1. Copyright © 2012 Pearson Education, Inc. All rights reserved Chapter 9 Statistics

2. Copyright © 2012 Pearson Education, Inc. All rights reserved 9.1 Frequency Distributions; Measures of Central Tendency

3. 9 - 3 Example  A survey asked a random sample of 30 business executives for their recommendations as to the number of college credits in management that a business major should have. The result are shown as below. (1)Group the data into intervals and find the frequency of each interval. (2) Draw a histogram and a frequency polygon this distribution.  3, 25, 22, 16, 0, 9, 14, 8, 34, 21,  15,12, 9, 3, 8,15,20,12,28,19  17,16,23,19,12,14,29,13,24,18 © 2012 Pearson Education, Inc.. All rights reserved.

4. 9 - 4 © 2012 Pearson Education, Inc.. All rights reserved.

5. 9 - 5 Figure 1 © 2012 Pearson Education, Inc.. All rights reserved.

6. 9 - 6 © 2012 Pearson Education, Inc.. All rights reserved.

7. 9 - 7 Your Turn 1 Find the mean of the following data: 12, 17, 21, 25, 27, 38, 49. © 2012 Pearson Education, Inc.. All rights reserved.

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9. 9 - 9 Your Turn 2 Find the mean of the following grouped frequency. IntervalMidpoint, x Frequency f Product x f 0-6326 7-1310440 14-20177119 21-272410240 28-3431393 35-41381 Total = 27Total = 536 © 2012 Pearson Education, Inc.. All rights reserved.

10. 9 - 10 © 2012 Pearson Education, Inc.. All rights reserved.

11. 9 - 11 Your Turn: Find the median of the data 12, 17, 21, 25, 27, 38, 49. © 2012 Pearson Education, Inc.. All rights reserved.

12. Copyright © 2012 Pearson Education, Inc. All rights reserved 9.2 Measures of Variation

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14. 9 - 14 Variation  Variation  is the heart of statistics  no variation, no need to do statistical analysis  the mean would describe the distribution

15. 9 - 15 Quest for Variation  Measures of dispersion  consider the spread between scores  Calculations include  range  variance  standard deviation

16. 9 - 16 The Range  Range  distance between the highest and lowest score in a distribution.  Calculation  Range = H minus L  H = the highest score in the data set  L = the lowest score in the data set

17. 9 - 17 Example: Find the range of each data set  Data Set A: 5,3,8,2,7,4,14,3,12,10  Data Set B: 1,1,1,1,1,1,1,1,1,100 © 2012 Pearson Education, Inc.. All rights reserved.

18. 9 - 18 Variance and Standard Deviation  Most commonly used measures of dispersion  based upon the distance of scores in a distribution from the mean  the mean is used as the central point The Variance is defined as: The average of the squared differences from the Mean.

19. 9 - 19 Variance  The variance is  a measure of the spread of scores in a distribution around its mean  The larger the variance  the greater the spread of scores around the mean  The smaller the variance  the more closely the scores are distributed around the mean

20. 9 - 20  Standard deviation  is a measure of dispersion of the scores around the mean  The higher the standard deviation  the greater the spread in the scores  The lower the standard deviation  the closer the scores are on average from the mean of the distribution

21. 9 - 21 Example: Variance and Standard Deviation of a population.  http://www.mathsisfun.com/data/standard- deviation.html © 2012 Pearson Education, Inc.. All rights reserved.

22. 9 - 22 Example 1 © 2012 Pearson Education, Inc.. All rights reserved. Find the range, variance, and standard deviation for the list of numbers: 7, 11, 16, 17, 19, 35.

23. 9 - 23 © 2012 Pearson Education, Inc.. All rights reserved. Example: Variance and Standard Deviation of a sample

24. 9 - 24 © 2012 Pearson Education, Inc.. All rights reserved.

25. 9 - 25 Example 2 Find the standard deviation for the grouped frequency distribution. Intervalxx2x2 ffx 2 0-6 7-13 14-20 21-27 28-34 35-41 TotalTotal = 12,528 © 2012 Pearson Education, Inc.. All rights reserved.

26. 9 - 26 Group Class Work © 2012 Pearson Education, Inc.. All rights reserved. Suppose we were interested in how many siblings are in statistics students' families. (1)Find the mean of the data. (2)Find the standard deviation.

27. 9 - 27 Summary  Explaining variation is the basis for statistical analysis  We begin with basic measures of dispersion  range  variance  standard deviation

28. Copyright © 2012 Pearson Education, Inc. All rights reserved 9.3 The Normal Distribution

29. 9 - 29 Normal Distribution Curve A normal distribution curve is symmetrical, bell-shaped curve defined by the mean and standard deviation of a data set.

30. 9 - 30 Right Skewed, Left Skewed, and the Bell -Shaped © 2012 Pearson Education, Inc.. All rights reserved.

31. 9 - 31 One standard deviation away from the mean ( ) in either direction on the horizontal axis accounts for around 68 percent of the data. Two standard deviations away from the mean accounts for roughly 95 percent of the data with three standard deviations representing about 99.7 percent of the data.

32. 9 - 32 Standard Normal Distribution © 2012 Pearson Education, Inc.. All rights reserved.

33. 9 - 33 z-scores When a set of data values are normally distributed, we can standardize each score by converting it into a z-score. z -scores make it easier to compare data values measured on different scales.

34. 9 - 34 Standard Normal Distribution A standard normal distribution is the set of all z -scores. The mean of the data in a standard normal distribution is 0 and the standard deviation is 1.

35. 9 - 35 A z-score reflects how many standard deviations above or below the mean a raw score is. The z-score is positive if the data value lies ___________ the mean and negative if the data value lies ___________ the mean.

36. 9 - 36 z-score formula Where x represents an element of the data set, the mean is represented by and standard deviation by.

37. 9 - 37 Analyzing the data Suppose SAT scores among college students are normally distributed with a mean of 500 and a standard deviation of 100. If a student scores a 700, what would be her z-score?

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39. 9 - 39 Your Turn 2 Find a value of z satisfying the following conditions. (a) 2.5% of the area is to the left of z. (b) 20.9% of the area is to the right of z. © 2012 Pearson Education, Inc.. All rights reserved.

40. 9 - 40 Your Turn 3 Find the z-score for x = 20 if a normal distribution has a mean 35 and standard deviation 20. © 2012 Pearson Education, Inc.. All rights reserved.

41. 9 - 41 © 2012 Pearson Education, Inc.. All rights reserved.

42. 9 - 42 © 2012 Pearson Education, Inc.. All rights reserved.

43. 9 - 43 Your Turn 4 Dixie Office Supplies finds that its sales force drives an average of 1200 miles per month per person, with a standard deviation of 150 miles. Assume that the number of miles driven by a salesperson is closely approximated by a normal distribution. Find the probability that a sales person drives between 1275 and 1425 miles. © 2012 Pearson Education, Inc.. All rights reserved.