Mathematics and Biostatistics (MS101) Level 1 Lecture (3) PDF

2024/2025 Mathematics and Biostatistics (MS101) Level 1 Lecture (3) Assoc. Prof. Ahmed R. Elsheakh Director of the PharmD Clinical Pharmacy Program Faculty of Pharmacy - Mansoura National University E-mail: [email protected]...

2024/2025 Mathematics and Biostatistics (MS101) Level 1 Lecture (3) Assoc. Prof. Ahmed R. Elsheakh Director of the PharmD Clinical Pharmacy Program Faculty of Pharmacy - Mansoura National University E-mail: [email protected] 1  Part I Biostatistics 2  Let’s Recap Why we need to learn statistics as pharmacists Lecture 1 Role of biostatistician Definitions Descriptive vs inferential statistics Sampling methods Types of variables Lecture 2 Descriptive statistics and data summarization Measurements of quantitative data (part I: central tendency) 3 From this lecture, each student will be able to: Deal with descriptive statistics and data summarization: Measurement of variability (spread) Define and calculate: 1. Range 2. Quartile & Interquartile range 3. Variance & Standard deviation 4 5 Why we measure dispersion together with central tendency??? Company #1: 47, 38, 35, 40, 36, 45, 39 Mean for both 40 Company #2: 70, 33, 18, 52, 27 6  1. Range  Range = Largest value - Smallest value  Greatly affected by outliers.  Its calculation is based on two values only. May be misleading in the presence of outliers.  Method for identification: 1. Sort or organize the data. 2. Find maximum and minimum values. 3. Subtract minimum from maximum. *In the health sciences, range is often reported as the actual values rather than the difference between the two extreme values (Min, Max). 7  1. Range Ex.: Suppose a researcher is studying the height of individuals in a population. The heights of 10 individuals are measured and recorded as follows: 165 cm, 170 cm, 175 cm, 180 cm, 182 cm, 185 cm, 190 cm, 195 cm, 200 cm, 205 cm. Solution The: lowest value= 165 cm, the highest value= 205 cm The range= 205 cm - 165 cm = 40 cm * This means that the tallest individual in the population is 40 cm taller than the shortest individual. 8  2. Quartiles and Interquartile Range Quartiles: Summary measures that divide a ranked data set into four equal parts. Interquartile range (IQR): The difference between the third quartile and the first quartile for a data. IQR= Q3 -Q1 9 Steps: 1. Sort (done) 2. Median = 30 3. Exclude median 4. Q1: middle position of lower half of data = 5th position Q1 = 29 5. Q3: middle position of upper half of data = 15th position Q3 = 33 So, IQR = 33 – 29= 4 years 10 Example #2: The following are the ages (in years) of nine patients in the intensive care unit at a specific hospital: 47, 28, 39, 51, 33, 37, 59, 24, 33 Find the values of the three quartiles. Where does the age of 28 years fall in relation to the ages of these employees? Find the IQR. 11 Solution First: rank the given data in increasing order. Then, calculate the three quartiles: A #1: Q1 = 30.5 years, Q2 = 37 years, and Q3 = 49 years A #2: The age of 28 falls in the lowest 25% of the ages. A #3: IQR= Q3-Q1= 49-30.5=18.5 years. 12 2. 3. Variance & Standard deviation (SD) SD: The most-used measure of dispersion. Its value tells how closely the values clustered around the mean. Obtained by taking the square root of the variance. ( xi  x ) 2 s n 1 13  3. Variance Steps to Calculate Variance and Standard Deviation 1. Calculate the mean. 2. Subtract the mean from each observation. 3. Square the difference. 4. Sum the squared differences 5. Divide the sum of the squared differences by n – 1(= variance) 6. Take the square root of the variance (= standard deviation) 14 Obs Age 1 27 2 28 3 28 Calculate Variance, Standard 4 29 Deviation 5 29 1. Calculate mean = 589/19 = 31 6 29 7 30 2. Subtract mean from each value 8 30 9 30 3. Square each difference 4. Sum the squared differences (“sum of 10 30 11 30 12 31 squares”) 13 31 14 32 5. Variance: Divide the sum of squares 15 33 by n – 1 16 34 17 35 6. Standard deviation: Take the square 18 36 root of the variance 19 37 Sum: 589 15 Obs Age Mean Diff 1 27 31 2 28 31 3 28 31 Calculate Variance, Standard 4 29 31 Deviation 5 29 31 1. Calculate mean 6 29 31 7 30 31 2. Subtract mean from each value 8 30 31 9 30 31 3. Square each difference 4. Sum the squared differences (“sum of 10 30 31 11 30 31 12 31 31 squares”) 13 31 31 14 32 31 5. Variance: Divide the sum of squares 15 33 31 by n – 1 16 34 31 17 35 31 6. Standard deviation: Take the square 18 36 31 root of the variance 19 37 31 Sum: 589 16 Obs Age Mean Diff 1 27 31 -4 2 28 31 -3 3 28 31 -3 Calculate Variance, Standard 4 29 31 -2 Deviation 5 29 31 -2 1. Calculate mean 6 29 31 -2 7 30 31 -1 2. Subtract mean from each value 8 30 31 -1 9 30 31 -1 3. Square each difference 4. Sum the squared differences (“sum of 10 30 31 -1 11 30 31 -1 12 31 31 0 squares”) 13 31 31 0 14 32 31 1 5. Variance: Divide the sum of squares by 15 33 31 2 n–1 16 34 31 3 17 35 31 4 6. Standard deviation: Take the square root 18 36 31 5 of the variance 19 37 31 6 Sum: 589 17 Obs Age Mean Diff Diff Sq’d 1 27 31 -4 16 2 28 31 -3 9 3 28 31 -3 9 Calculate Variance, Standard 4 29 31 -2 4 Deviation 5 29 31 -2 4 1. Calculate mean 6 29 31 -2 4 7 30 31 -1 1 2. Subtract mean from each value 8 30 31 -1 1 9 30 31 -1 1 3. Square each difference 4. Sum the squared differences (“sum 10 30 31 -1 1 11 30 31 -1 1 12 31 31 0 0 of squares”) 13 31 31 0 0 14 32 31 1 1 5. Variance: Divide the sum of squares 15 33 31 2 4 by n – 1 16 34 31 3 9 17 35 31 4 16 6. Standard deviation: Take the square 18 36 31 5 25 root of the variance 19 37 31 6 36 Sum: 589 18 Obs Age Mean Diff Diff Sq’d 1 27 31 -4 16 2 28 31 -3 9 3 28 31 -3 9 Calculate Variance, Standard 4 29 31 -2 4 Deviation 5 29 31 -2 4 1. Calculate mean 6 29 31 -2 4 7 30 31 -1 1 2. Subtract mean from each value 8 30 31 -1 1 9 30 31 -1 1 3. Square each differences 10 30 31 -1 1 11 30 31 -1 1 4. Sum the squared differences 12 31 31 0 0 (“sum of squares”) 13 31 31 0 0 14 32 31 1 1 5. Variance: Divide the sum of squares 15 33 31 2 4 by n – 1 16 34 31 3 9 17 35 31 4 16 6. Standard deviation: Take the square 18 36 31 5 25 root of the variance 19 37 31 6 36 Sum: 589 Sum: 142 19 Obs Age Mean Diff Diff Sq’d 1 27 31 -4 16 2 28 31 -3 9 3 28 31 -3 9 Calculate Variance, Standard 4 29 31 -2 4 Deviation 5 29 31 -2 4 1. Calculate mean 6 29 31 -2 4 7 30 31 -1 1 2. Subtract mean from each value 8 30 31 -1 1 9 30 31 -1 1 3. Square each differences 10 30 31 -1 1 11 30 31 -1 1 4. Sum the squared differences 12 31 31 0 0 (“sum of squares”) 13 31 31 0 0 14 32 31 1 1 5. Variance: Divide the sum of 15 33 31 2 4 squares by n – 1 16 34 31 3 9 17 35 31 4 16 Variance = 142 / 18 = 7.889 18 36 31 5 25 19 37 31 6 36 6. Standard deviation: Take the Sum: 589 Sum: 142 square root of the variance 20 Obs Age Mean Diff Diff Sq’d 1 27 31 -4 16 2 28 31 -3 9 3 28 31 -3 9 Calculate Variance, Standard 4 29 31 -2 4 Deviation 5 29 31 -2 4 1. Calculate mean 6 29 31 -2 4 7 30 31 -1 1 2. Subtract mean from each value 8 30 31 -1 1 9 30 31 -1 1 3. Square each differences 4. Sum the squared differences (“sum 10 30 31 -1 1 11 30 31 -1 1 12 31 31 0 0 of squares”) 13 31 31 0 0 14 32 31 1 1 5. Variance: Divide the sum of 15 33 31 2 4 squares by n – 1 16 34 31 3 9 17 35 31 4 16 Variance = 142 / 18 = 7.889 18 36 31 5 25 19 37 31 6 36 6. Standard deviation: Take the Sum: 589 Sum: 142 square root of the variance SD = 7.889 = 2.81 21 Q: Why do we square the deviations to calculate the variance and standard deviation?? 22  Summary Range IQR SD Max - Min Q3 – Q1 ( xi  x ) 2 s n 1 23 24  References Rose, M. R. (2024). Biostatistics and Evidence-Based Medicine. In Harriet Lane Handbook (23th ed. Chapter 29, pp. 746-757.e1). Elsevier. Weiss, N. A. (2014). Introductory statistics (7th ed.). Pearson. Rosner, B. (2015). Fundamentals of biostatistics (8th ed.). Brooks/Cole, Cengage Learning. Kirkwood, B. R., & Sterne, J. A. C. (2003). Essential medical statistics (8th ed.). Wiley-Blackwell. 25 Helpful Textbook on Biostatistics for further reading 26 27