1. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 1 Averages and Variation

2. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 2 Today Check in –Quiz Wednesday –Will NOT NOT NOT include today’s material focus on other lectures –Proposal draft assigned It’s a draft of the full proposal Due Wednesday november 23rd

3. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 3 Measures of Central Tendency Mode Median Mean

4. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 4 The Mode the value or property that occurs most frequently in the data

5. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 5 Find the mode: 6, 7, 2, 3, 4, 6, 2, 6 The mode is 6.

6. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 6 Find the mode: 6, 7, 2, 3, 4, 5, 9, 8 There is no mode for this data.

7. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 7 The Median the central value of an ordered distribution

8. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 8 To find the median of raw data: Order the data from smallest to largest. For an odd number of data values, the median is the middle value. For an even number of data values, the median is found by dividing the sum of the two middle values by two.

9. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 9 Find the median: Data:5, 2, 7, 1, 4, 3, 2 Rearrange:1, 2, 2, 3, 4, 5, 7 The median is 3.

10. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 10 Find the median: Data:31, 57, 12, 22, 43, 50 Rearrange:12, 22, 31, 43, 50, 57 The median is the average of the middle two values =

11. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 11 The Mean The mean of a collection of data is found by: summing all the entries dividing by the number of entries

12. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 12 Find the mean: 6, 7, 2, 3, 4, 5, 2, 8

13. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 13 Sigma Notation The symbol  means “sum the following.”  is the Greek letter (capital) sigma.

14. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 14 Notations for mean Sample mean “x bar” Population mean Greek letter (mu)

15. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 15 Number of entries in a set of data If the data represents a sample, the number of entries = n. If the data represents an entire population, the number of entries = N.

16. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 16 Sample mean

17. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 17 Population mean

18. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 18 Resistant Measure a measure that is not influenced by extremely high or low data values

19. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 19 Which is less resistant? Mean Median The mean is less resistant. It can be made arbitrarily large by increasing the size of one value.

20. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 20 Weighted Average Average calculated where some of the numbers are assigned more importance or weight

21. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 21 Weighted Average

22. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 22 Compute the Weighted Average: Midterm grade = 92 Term Paper grade = 80 Final exam grade = 88 Midterm weight = 25% Term paper weight = 25% Final exam weight = 50%

23. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 23 Compute the Weighted Average: xwxw Midterm 92.2523 Term Paper 80.2520 Final exam 88.5044 1.0087

24. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 24 Percentiles For any whole number P (between 1 and 99), the Pth percentile of a distribution is a value such that P% of the data fall at or below it. The percent falling above the Pth percentile will be (100 – P)%.

25. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 25 Percentiles 40% of data Lowest value Highest value P 40 60% of data

26. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 26 Quartiles Percentiles that divide the data into fourths Q 1 = 25th percentile Q 2 = the median Q 3 = 75th percentile

27. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 27 Computing Quartiles Order the data from smallest to largest. Find the median, the second quartile. Find the median of the data falling below Q 2. This is the first quartile. Find the median of the data falling above Q 2. This is the third quartile.

28. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 28 Find the quartiles: 12 15 16 16 17 18 22 22 23 24 25 30 32 33 33 34 41 45 51 The data has been ordered. The median is 24.

29. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 29 Find the quartiles: 12 15 16 16 17 18 22 22 23 24 25 30 32 33 33 34 41 45 51 The data has been ordered. The median is 24.

30. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 30 Find the quartiles: 12 15 16 16 17 18 22 22 23 24 25 30 32 33 33 34 41 45 51 For the data below the median, the median is 17. 17 is the first quartile.

31. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 31 Find the quartiles: 12 15 16 16 17 18 22 22 23 24 25 30 32 33 33 34 41 45 51 For the data above the median, the median is 33. 33 is the third quartile.

32. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 32 Find the interquartile range: 12 15 16 16 17 18 22 22 23 24 25 30 32 33 33 34 41 45 51 IQR = Q 3 – Q 1 = 33 – 17 = 16

33. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 33 Measures of Variation Range Standard Deviation Variance—but we won’t talk about this

34. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 34 The Range the difference between the largest and smallest values of a distribution

35. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 35 Find the range: 10, 13, 17, 17, 18 The range = largest minus smallest = 18 minus 10 = 8

36. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 36 The standard deviation a measure of the average variation of the data entries from the mean

37. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 37 Standard Deviation Tells us how much data entries differ from the mean Why do we care? Can’t we just calculate the mean and the range?

38. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 38 Standard Deviation—why? Suppose 2 data sets: 1, 4, 4, 5, 6, 7, 8, 9, 10; range = 10-1=9 Mean = 54/9 = 6 Or 1, 2, 5, 6, 7, 7, 7, 9, 10; range = 10-1=9 Mean = 54/9 = 6 Data sets are different, but the mean and range are the same.

39. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 39 Standard Deviation Knowing HOW the data are arranged (distributed) tells us more than the mean and range. A lot of variability, or not very much variability? Especially important in large data sets where it may be impossible to ‘eyeball’ the variability.

40. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 40 Standard deviation of a sample n = sample size mean of the sample

41. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 41 To calculate standard deviation of a sample Calculate the mean of the sample. Find the difference between each entry (x) and the mean. These differences will add up to zero. Square the deviations from the mean. Sum the squares of the deviations from the mean. Divide the sum by (n  1) to get the variance. Take the square root of the variance to get the standard deviation.

42. Copyright (C) 2002 Houghton Mifflin Company. All rights reserved. 42 Find the standard deviation x 30 26 22 4 0  4 16 0 16 ___ 32 78 mean= 26 Sum = 0