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# MEASURES OF CENTRAL TENDENCY AND VARIABILITY

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### MEASURES OF CENTRAL TENDENCY AND VARIABILITY

1. 1. Introduction and Focus Questions Have you ever wondered why a certain size of shoe or brand of shirt is made more available than other sizes? Have you ever asked yourself why a certain basketball player gets more playing time than the rest of his team mates? Have you ever thought of comparing your academic performance with your classmates? Have you ever wondered what score you need for each subject to qualify for honors? Have you at certain time asked yourself how norms and standards are made?
2. 2. Descriptive Statistics Ungrouped Data Measures of Central Tendency Measures of Variability Grouped Data Measures of Central Tendency Measures of Variability
3. 3. PRE-ASSESSMENT (Anticipation-Reaction Guide) Before Questions Which measure of central tendency is generally used in determining the size of the most saleable shoes in a department store? What is the most reliable measure of variability? Which measure of central tendency is greatly affected by extreme scores? Margie has grades 86, 68, and 79 in her first three tests in Algebra. What grade must she obtain on the 4th test to get an average of 78? What is the median age of a group of employees whose ages are 36, 38, 24, 21, and 27? If the range of a set of scores is 14 and the lowest score is 7, what is the highest score? What is the standard deviation of the scores 5, 4, 3, 6, and 2? After
4. 4. The Situation… You are one of the winners in a contest where the prizes are gift certificates from the famous stores in the city. The sponsors are the following: BENCH PENSHOPPE FOLDED & HUNG SM DEPARTMENT STORE You are to choose only one store. Which among the four stores will you choose?
5. 5. The Mode The mode is the value or element which occurs most frequently in a set of data. It is the value or element with the greatest frequency. Mode can be quantitative or qualitative. To find the mode for a set of data: 1. select the measure that appears most often in the set; 2. if two or more measures appear the same number of times, then each of these values is a mode; 3. if every measure appears the same number of times, then the set of data has no mode.
6. 6. Try answering these items… Find the mode in the given sets of scores. 1. {10, 12, 9, 10, 13, 11, 10} 2. {15, 20, 18, 19, 18, 16, 20, 18} 3. {5, 8, 7, 9, 6, 8, 5} 4. {7, 10, 8, 5, 9, 6, 4} 5. {12, 16, 14, 15, 16, 13, 14} 6. {smart, globe, sun, sun, sun, globe, smart} 7. {1, 1, 1, 2, 2, 2, 3, 3, 3}
7. 7. Pinklace sells ice cream. For five days, they sold the following: Number of cups of ice cream sold Monday 13 Tuesday 27 Wednesday 15 Thursday 15 Friday 30 Questions: 1. What is the total number of cups of ice cream sold during the whole week? 2. If Pinklace will be able to sell same number of cups of ice cream each day, how many will it be?
8. 8. The Mean The mean (also known as arithmetic mean) is the most commonly used measure of central position. It is used to describe a set of data where the measures cluster or concentrate at a point. As the measures cluster around each other, a single value appears to represent distinctively the typical value.
9. 9. How do we compute for the mean? It is the sum of measures x divided by the number N of measures. It is symbolized as x (read as x bar). To find the mean of an ungrouped data, use the formula where = summation of x (sum of measures) and N= number of values of x.
10. 10. Let’s practice… The grades in Geometry of 10 students are 87, 84, 85, 85, 86, 90, 79, 82, 78, 76. What is the average grade of 10 students? Solution:
11. 11. Let’s Practice. Find the mean of the following numbers: 1. 9, 15, 12, 10, 20 2. 100, 121, 132, 143 3. 54, 58, 61, 72, 81, 65
12. 12. WORK IN PAIRS The first three test scores of each of the four students are shown. Each student hopes to maintain an average of 85. Find the score needed by each student on the fourth test to have an average of 85, or explain why such average is not possible. a. Lisa: 78, 80, 100 82 b. Mary: 90, 92, 95 63 Lina: 79, 80, 81 d. Willie: 65, 80, 80 c. 100 115
13. 13. The situation… Sonya’s Kitchen received an invitation in a Food Exposition. All the seven service crew are eager to go but only one can represent the restaurant. To be fair, Sonya thought of sending the crew whose age is in the middle of the ages of the seven crews.
14. 14. She made a list of the service crews and their ages: Service Crew Age Michelle Sheryl Karen Mark Jason Oliver Eliza 47 21 20 19 18 18 18 Guide Questions: 1. What is the mean age of the service crew? 2. Is there someone in this group who has this age? 3. How many persons are older than the mean age? How many are younger? 4. Do you think this is the best measure of central tendency to use? Explain.
15. 15. Looking at the same list… Service Crew Michelle Sheryl Karen Mark Jason Oliver Eliza Age 47 21 20 19 18 18 18 Guide Questions: 1. Arrange the ages in numerical order. 2. What is the middle value? 3. Is there a crew with this representative age? 4. How many crew are younger than this age? Older than this age? 5. Compare the result with the previous activity. Which result do you think is a better basis of choosing the representative? 6. Who is now the representative of Sonya’s Kitchen in the Food Fair?
16. 16. The Median The median is the middlemost value or term in a set of data arranged according to size/magnitude (either increasing or decreasing). If the number of values is even, the median is the average of the two middlemost values.
17. 17. Let’s practice… Andrea’s scores in 9 quizzes during the first quarter are 8, 7, 6, 10, 9, 5, 9, 6, and 10. Find the median. Solution Arrange the scores in increasing order. 5, 6, 6, 7, 8, 9, 9, 10, 10 The median is 8.
18. 18. Find the median of the following sets of data: 32, 45, 22, 21, 18, 36, 50 2. 95, 95, 96, 88, 82, 100 3. 221, 332, 421, 326, 281, 220, 341, 109, 112 1.
19. 19. What measure of central tendency is used in the following situations?  Kevin noticed that half of the cereal brands in the     store cost more than Php 150.00. median The average score on the last Pre-Algebra test was 85. mean The most common height on the basketball team is 6 ft 11 in. mode One-half of the cars at a dealership cost less than Php 700, 000.00. median The average amount spent per customer in a department store is Php 2, 500.00. mean
20. 20. Calculate the mean, median, and mode of each set of numbers. 1. 4, 14, 29, 44, 46, 52, 55 2. 42, 49, 49, 49, 49 3. 22, 34, 34, 34, 45, 61 4. 20, 22, 56, 62, 63, 67 5. 11, 33, 54, 54, 71, 84, 93 1. Mean = 34. 86 Median = 44 Mode = none 2. Mean = 47.6 Median = 49 Mode = 49 3. Mean = 38.33 Median = 34 Mode = 34
21. 21. Solve the following problems. Andy has grades of 84, 65, and 76 on three math tests. What grade must he obtain on the next test to have an average of exactly 80 for the four tests? 2. A storeowner kept a tally of the sizes of suits purchased in her store. Which measure of central tendency should the storeowner use to describe the most saleable suit? 3. A tally was made of the number of times each color of crayon was used by a kindergarten class. Which measure of central tendency should the teacher use to determine which color is the favorite color of her class? 1.
22. 22. Continuation… 4. In January of 2006, your family moved to a tropical climate. For the year that followed, you recorded the number of rainy days that occurred each month. Your data contained 14, 14, 10, 12, 11, 13, 11, 11, 14, 10, 13, 12. a. Find the mean, mode, and the median for your data set of rainy days. b. If the number of rainy days doubles each month in the year 2007, what will be the mean, mode, median? c. If, instead, there are three more rainy days per month in the year 2007, what will be the mean, mode, median?
23. 23. Continuation… 5. The values of 11 houses on Washington Street are shown in the table. a. Find the mean value of these houses in dollars. b. Find the median value of these houses in dollars. c. State which measure of central tendency, the mean or the median, best represents the values of these 11 houses. Justify your answer.
24. 24. The situation… A testing laboratory wishes to test two experimental brands of outdoor paint to see how long each paint will last before fading. The testing lab makes use of six gallons of paint for each brand name to test.
25. 25. The results (in months) are as follows: Brand A: Brand B: 10 35 60 45 50 30 30 35 40 40 20 25 Guide Questions: 1. What is the mean score of each brand? 2. Can the mean of each brand be a good basis for comparing them? 3. Which brand has results closer to the mean? 4. If you are to choose from these two brands, which would you prefer? Why?
26. 26. Measures of Dispersion or Variability -refer to the spread of the values about the mean. These are important quantities used by statisticians in evaluation. Smaller dispersion of scores arising from the comparison often indicates more consistency and more reliability. The most commonly used measures of dispersion are the range, the average deviation, the standard deviation, and variance.
27. 27. The Range The range is the simplest measure of variability. It is the difference between the largest value and the smallest value. Range= Largest Value – Smallest Value
28. 28. Going back to the activity… Brand A: 10 60 Largest Value = 60 50 30 40 20 Smallest Value = 10 RANGE = Largest Value – Smallest Value = 60 – 10 = 50 Brand B: 35 45 Largest Value = 45 30 35 40 25 Smallest Value = 25 RANGE = Largest Value – Smallest Value = 45 – 25 = 20
29. 29. Now consider another situation The following are the daily wages of 8 factory workers of two garments factories A and B. Find the range of salaries in peso (Php). Factory A: Factory B: 400, 450, 520, 380, 482, 495, 575, 450 450, 460, 462, 480, 450, 450, 400, 600 Questions: 1. What is the mean wage of each group of workers? 2. What is the range of wages of each group of workers? 3. For this case, are you convinced that the group with lower range has more consistent wages?
30. 30. Though the range is the simplest and easiest to find measure of variability, it is not a stable measure. Its value can fluctuate greatly even with a change in just a single value, either the highest or lowest.
31. 31. The Average/Mean Deviation The dispersion of a set of data about the average of these data is the average deviation or the mean deviation.
32. 32. How to find the average/mean deviation:
33. 33. Procedure in computing the average deviation (Refer to the activity about wages of factory workers) 1. Find the mean of the scores. Factory A: 400, 450, 520, 380, 482, 495, 575, 450 2. Find the absolute difference between each score and the mean.
34. 34. 3. Find the sum of the absolute differences and then divide by N.
35. 35. or you may present using a table… 400 469 -69 69 450 469 -19 19 520 469 51 51 380 469 -89 89 482 469 13 13 495 469 26 26 575 469 106 106 450 469 -19 19 Total 392
36. 36. (Refer to the activity about wages of factory workers) 1. Find the mean of the scores. Factory B: 450, 460, 462, 480, 450, 450, 400, 600 2. Find the absolute difference between each score and the mean.
37. 37. 3. Find the sum of the absolute differences and then divide by N. Lower Average Deviation means more consistent scores.
38. 38. 450 469 -19 19 460 469 -9 9 462 469 -7 7 480 469 11 11 450 469 -19 19 450 469 -19 19 400 469 -69 69 600 469 131 131 Total 284
39. 39. Find the Average Deviation of the following: 1. Science Achievement Scores: 60, 75, 80, 85, 90, 95 2. The weights in kilogram of 10 students are: 52, 55, 50, 55, 43, 45, 40, 48, 45, and 47
40. 40. The average deviation gives a better approximate than the range. However, it does not lend itself readily to mathematical treatment for deeper analysis. It’s the standard deviation. Then what measure of variability is the most reliable?
41. 41. The Standard Deviation  1. Find the mean.
42. 42. 2. Find the deviation from the mean. 39 5. Compute for the standard deviation. 289 10 22 -12 144 22 2 4 16 22 -6 36 19 4. Add all the squared deviations. 17 24 3. Square the deviations. 22 22 -3 9 26 22 4 16 29 22 7 49 30 22 8 64 5 22 -17 289 SUM 900
43. 43. What does a standard deviation of 10 imply? So does that mean that a lower standard deviation means less varied scores? It means that most of the scores are within 10 units from the mean. That’s correct! Lower standard deviation shows more consistent scores.
44. 44. Let’s practice… Compare the standard deviation of the scores of the three students in their Mathematics quizzes. Student Mathematics Quizzes A 97, 92, 96, 95, 90 B 94, 94, 92, 94, 96 C 95. 94, 93, 96, 92
45. 45. Anticipation-Reaction Guide Before Questions Which measure of central tendency is generally used in determining the size of the most saleable shoes in a department store? What is the most reliable measure of variability? Which measure of central tendency is greatly affected by extreme scores? Margie has grades 86, 68, and 79 in her first three tests in Algebra. What grade must she obtain on the 4th test to get an average of 78? What is the median age of a group of employees whose ages are 36, 38, 24, 21, and 27? If the range of a set of scores is 14 and the lowest score is 7, what is the highest score? What is the standard deviation of the scores 5, 4, 3, 6, and 2? After
46. 46. You were asked to find the mean, median and mode of the Math grades of all the students in 2 Learning Groups. The grades of 72 students are as follows: 85 78 82 88 89 92 90 82 85 85 83 79 80 86 75 92 91 88 87 87 78 79 80 81 79 88 91 92 81 90 85 84 83 82 82 83 90 91 92 95 75 78 89 80 81 82 82 76 77 90 87 88 83 83 95 92 79 79 79 95 90 79 90 91 79 95 73 85 97 78 91 76
47. 47. Organizing data STEM-LEAF DIAGRAM First digits (stem) Second digits (leaf) 7 8 9 Grades Frequency 70-79 18 80-89 33 90-99 21 GROUPED DATA
48. 48. RULES FOR GROUPING 1. 2. 3. 4. 5. The intervals must cover the complete range of values. The intervals need not begin nor end with the lowest or highest values. The intervals must be of equal size. For effective grouping, the number of intervals should be between 5 and 15. Every score must be tallied from highest to lowest or from lowest to highest. Thus, the intervals should not overlap. When an interval ends with a counting number, the next intervals begins with the next counting number.
49. 49. First digits (stem) 7 Second digits (leaf) (0-4) (5-9) 8 (0-4) (5-9) 9 (0-4) (5-9)
50. 50. The Grouped Data Grades Frequency 70 – 74 75 – 79 80 – 84 85 – 89 90 – 94 95 – 99 1 17 18 15 16 5
51. 51. Steps in Finding the Mean
52. 52. Find the mean and the mode. 1 2 3 4 5 6 TOTAL 2 3 5 1 2 4
53. 53. Find the mean , modal class, and the mode. 80 – 84 85 – 89 90 – 94 95 – 99 100 – 104 TOTAL 2 8 11 3 1
54. 54. Steps in Finding the Median 1. Obtain the cumulative frequencies.
55. 55. Classes Position of the Data 1–4 16 16 1 – 16 5–8 20 36 17 – 36 9 – 12 28 64 37 – 64 13 – 16 24 88 65 – 88 17 – 20 16 104 89 – 104 21 – 24 11 115 105 – 115 25 – 28 5 120 116 – 120 TOTAL 120
56. 56. Mode The mode is the midpoint of the modal class.
57. 57. RANGE 100 – 199 200 – 299 300 – 399 400 – 499 6 10 30 20
58. 58. AVERAGE DEVIATION Example: Calculate the mean deviation for the 30 marathon times in the grouped distribution as follows: Time (min) Frequency 128-130 131-133 134-136 3 1 4 137-139 140-142 3 7 143-145 12
59. 59. Finding the Average Deviation Time (min) Class Mark (CM) Frequency 128 – 130 129 3 387 131 – 133 130 1 132 134 – 136 135 4 540 137 – 139 138 3 414 140 – 142 141 7 987 143 – 145 144 12 1728 30 4188 min Sum
60. 60. Time (min) Class Mark (CM) Frequency fCM f 128-130 129 3 387 10.6 31.8 131-133 132 1 132 7.6 7.6 134-136 135 4 540 4.6 18.4 137-139 138 3 414 1.6 4.8 140-142 141 7 987 1.4 9.8 143-145 144 12 1728 4.4 52.8 30 4188 125.2 min
61. 61. Variance and Standard Deviation x 40-44 42 1 42 1764 45-49 47 7 329 15463 50-54 52 12 624 32448 55-59 57 24 1368 77976 60-64 62 29 1798 111476 65-69 67 14 938 62846 70-74 72 5 360 25920 75-79 77 3 231 17787 95 5690 345680 SUM
62. 62. THANK YOU!