1. MR. MARK ANTHONY GARCIA, M.S. MATHEMATICS DEPARTMENT DE LA SALLE UNIVERSITY

2. MEASURES OF CENTRAL TENDENCY The measure of central tendency measures the centrality or center of a data set. These measures are the mean, median and the mode.

3. Mean The mean of a numeric variable is calculated by adding the values of all observations in a data set and then dividing that sum by the number of observations in the set. This provides the average value of all the data.

4. Mean: Sample and Population Mean = sum of all observations divided by the number of observations

5. Example: Mean Mount Rival hosts a soccer tournament each year. This season, in all their 10 games, the lead scorer for the home team scored 7, 5, 0, 7, 8, 5, 5, 4, 1 and 5 goals. What was the mean score?

6. Example: Mean

7. A marathon race was completed by more than 100 participants. A sample of 5 participants were taken and their race times were recorded (in hours) as follows: 2.7, 8.3, 3.5, 5.1 and 4.9. What is the mean race time of the 5 participants?

8. Example: Mean

9. Interpreting the Mean Consider the mean first quiz scores for two different freshman sections in Science: Section XSection Y 72.7 69.8 Which of the two sections got a higher score?

10. Interpreting the Mean The following are the times (in minutes) that new employees need to learn a job from two companies and their means: Mean Company A: 25, 19, 30, 27, 22 24.6 Company B: 24, 23, 27, 29, 24 25.4 Which company has fast-learning new employees?

11. Remark: Mean The mean is the most commonly used measure of central tendency and is the best measure when comparing two or more sets of data. However, it is affected by extreme values.

12. Median The median of a set of observations arranged in an increasing or decreasing order of magnitude is the middle value when the number of observations is odd or the arithmetic mean of the two middle values when the number of observations is even.

13. Example: Median On 5 term tests in sociology, a student has made grades of 82, 93, 86, 92, and 79. Find the median for this test grades. No. of observations: n = 5 (Odd) Ascending Order: 79, 82, 86, 92, 93 Median: 86

14. Example: Median The nicotine contents for a random sample of 6 cigarettes of a certain brand are found to be 2.3, 2.7, 2.5, 2.9, 3.1, and 1.9 milligrams. Find the median.

15. Example: Median

16. Position of the Middle Values

17. Interpreting the Median Suppose that the median of the final exam score of a particular class in Mathematics is 62 points. What is the implication of this value? This means that 50% of the class scored 62 or more in the Mathematics final exam.

18. Remark: Median The median is not affected by extreme values. It is the best measure of center in terms of position in an arranged sequence. Often, it is used for curving or adjusting values to fit in a normal distribution.

19. Mode The mode of a set of observations is that value which occurs most often or with the greatest frequency.

20. Example: Mode The following are the IQ scores of 10 teenagers: 89, 82, 84, 82, 87, 95, 79, 84, 82, 87 The mode of the data set is 82 with the highest frequency 3.

21. Remark: Mode  For some sets of data, there may be several values occurring with the greatest frequency in which case we have more than one mode.  The mode does not always exist. This is certainly true if each distinct observation occur with the same frequency.

22. Example: Mode Consider the following sets of data: Data Set 1: 10, 20, 20, 30, 40, 40, 50, 60 Data Set 2: 7, 3, 6, 4, 6, 4, 3, 7, 4, 6, 3 For data set 1, there are two modes, 20 and 40. For data set 2, there are three modes, 3, 4 and 6.

23. Example: Mode Consider the following sets of data: Data Set 1: 90, 97, 98, 97, 90, 98 Data Set 2: 89, 88, 92, 95, 98, 97, 91, 94 The two sets of data have no modes.

24. Remark: Mode The mode is not the best measure of center since not all data sets can possess this value. However, the mode is the only measure that may also be used for qualitative data.

25. Comparing two sets of data Consider the following sets of data: 1 st set of data: 9, 10 and 11 Mean = 10 2 nd set of data: 1, 10 and 19 Mean = 10 What is the difference between the two sets of data?

26. Measures of Variation Any measure describing how spread the given observations relative from the mean is a measure of variation. These measures are the range, variance and standard deviation.

27. Measures of Variation: Situation Consider the following measurements, in liters, for two samples of orange juice bottled by companies A and B. In which company would you buy based on the following values? Sample A 0.971.000.941.031.11 Sample B 1.061.010.880.911.14

28. Range  The range is the difference between the highest and lowest value of the data set.  However, it is not a good measure of variability.  In the previous table, the range of the orange juice bottle contents for companies A and B are 0.17 and 0.26 respectively.

29. Variance and Standard Deviation Population Variance or squared deviation

30. Variance and Standard Deviation Sample Variance

31. Variance and Standard Deviation

32. Example: Variance and Standard Deviation Sample A 0.971.000.941.031.11 Sample B 1.061.010.880.911.14

33. Example: Variance and Standard Deviation Based from the variance and standard deviation, company A consistently bottles orange juice according to its advertised volumes because it has a lower standard deviation and lower variance.

34. Interpreting Variance and Standard Deviation Given the variances and standard deviations of two or more sets of data, we say that a set of data is consistent or has less variability or is less dispersed whenever it has the lowest variance or standard deviation.

35. Measures of Position The measures that describe or locate the position of certain non-central pieces of data relative to the entire set of data is measure of position or measure of relative standing. Some of these measures are the percentiles and quartiles.

36. Percentiles

38. Example: Percentiles

39. Computing Percentiles

40. Example: Percentiles Arrangement of the set of data in increasing order: 54, 62, 65, 69, 70, 72, 76, 78, 83, 85, 98

41. Example: Percentiles

42. Interpreting Percentiles

43. Example: Percentiles

45. Percentiles A percentile is best described as a comparison score. It’s a common term in all kinds of testing of data, but many will be most familiar with percentiles as they relate to standardized testing in schools

46. Quartiles

48. Computing Quartiles

49. Example: Quartiles

52. Interquartile Range

53. Example: IQR Again, consider the following scores of 11 students in an exam 54, 62, 65, 69, 70, 72, 76, 78, 83, 85, 98 Compute the IQR.

54. Example: IQR

55. BOX PLOT A box plot or boxplot is a convenient way of graphically depicting groups of numerical data through their quartiles.

56. BOX PLOT  If the median line is at the center of the box, then the data set is symmetric.  If the median line is closer to the first quartile, then data set is skewed to the right.  If the median line is closer to the third quartile, then the data set is skewed to the left.

57. Measure of Skewness A set of observations is symmetrically distributed if its graphical representation (histogram, bar chart) is symmetric with respect to a vertical axis passing through the mean. For a symmetrically distributed population or sample, the mean, median and mode have the same value. Half of all measurements are greater than the mean, while half are less than the mean.

58. Histogram: Symmetric

59. Measure of Skewness A set of observations that is not symmetrically distributed is said to be skewed. It is positively skewed if a greater proportion of the observations are less than or equal to the mean; this indicates that the mean is larger than the median. The histogram of a positively skewed distribution will generally have a long right tail; thus, this distribution is also known as being skewed to the right.

60. Histogram: Skewed to the Right

61. Measure of Skewness On the other hand, a negatively skewed distribution has more observations that are greater than or equal to the mean. Such a distribution has a mean that is less than the median. The histogram of a negatively skewed distribution will generally have a long left tail; thus, the phrase skewed to the left is applied here.

62. Histogram: Skewed to the Left

63. Measure of Skewness

64. Interpreting Skewness  If SK = 0, the data has a symmetric distribution.  If SK > 0, the data set has a positively skewed distribution. This means that more than 50% in the data set that are less than the mean.  If SK < 0, the data set has a negatively skewed distribution and this means that there are more than 50% in the data set are greater than or equal to the mean.

65. Example: Skewness