Central tendency and dispersion

This lecture is based on post-graduate medical students of all subject those who are students MS/MD/FCPS of different subject on Central Tendency and Dispersion.

1. Central Tendency and Dispersion Prof Md Anisur Rahman MBBS, DO, FCPS (eye) Head of the department (Ophthalmology) DMC 5/7/2021 1 [email protected]

2. . Characteristic of central tendency Central tendency of a data set is the tendency of data to cluster around a central point of the series. Measures of central tendency is a single typical value of a data set that represent a set of data and around which other values of data set are found to cluster. To get a single value, that represents the entire data & describes the characteristic of whole set of data. 5/7/2021 2 [email protected]

3. What are the central tendencies? Mean Median To some extent 3. Mode 5/7/2021 3 [email protected]

4. Measures of central tendency: (Mean) What is mean? Advantages of mean Disadvantages of mean Formula to calculate the mean Solve the problem 5/7/2021 [email protected] 4

5. What is Mean? Mean is the sum of scores divided by the total number of Observations. It is commonly used in statistics. Sometimes mean is denoted by µ ( mui ) and sometimes by   (X bar) BUT WHY? 5/7/2021 5 [email protected]

6. Measures of central tendency (Mean) If we get the mean from population then it is called µ ( mui ) But when we get the mean from sample it is called (X bar) 5/7/2021 6 [email protected]

7. Measures of central tendency (Mean) Mean of a sample Mean of a population Solve the problem: What is the mean of 3, 4, 4 5, 7, 7, 8, 8, 8, 9. 9, 11 Add 4+4+5+7+7+8+8+8+9+9+11+12= 90. Here, n= 12 So mean = 90/12 = 7.5 5/7/2021 7 [email protected]

8. Advantages of mean Uniqueness: only one mean for a set of data. Simplicity: easy to calculate and understand. Sensitivity: sensitive to and affected by all values, that means it uses all the information in the distribution. 5/7/2021 8 [email protected]

9. Advantages of mean Can be applied in all normally distributed numerical data. Used as the basis of further most common and most powerful statistical computation. Means of sub-groups may be combined to get the mean of entire group. (Unlike median & mode). 5/7/2021 [email protected] 9

10. Disadvantages of mean Grossly affected by extreme values (outliers) of data. Not useful in skewed distribution of data. 5/7/2021 10 [email protected]

11. Measures of central tendency: (Median) When we use median? How to calculate median? Solve the problem Advantages of median Disadvantage of median Why median is not enough? 5/7/2021 11 [email protected]

12. Measures of central tendency: (Median) Have you ever think that, why some scientific article use median, in spite of mean? Remember when the data set is not in normal shaped, it is skewed in spite of mean we use median. 5/7/2021 12 [email protected]

13. How to calculate median? Middle most value of data set arranged in ascending or descending order. If odd number of values, middle most value is the median If even number of values, mean of the middle two values is the median. 5/7/2021 13 [email protected]

14. Measures of central tendency: (Median) What is the median of 1, 7, 7, 14, 11, 6, 5, 20, 17, 19, 19 First of all arrange it ascending or descending order. Say, we arrange it in ascending order; 1, 5, 6, 7, 7, 11, 14, 17, 19, 19, 20 Here n =11, so the median will be the 6th number which is 11. So median is 11 5/7/2021 14 [email protected]

15. Measures of central tendency: (Median) What is the median of 1, 7, 7, 14, 11, 6, 5, 20, 17, 19. Say, we arrange it in ascending order; 1, 5, 6, 7, 7, 11 , 14, 17, 19, 20 Here, n=10 which is an even number so the median 5/7/2021 15 [email protected] (5 th number + 6 th number)/2 = (7 + 11)/2 = 9

16. Advantages of median Not affected by extreme value. Good for ordinal data. Good for numerical skewed data. Uniqueness. Simplicity 5/7/2021 16 [email protected]

17. Disadvantage of median Ignore most of the information. Does not take into account all values. It requires ranking of all the scores and counting to find out the middle. Its use in further statistical computation is somewhat limited. 5/7/2021 17 [email protected]

18. Why median is not enough? The median is known as a measure of location; that is, it tells us where the data are. As stated in, we do not need to know all the exact values to calculate the median; if we made the smallest value even smaller or the largest value even larger, it would not change the value of the median. 5/7/2021 18 [email protected]

19. Why median is not enough? Thus the median does not use all the information in the data and so it can be shown to be less efficient than the mean or average, which does use all values of the data. 5/7/2021 19 [email protected]

20. Measures of central tendency (Mode) What is mode? Advantages of mode Disadvantages of mode Solve the problems 5/7/2021 [email protected] 20

21. What is mode? It is the most frequent and repeated values observed in a data set. It is the most common score in a frequency distribution e.g. in a data set: 1, 2, 2, 2, 3, 4, 5, 6; the mode is 2 5/7/2021 [email protected] 21

22. Advantages of mode Not affected by extreme values and the skewness of data. Simplicity. Good for bimodal distribution. 5/7/2021 [email protected] 22

23. Disadvantages of mode Often not clear defined. Not much used in statistics. Ignore most of the information 5/7/2021 [email protected] 23

24. Measures of Dispersion In statistics, dispersion denotes how stretched or squeezed a distribution is. Dispersion is contrasted with location or central tendency, and together they are the most used properties of distributions. 5/7/2021 [email protected] 24

25. Following are the measures of dispersion of individual observation Range Interquartile range Mean deviation Variance Standard deviation Co-efficient of variation 5/7/2021 [email protected] 25

26. Range (Variability) The range is equal to the high score minus the low score in a distribution Say in your study you take the age of 10 people. They are as follows 25, 48, 22, 34, 33, 34, 38, 40, 60, 29, What is the range? You arrange them in ascending or descending order 5/7/2021 [email protected] 26

27. Range (Variability) 22,25,29,33,34,34,38,40,48,60. So the range is (minimum 22, and maximum 60) = 60 – 22 = 38 Here only 10 data has taken so you can do it manually. But when the data is large enough it is very much tedious to calculate manually. We have to use SPSS. Or EXCEL file. 5/7/2021 [email protected] 27

28. Interquartile range Range is a measure based on two extreme observations and it fails to take account of the scatter within the range. In Interquartile range some extreme observations on two sides are discarded. 1/4 = 25% of observations at the lower end and another ¼ = 25% of observations at the upper end and Interquartile range include the middle 50% of observations 5/7/2021 [email protected] 28 25% 25% 25% 25%

29. Interquartile range Q1 Q2 Q3 5/7/2021 [email protected] 29 50% Interquartile range represents the difference between the third quartile and first quartile. Symbolically, Interquartile range = Q 3 ―Q 1

30. Mean deviation or average deviation It is the average of the deviation from arithmetic mean, Formula of mean deviation (MD) = Exercise: The diastolic pressure of 8 individuals are 82, 70, 75, 93, 95, 80, 85 and 76. Now, find the mean deviation 5/7/2021 [email protected] 30

31. Diastolic BP X Arithmetic mean Deviation from the mean X― 82 82 70 82 -12 75 82 -7 93 82 +11 95 82 [phone] +3 76 82 -6 5/7/2021 [email protected] 31 Mean deviation = 54/8 = 6.75

32. Variance and standard deviation In case of mean deviation we have problem of ignoring signs. We can overcome the problem by- Squaring the deviation Averaging this sums of squared deviation that is by dividing the sums of squared deviation with number of observations (n) which is called variance Now if we take square root the variance it will become standard deviation. 5/7/2021 [email protected] 32

33. Variance and standard deviation X X- 7 5 [phone] [phone] +1 1 7 5 +2 4 6 5 +1 1 5 5 5/7/2021 [email protected] 33 32/9-1=4 So Variance 4 SD = Square root of 4 = 2

34. Variance and standard deviation Variance is used most commonly with more advanced statistical procedures such as regression analysis , analysis of variance (ANOVA), and the determination of the reliability of a test The variance is also known as the mean square (MS) 5/7/2021 [email protected] 34

35. To calculate the standard deviation follows the following stages First of all to calculate the arithmetic mean of all deviations Now to take the deviation of each value from the arithmetic mean Then square each deviation To add up the squared deviation 5/7/2021 [email protected] 35

36. To divide the result by the number of observations n or n-1 (for population n, for sample size less than 30, n-1) Then to take the square root, which gives the standard deviation 5/7/2021 [email protected] 36

37. Example OF SD Consider two students, each of whom has taken five exams. Student A has scores 84, 86, 83, 85, and 87. Student B has scores 90, 75, 94, 68, and 98. Compute the SD for both Student A and Student B 5/7/2021 [email protected] 37

38. Here is the calculation for student A. MARKS OF “A” MEAN DIFFERENCE SQUARE 84 85 -1 1 86 85 [phone] 85 +2 4 10 5/7/2021 [email protected] 38 10/5-1 1.58

39. Here is the calculation for student B. MARKS OF “A” MEAN DIFFERENCE SQUARE 90 85 [phone] 85 [phone] 85 [phone]/7/2021 [email protected] 39 664/5-1 12.88

40. Since the standard deviation of Student B’s scores is greater than that of Student A’s (12.88 > 1.58), Student B’s scores are not as consistent as those of Student A. Standard deviation gives us an idea of the “spread” of the dispersion; that the larger the Standard deviation, the greater the dispersion of values about the mean. 5/7/2021 [email protected] 40

41. Exercise: 1 Average weight of baby at birth is 3.05 kg with the SD of 0.39 kg. If the birth is normally distributed would you regard as weight of 4 kg is abnormal? And weight of 2.5 kg is normal ? 5/7/2021 [email protected] 41

42. Solution: Normal limits of weight at ± 1.96 SD (3.05 ± 1.96 x 0.39) will be 2.29 kg and 3.81 kg. The weight of 4 kg falls outside the normal limits (since 4> 3.81) so it is taken as abnormal. The weight of 2.5 kg lies within the normal limits of 2.29 and 3.81 so it is not taken as abnormal . 5/7/2021 [email protected] 42

43. coefficient of variation (CV) The coefficient of variation is a measure of spread that describes the amount of variability relative to the mean. Because the coefficient of variation is unit less, we can use it instead of the standard deviation to compare the spread of data sets that have different units or different means. 5/7/2021 [email protected] 43

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45. EXAMPLE: Co-efficient of variation In a series of 40 adults, mean systolic blood pressure was 120 and SD was 10. In another series of 30 adults mean height and SD were 160 cm and 5 respectively. Now find which character show greater variation . 5/7/2021 [email protected] 45

46. 5/7/2021 [email protected] 46 We Know CV of BP = (10/120) X 100 = 8.33% CV of Height = (5/160) X100 = 3.13% Thus BP is found to be a more variable character than height (8.33/3.13) = 2.66 times

47. Distribution of Data

48. The list below shows the symbols used in certain statistical measures   = the sample mean- note the bar over the X. We can say 'the mean of X' or just 'X bar' when reading this. μ = the population mean (pronounced mew) S 2  = the sample variance (say S squared) ἀ 2 = the population variance (pronounced sigma) S = the sample standard deviation σ  = the population standard deviation (sigma )  

49. Population statistics are referred to using Greek symbols and sample statistics use letter from the Roman alphabet. 5/7/2021 [email protected] 49

50. Distribution of Data There are several types of data distribution in statistics. Normal distribution Binomial distribution Poisson distribution And many other types. Among them all, Normal distribution of data is widely used

51. Normal distribution A normal distribution has a bell-shaped density curve described by its mean µ (mu) and standard deviation σ The density curve is symmetrical, centered about its mean. The mean, mode and median are equal or near to equal, with its spread determined by its standard deviation

52. Binomial distribution model It is an important probability model that is used when there are two possible outcomes (hence "binomial").Each replication of the process results in one of two possible outcome (success or failure), The probability of success is the same for each replication, and the replications are independent, meaning here that a success in one patient does not influence the probability of success in another .

53. The Normal Distribution. Why it is important? It is very important to test data whether data is normally distributed or not, because statistical test depends upon the data distribution. If data is normally distributed then parametric test will be done. If data is not normally distributing then non-parametric test Parametric tests are more powerful than non-parametric test

54. Parametric & Non-parametric Test t test, ANOVA test Wilcoxon test, sign test, Mann-Whitney test, 5/8/2021 [email protected] 54

55. Properties of a normal distribution The mean, median and mode are all equal. The curve is symmetric at the center (i.e. around the mean, μ ). Exactly half of the values are to the left of center and exactly half the values are to the right. The total area under the curve is 1 .

56. Describing the normal distribution : A normal distribution is more commonly known as a bell curve.  This type of curve shows up throughout statistics and the real world.  For example, after we give a test in any of our classes, one thing that we like to do is to make a graph of all the scores. We typically write down 10 point ranges such as 60-69, 70-79, and 80-89, then put a tally mark for each test score in that range. Almost every time we do this, a familiar shape emerges .

57. A few students do very well and a few do very poorly. A bunch of scores end up clumped around the mean score. Different tests may result in different means and standard deviations, but the shape of the graph is nearly always the same. This shape is commonly called the bell curve .

58. Important features of bell curve: There are several features of bell curve that is important and distinguishes them from other curves in statistics: A bell curve has one mode, which coincides with the mean and median. This is the center of the curve where it is at its highest. A bell curve is symmetric. If it were folded along a vertical line at the mean, both halves would match perfectly because they are mirror images of each other .

59. A bell curve follows the 68-95-99.7 rule, A bell curve follows the 68-95-99.7 rule, which provides a convenient way to carry out estimated calculations: Approximately 68% of all of the data lies within one standard deviation of the mean. Approximately 95% of all the data is within two standard deviations of the mean. Approximately 99.7% of the data is within three standard deviations of the mean.

60. An example: S uppose we have 100 students who took a statistics test with mean score of 70 and standard deviation of 10. The standard deviation is 10. Subtract and add 10 to the mean. This gives us 60 and 80. By the 68-95-99.7 rule we would expect about 68% of 100, or 68 students to score between 60 and 80 on the test .

61. Two times the standard deviation is 20. If we subtract and add 20 to the mean we have 50 and 90. We would expect about 95% of 100, or 95 students to score between 50 and 90 on the test. A similar calculation tells us that effectively everyone scored between 40 and 100 on the test .

62. Average weight of baby at birth is 3.05 kg with the SD of 0.39 kg. If the birth is normally distributed would you regard as weight of 4 kg is abnormal? And weight of 2.5 kg is normal? Solution: Normal limits of weight at ± 1.96 SD (3.05 ± 1.96 x 0.39) will be 2.29 kg and 3.81 kg. The weight of 4 kg falls outside the normal limits (since 4> 3.81) so it is taken as abnormal. The weight of 2.5 kg lies within the normal limits of 2.29 and 3.81 so it is not taken as abnormal.

63. Asymmetrical Distribution of data: Skewness/Kurtosis Skewness is the degree of departure from symmetry of a distribution. A skewed data distribution or bell curve can be either positive or negative. A positive skew means that the extreme data results are larger. This skews the data in that it brings the mean (average) up. The mean will be larger than the median in a skewed data set. A negative skew means the opposite: that the extreme data results are smaller. This means that the mean is brought down, and the median is larger than the mean .