of significance depends essentially on the type of data that we have. 1-Quantitative Data ( Means & SD): t test , paired t test and , ANOVA 2-Qualitative Data>>> Chi , and Z test .Tests of significance: DEPT OF PHARMACOLOGY Tests of significance Comparison of means : 1-comparing two means of large samples using the normal distribution: (z test or SND standard normal deviate) If we have a large sample size i.e. 60 or more and it follows a normal distribution then we have to use the z-test. z = (population mean — sample mean) / SD. If the result of z >2 then there is significant difference.Tests of significance: DEPT OF PHARMACOLOGY Tests of significance Since the normal range for any biological reading lies between the mean value of the population reading ± 2 SD. (this range includes 95% of the area under the normal distribution curve). Student’s t-test: DEPT OF PHARMACOLOGY Student’s t-test 2-Comparing two means of small samples using t-test: If we have a small sample size (less than 60), we can use the t distribution instead of the normal distribution. T = mean1 — mean2 / (SD 1 2 / n1) + (SD 2 2 / n2)PowerPoint Presentation: DEPT OF PHARMACOLOGY T-Test Case 1: The variable has a normal distribution and is known. In this case the test statistic is which has a standard normal distribution if . Case 2: The variable has a normal distribution and is unknown. The test statistic is which has a distribution if is true.PowerPoint Presentation: DEPT OF PHARMACOLOGY T-Test Case 3: The variable is not normal but n is large (which n>30), may be known or unknown. The test statistic is By central limit theorem it has approximately standard normal distribution (0,1) if is true.t-test: DEPT OF PHARMACOLOGY The value of t will be compared to values in the specific table of "t distribution test" at the value of the degree of freedom. If the value of t is less than that in the table , then the difference between samples is insignificant. If the t value is larger than that in the table so the difference is significant i.e. the null hypothesis is rejected. t-testt-test: DEPT OF PHARMACOLOGY 2-Comparing two means of small samples using t-test: If we have a small sample size (less than 60), we can use the t distribution instead of the normal distribution. T = mean1 — mean2 / (SD 1 2 / n1) + (SD 2 2 / n2) t-testPaired t-test: DEPT OF PHARMACOLOGY 3-paired t-test: If we are comparing repeated observation in the same individual or difference between paired data, we have to use paired t-test where the analysis is carried out using the mean and standard deviation of the difference between each pair. Paired t-testANOVA: DEPT OF PHARMACOLOGY 4-comparing several means: Sometimes we need to compare more than two means, this can be done by the use of several t-test which is not only tedious but can lead to spurious significant results. Therefore we have to use what we call analysis of variance or ANOVA. ANOVAANOVA: DEPT OF PHARMACOLOGY 4-comparing several means: There are two main types: one-way analysis of variance and two-way analysis of variance. One-way analysis of variance is appropriate when the subgroups to be compared are defined by just one factor, for example comparison between means of different socio-economic classes. The two-way analysis of variables is used when the subdivision is based upon more than one factor ANOVAANOVA: DEPT OF PHARMACOLOGY The main idea in the analysis of variance is that we have to take into account the variability within the groups and between the groups and value of F is equal to the ratio between the means sum square of between the groups and within the groups. F = between-groups MS / within-groups MS ANOVAChi-Squared Test: DEPT OF PHARMACOLOGY b-Qualitative variables: 1)Chi -squared test : Qualitative data are arranged in table formed by rows and columns. One variable define the rows and the categories of the other variable define the column. Chi-Squared TestChi-Squared Test: DEPT OF PHARMACOLOGY A chi-squared test is used to test whether there is an association between the row variable and the column variable or, in other words whether the distribution of individuals among the categories of one variable is independent of their distribution among the categories of the other. X 2 = (O-E) 2 / E Chi-Squared TestChi-Squared Test: DEPT OF PHARMACOLOGY 1)Chi -squared test : degree of freedom = (row - 1) (column - 1) O = observed value in the table E = expected value calculated as follows: E = Rt x Ct / GT total of row x total of column / grand total Chi-Squared TestSelected nonparametric tests Chi-Square test. Example: DEPT OF PHARMACOLOGY Selected nonparametric tests Chi-Square test. Example Question: whether men are treated more aggressively for cardiovascular problems than women? Sample: people have similar results on initial testing Response: whether or not a cardiac catheterization was recommended Independent: sex of the patientSelected nonparametric tests Chi-Square test. Example: DEPT OF PHARMACOLOGY Selected nonparametric tests Chi-Square test. Example Result : observed frequenciesSelected nonparametric tests Chi-Square test. Example: DEPT OF PHARMACOLOGY Selected nonparametric tests Chi-Square test. Example Result : expected frequenciesSelected nonparametric tests Chi-Square test. Example: DEPT OF PHARMACOLOGY Selected nonparametric tests Chi-Square test. Example Result : 2 = 2.52, df=1 (2-1) (2-1) p > 0.05 Null hypothesis is accepted at 5% level Conclusion: Recommendation for cardiac catheterization is not related to the sex of the patientTHANK YOU: DEPT OF PHARMACOLOGY THANK YOU