UNIT III Central tendency measure of dispersion.pptx

UNIT III Central tendency measure of dispersion.pptx

1. UNIT III Central tendency measure of dispersion DR PRASANNA MOHAN PROFESSOR/RESEARCH HEAD KRUPANIDHI COLLEGE OF PHYSIOTHERAPY

2. Summary of Measures Central Tendency Mean Median Mode Quartile Summary Measures Variation Variance Standard Deviation Coefficient of Variation Range

3. Measures of Central Tendency A measure of central tendency is a descriptive statistic that describes the average, or typical value of a set of scores. There are three common measures of central tendency: The mean The median The mode

4. The mean is: the arithmetic average of all the scores (  X)/N the number, m, that makes  (X - m) equal to 0 the number, m, that makes  (X - m) 2 a minimum The mean of a population is represented by the Greek letter  ; the mean of a sample is represented by X The Mean

5. Calculating the Mean for Grouped Data where: f X = a score multiplied by its frequency 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 12 14 Mean = 5 Mean = 6 Mean affected by extreme values

6. When To Use the Mean You should use the mean when the data are interval or ratio scaled Many people will use the mean with ordinally scaled data too and the data are not skewed The mean is preferred because it is sensitive to every score If you change one score in the data set, the mean will change

7. Calculating the Mean Calculate the mean of the following data: 1 5 4 3 2 Sum the scores ( X) : 1 + 5 + 4 + 3 + 2 = 15 Divide the sum ( X = 15) by the number of scores (N = 5): 15 / 5 = 3 Mean = X = 3

8. Find the mean of the following data: Mean = [3(10)+10(9)+9(8)+8(7)+10(6)+ 2(5)]/42 = 7.57 Score Number of students 10 3 9 10 8 9 7 8 6 10 5 2 Calculating the Mean for Grouped Data

9. Activity

10. The Median The median is simply another name for the 50 th percentile It is the score in the middle; half of the scores are larger than the median and half of the scores are smaller than the median Not affected by extreme values 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 12 14 Median = 5 Median = 5

11. How To Calculate the Median Conceptually, it is easy to calculate the median There are many minor problems that can occur; it is best to let a computer do it Sort the data from highest to lowest Find the score in the middle middle = (n + 1) / 2 If n, the number of scores, is even the median is the average of the middle two scores

12. Calculating the Median for Grouped Data To use this formula first determine median class. Median class is that class whose less than type cumulative frequency is just more than N / 2 ; l = lower limit of median class ; cf = less than type cumulative frequency of premedian class; f = frequency of median class h = class width.

13. When To Use the Median The median is often used when the distribution of scores is either positively or negatively skewed The few really large scores (positively skewed) or really small scores (negatively skewed) will not overly influence the median

14. Median Example What is the median of the following scores: 10 8 14 15 7 3 3 8 12 10 9 Sort the scores: 15 14 12 10 10 9 8 8 7 3 3 Determine the middle score: middle = (n + 1) / 2 = (11 + 1) / 2 = 6 Middle score = median = 9

15. Median Example What is the median of the following scores: 24 18 19 42 16 12 Sort the scores: 42 24 19 18 16 12 Determine the middle score: middle = (n + 1) / 2 = (6 + 1) / 2 = 3.5 Median = average of 3 rd and 4 th scores: (19 + 18) / 2 = 18.5

16. The Mode The mode is the score that occurs most frequently in a set of data Not Affected by Extreme Values There May Not be a Mode There May be Several Modes Used for Either Numerical or Categorical Data 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

17. Activity

18. Calculating the Mode for Grouped Data To use this formula first determine modal class. Modal class is that class which has maximum frequency ; l = lower limit of modal class; f m = maximum frequency; f 1 = frequency of pre modal class ; f 2 = frequency of post modal class h = class width.

19. When To Use the Mode The mode is not a very useful measure of central tendency It is insensitive to large changes in the data set That is, two data sets that are very different from each other can have the same mode The mode is primarily used with nominally scaled data It is the only measure of central tendency that is appropriate for nominally scaled data

20. Activity

21. Mean, Median, and Mode Calculations Sample Data for Individual and Group Data

22. Individual Data: Sample and Calculations

23. Mean Calculation: Individual Data

24. Median Calculation: Individual Data

25. Mode Calculation: Individual Data

26. Where: f = frequency x = mid-point of the class interval ∑f = total frequency

27. Grouped Data: Sample and Calculations Sample Data: Hours studied by 50 students Hours Studied (Class Interval) Number of Students (Frequency) 0 - 2 5 2 - 4 8 4 - 6 12 6 - 8 15 8 - 10 7 10 - 12 3

28. Mean Calculation: Grouped Data

29. Median Calculation: Grouped Data

30. Class Interval Frequency Cumulative Frequency 0 - 2 5 5 2 - 4 8 13 4 - 6 12 25 6 - 8 15 40 8 - 10 7 47 10 - 12 3 50

31. Mode Calculation: Grouped Data L= lower boundary of the modal class f1​ = frequency of the modal class f0 = frequency of the class before the modal class f2​ = frequency of the class after the modal class h = class width (2 hours)

32. Mode

33. Wages ( C.I.) 40-60 60-80 80-100 100-120 120-140 140-160 No.of workers (freq) 50 80 30 20 50 20 X 1 2 3 4 5 Total No. of Families (freq) 20 50 20 5 5 100 Problem 1 : Wages (in Rs) paid to workers of an organization are given below. Calculate Mean, Median and Mode. Problem 2 : Weekly demand for marine fish (in kg) (x) for 100 families is given below. Calculate Mean, Median and Mode. Calculate Mean, Median & Mode

34. In symmetrical distributions, the median and mean are equal For normal distributions, mean = median = mode In positively skewed distributions, the mean is greater than the median In negatively skewed distributions, the mean is smaller than the median Relation Between Mean, Median & Mode

35. Normal and skewed distribution

36. Guidelines for Measures of Central Tendency in Physiotherapy Research Understanding Mean, Median, and Mode in Clinical Studies

37. 1. Mean (Arithmetic Average) • Best for continuous data (e.g., pain scores, range of motion) • Use when data is symmetrically distributed • Avoid if there are extreme outliers (e.g., recovery time) Example: Average pain reduction across patients.

38. 2. Median (Middle Value) • Best for skewed data or when outliers are present (e.g., recovery time) • Use with ordinal data (e.g., satisfaction scores) • Avoid with symmetrical data where the mean is more appropriate Example: Median recovery time in post-surgery patients.

39. 3. Mode (Most Frequent Value) • Best for categorical data (e.g., most common injury type) • Use to find the most frequent values • Avoid for continuous data with unique values Example: Most common rehabilitation exercise chosen by patients.

40. Physiotherapy Research Example Study: Effect of ultrasound therapy on pain and range of motion • Mean: Average improvement in shoulder range of motion • Median: Median recovery time for patients • Mode: Most common type of injury treated

41. Summary of Guidelines • Mean: Use for continuous, symmetrical data • Median: Use for skewed data or ordinal data • Mode: Use for categorical data or finding the most frequent values

42. Measure Physiotherapy Research Use Case When to Use Avoid When Mean Average pain reduction, range of motion after treatment Continuous, symmetrical data (e.g., range of motion) Skewed data with extreme outliers (e.g., recovery time) Median Recovery time after surgery, patient satisfaction Skewed data, ordinal data (e.g., satisfaction scores) Symmetrical distribution, nominal data Mode Most common injury type, most common treatment exercise Categorical data, most frequent values Continuous data without repeated values

43. Important Measure of Variation Shows Variation About the Mean: For the Population: For the Sample: Variance For the Population: use N in the denominator. For the Sample : use n - 1 in the denominator.

44. Important Measure of Variation Shows Variation About the Mean: For the Population: For the Sample: Standard Deviation

45. Coefficient of Variation Measure of Relative Variation Always a % Shows Variation Relative to Mean Used to Compare 2 or More Groups Formula (for Sample):

46. Stock A: Average Price last year = $50 Standard Deviation = $5 Stock B: Average Price last year = $100 Standard Deviation = $5 Coefficient of Variation: Stock A: CV = 10% Stock B: CV = 5% Comparing Coefficient of Variation

47. Shape of Curve Describes How Data Are Distributed Measures of Shape: Symmetric or skewed Right-Skewed Left-Skewed Symmetric Mean = Median = Mode Mean Median Mode Median Mean Mode

48. 5 test scores for Calculus I are 95, 83, 92, 81, 75. Consider this dataset showing the retirement age of 11 people, in whole years: 54, 54, 54, 55, 56, 57, 57, 58, 58, 60, 60 3. Here are a bunch of 10 point quizzes from MAT117: 9, 6, 7, 10, 9, 4, 9, 2, 9, 10, 7, 7, 5, 6, 7 4. 11, 140, 98, 23, 45, 14, 56, 78, 93, 200, 123, 165 Find the Variance, SD & CV

49. Class Interval Frequency 2 -< 4 3 4 -< 6 18 6 -< 8 9 8 -< 10 7 Find the Variance, SD & CV Example A: 3, 10, 8, 8, 7, 8, 10, 3, 3, 3 Example B: 2, 5, 1, 5, 1, 2 Example C: 5, 7, 9, 1, 7, 5, 0, 4

50. Exam marks for 60 students (marked out of 65) mean = 30.3 sd = 14.46 Find the Mean, Median, Mode Variance, SD & CV

51. Group Frequency Table Frequency Percent 0 but less than 10 4 6.7 10 but less than 20 9 15.0 20 but less than 30 17 28.3 30 but less than 40 15 25.0 40 but less than 50 9 15.0 50 but less than 60 5 8.3 60 or over 1 1.7 Total 60 100.0