1. Adapted from Pearson Education, Inc. Copyright © 2009 Pearson Education, Inc. Welcome to MM150! Kirsten Meymaris Thursday, Mar. 31st Plan for the hour Final Project Reminder – due Tuesday, April 5th What is “average”? (Measures of Central Tendency, 9.1) How far apart are the data? (Measures of Dispersion, 9.2) Completing this course

2. Adapted from Pearson Education, Inc. Copyright © 2009 Pearson Education, Inc. Final Project You will need to create 5 slides or pages: On slide or page 1: provide your name, the project title, and the course and section number. On slides or pages 2, 3 and 4: These three slides (or pages) are the main part of your project. Introduce your chosen profession and give a brief overview of the math concept (or concepts) you will apply to the profession. Then go into more detail, and describe how the concept can apply to your chosen profession. You do not have to “do the math.” You can simply describe how you would use it. Make sure you offer an example or examples of situations in which you would use the concept you have chosen. Provide detail – actively discuss your example. On slide or page 5, provide any resources you have used to give credit to others’ ideas and information. Every student must have a reference page. If the textbook in your only “source,” that’s fine!

3. Adapted from Pearson Education, Inc. Copyright © 2009 Pearson Education, Inc. Final Project Reminders Choose either Microsoft Word or Microsoft PowerPoint in which to create your final project. The final project must be submitted via the Unit 9 drop box - Due April 5 th !! Students MUST include a reference page! If your examples are based on your actual real-life day at the job (i.e., you did no “research”), then list the textbook as your reference. Every student can list the textbook as a reference. Check spelling and grammar and visit the Writing Center if needed. Submit your (ungraded) final project into the Unit 10 Math Fair. Spend some time there, reading your classmates’ projects. I always learn something new from my students’ projects.

4. Adapted from Pearson Education, Inc. Copyright © 2009 Pearson Education, Inc. 9.1 Measures of Central Tendency

5. Adapted from Pearson Education, Inc. Copyright © 2009 Pearson Education, Inc. Measures of Central Tendency MeasureWhatHow Mean Sum of values divided by number in set Median middle value of set of ranked data Rank data, Select middle value (* in some cases, you’ll calculate mean of two middle values) Mode value that occurs most frequently in set Rank data, Select value(s) that occurs the most (in some cases, there isn’t one!) Midrange value halfway between lowest (L) and highest (H) values in set

6. Adapted from Pearson Education, Inc. Copyright © 2009 Pearson Education, Inc. Questions to ask When are different measures of central tendency used? How different can these measures be? Can they ever be the same? Does every set always have each measure?

7. Adapted from Pearson Education, Inc. Copyright © 2009 Pearson Education, Inc. What is the average age of students here at Kaplan? 222526 282930 33 37 556919 20 50 383632 272444 414521 423979 344748 Last week, we surveyed (took a sample) of students in M150-20AU (30 students) to estimate the whole Kaplan student body.

8. Adapted from Pearson Education, Inc. Copyright © 2009 Pearson Education, Inc. Mean The mean, is the sum of the data divided by the number of pieces of data. The formula for calculating the mean is where  x represents the sum of all the data and n represents the number of pieces of data. Note: The arithmetic mean, or simply the mean is symbolized by “x bar” when it is a sample of a population or by the Greek letter mu, , when it is the entire population.

9. Adapted from Pearson Education, Inc. Copyright © 2009 Pearson Education, Inc. Example - Mean What is the mean age of students in MM150? 222526 282930 33 37 556919 20 50 383632 272444 414521 423979 344748

10. Adapted from Pearson Education, Inc. Copyright © 2009 Pearson Education, Inc. Solution

11. Adapted from Pearson Education, Inc. Copyright © 2009 Pearson Education, Inc. Example - Median The median is the value in the middle of a set of ranked data. 192941 203042 203244 213345 223347 243448 253650 263755 273869 283979

12. Adapted from Pearson Education, Inc. Copyright © 2009 Pearson Education, Inc. Example - Median The median is the value in the middle of a set of ranked data. 192941 203042 203244 213345 223347 243448 253650 263755 273869 283979 Middle value (median) In this case needs to be averaged

13. Adapted from Pearson Education, Inc. Copyright © 2009 Pearson Education, Inc. Example - Median The median is the value in the middle of a set of ranked data. 192941 203042 203244 213345 223347 243448 253650 263755 273869 283979

14. Adapted from Pearson Education, Inc. Copyright © 2009 Pearson Education, Inc. Example - Mode The mode is the piece of data that occurs most frequently. 192941 203042 203244 213345 223347 243448 253650 263755 273869 283979

15. Adapted from Pearson Education, Inc. Copyright © 2009 Pearson Education, Inc. Example - Mode The mode is the piece of data that occurs most frequently. Mode = 20 and 33 (bimodal!) 192941 203042 203244 213345 223347 243448 253650 263755 273869 283979 Note: A set does not necessarily have a mode. Also, the set could have more than one mode!

16. Adapted from Pearson Education, Inc. Copyright © 2009 Pearson Education, Inc. Example - Midrange The midrange is the value halfway between the lowest (L) and highest (H) values in a set of data. 192941 203042 203244 213345 223347 243448 253650 263755 273869 283979

17. Adapted from Pearson Education, Inc. Copyright © 2009 Pearson Education, Inc. Example - Midrange The midrange is the value halfway between the lowest (L) and highest (H) values in a set of data. 192941 203042 203244 213345 223347 243448 253650 263755 273869 283979

18. Adapted from Pearson Education, Inc. Copyright © 2009 Pearson Education, Inc. Example – Review of Measures What is the average age of students in MM150? Mean = 36.43 Median = 33.5 Mode = 20 and 33 (bimodal!) Midrange = 49 192941 203042 203244 213345 223347 243448 253650 263755 273869 283979

19. Adapted from Pearson Education, Inc. Copyright © 2009 Pearson Education, Inc. Measures of Position Measures of position are often used to make comparisons. Two measures of position are percentiles and quartiles. The nth percentile means that you outscored

20. Adapted from Pearson Education, Inc. Copyright © 2009 Pearson Education, Inc. Percentiles The nth percentile means that you are older than n% of the students and younger than (100-nth)% of the students. For example, age = 27 You are in the 45 th percentile. “45% of the people are younger than you and 65% of the students are older than you!”

21. Adapted from Pearson Education, Inc. Copyright © 2009 Pearson Education, Inc. To Find the Quartiles of a Set of Data 1.Order the data from smallest to largest. 2.Find median of the set, Q 2 3.Find the median of the first (lower) half, Q 1 4.Find the median of the last (upper) half, Q 3

22. Adapted from Pearson Education, Inc. Copyright © 2009 Pearson Education, Inc. Example: Quartiles The ages of Kaplan students in one section of MM150. Determine Q 1, Q 2, and Q 3. 192941 203042 203244 213345 223347 243448 253650 263755 273869 283979

23. Adapted from Pearson Education, Inc. Copyright © 2009 Pearson Education, Inc. Example: Quartiles Q 1 = 26 Q 2 = 33.5 Q 3 = 44 192941 203042 203244 213345 223347 243448 253650 263755 273869 283979 33.5

24. Adapted from Pearson Education, Inc. Copyright © 2009 Pearson Education, Inc. Using Excel to Calculate Stats

25. Adapted from Pearson Education, Inc. Copyright © 2009 Pearson Education, Inc. 9.2 Measures of Dispersion

26. Adapted from Pearson Education, Inc. Copyright © 2009 Pearson Education, Inc. Measures of Dispersion Measures of dispersion are used to indicate the spread of the data.

27. Adapted from Pearson Education, Inc. Copyright © 2009 Pearson Education, Inc. Measures of Dispersion MeasureWhatHow Range Total spread of data in set Rank the data, Highest(H) – Lowest(L) Standard Deviation How different from the mean

28. Adapted from Pearson Education, Inc. Copyright © 2009 Pearson Education, Inc. Example: Range What is the range of ages of students at Kaplan? 192941 203042 203244 213345 223347 243448 253650 263755 273869 283979

29. Adapted from Pearson Education, Inc. Copyright © 2009 Pearson Education, Inc. Example: Range What is the range of ages of students at Kaplan? Range = 79 - 19 = 60 192941 203042 203244 213345 223347 243448 253650 263755 273869 283979

30. Adapted from Pearson Education, Inc. Copyright © 2009 Pearson Education, Inc. Standard Deviation The standard deviation measures how much the data differ from the mean. It is symbolized with s when it is calculated for a sample, and with  (Greek letter sigma) when it is calculated for a population.

31. Adapted from Pearson Education, Inc. Copyright © 2009 Pearson Education, Inc. What does Standard Deviation mean? Consider two data sets with mean = 9.5 {5, 8, 9, 10, 12, 13} {8, 9, 9,10, 10, 11}

32. Adapted from Pearson Education, Inc. Copyright © 2009 Pearson Education, Inc. What does Standard Deviation mean? The standard deviation measures how much the data differ from the mean. Sometimes a small standard deviation is good For example – A production line wanting all products of roughly the same size. Sometimes a large standard deviation is good For example – Ages of students at Kaplan – a wide spread of ages can be a good thing!

33. Adapted from Pearson Education, Inc. Copyright © 2009 Pearson Education, Inc. To Find the Standard Deviation of a Set 1.Find the mean 2.Setup table 3.Sum last column (Data-Mean) 2 4.Divide by (n-1) ( n=number of data in set) 5.Take the square root – Voila! DataData-Mean(Data-Mean) 2

34. Adapted from Pearson Education, Inc. Copyright © 2009 Pearson Education, Inc. Example Find the standard deviation of the following prices of selected washing machines: $280, $217, $665, $684, $939, $299 Find the mean.

35. Adapted from Pearson Education, Inc. Copyright © 2009 Pearson Education, Inc. Example Find the standard deviation of the following prices of selected washing machines: $280, $217, $665, $684, $939, $299 Find the mean.

36. Adapted from Pearson Education, Inc. Copyright © 2009 Pearson Education, Inc. Example continued, mean = 514 939 684 665 299 280 217 (Data  Mean) 2 Data  Mean Data

37. Adapted from Pearson Education, Inc. Copyright © 2009 Pearson Education, Inc. Example continued, mean = 514 421,5160 180,625425939 28,900170684 22,801151665 46,225  215 299 54,756  234 280 (  297) 2 = 88,209  297 217 (Data  Mean) 2 Data  Mean Data

38. Adapted from Pearson Education, Inc. Copyright © 2009 Pearson Education, Inc. Example continued, mean = 514 4.Divide by (n-1), n=6 5.Take square root

39. Adapted from Pearson Education, Inc. Copyright © 2009 Pearson Education, Inc. Example continued, mean = 514 The standard deviation is $290.35.

40. Adapted from Pearson Education, Inc. Copyright © 2009 Pearson Education, Inc. Example – Standard Deviation Find the standard deviation of ages of Kaplan students in MM150. Mean = 36.43 192941 203042 203244 213345 223347 243448 253650 263755 273869 283979

41. Adapted from Pearson Education, Inc. Copyright © 2009 Pearson Education, Inc. DataData-Mean(Data-Mean) 2 19 -17.43999.058886 20 -16.43269.9449 20 -16.43269.9449 21 -15.43238.0849 22 -14.43208.2249 24 -12.43154.5049 25 -11.43130.6449 26 -10.43108.7849 27 -9.4388.9249 28 -8.4371.0649 29 -7.4355.2049 30 -6.4341.3449 32 -4.4319.6249 33 -3.4311.7649 33 -3.4311.7649 34 -2.435.9049 36 -0.430.1849 37 0.570.3249 38 1.572.4649 39 2.576.6049 41 4.5720.8849 42 5.5731.0249 44 7.5757.3049 45 8.5773.4449 47 10.57111.7249 48 11.57133.8649 50 13.57184.1449 55 18.57344.8449 69 32.571060.8049 79 42.571812.2049

42. Adapted from Pearson Education, Inc. Copyright © 2009 Pearson Education, Inc. Example – Standard Deviation Find the standard deviation of ages of Kaplan students in MM150. 1. Mean = 36.43 2. Make table 3. Sum (Data-Mean) 2 = 6524.62 4. Divide by (n-1) = 29 students = 6524.62/29 5. = 192941 203042 203244 213345 223347 243448 253650 263755 273869 283979

43. Adapted from Pearson Education, Inc. Copyright © 2009 Pearson Education, Inc. Example – Standard Deviation Find the standard deviation of ages of Kaplan students in MM150. 1. Mean = 36.43 2. Make table 3. Sum (Data-Mean) 2 = 6524.62 4. Divide by (n-1) = 29 students = 6524.62/29 = 224.99 5. = 14.99 ~ 15 192941 203042 203244 213345 223347 243448 253650 263755 273869 283979

44. Adapted from Pearson Education, Inc. Copyright © 2009 Pearson Education, Inc. Thank You! Ask, Ask, Ask any questions you have. I hope to “see” you again in another math class! kmeymaris@kaplan.edu