2. Copyright © 2011 Pearson Education, Inc. Putting Statistics to Work

3. Copyright © 2011 Pearson Education, Inc. Slide 6-3 Unit 6B Measures of Variation

4. 6-B Copyright © 2011 Pearson Education, Inc. Slide 6-4 Why Variation Matters Consider the following waiting times for 11 customers at 2 banks. Big Bank (three lines): 4.1 5.2 5.6 6.2 6.7 7.2 7.7 7.7 8.5 9.3 11.0 Best Bank (one line): 6.6 6.7 6.7 6.9 7.1 7.2 7.3 7.4 7.7 7.8 7.8 Which bank is likely to have more unhappy customers? → Big Bank, due to more surprise long waits

5. 6-B Copyright © 2011 Pearson Education, Inc. Slide 6-5 Range The range of a data set is the difference between its highest and lowest data values. range = highest value (max) – lowest value (min)

6. 6-B Copyright © 2011 Pearson Education, Inc. Slide 6-6 Quartiles The median of the data set divides the data set into the lower half and the upper half. The lower quartile (or first quartile) divides the lowest fourth of a data set from the upper three-fourths. It is the median of the data values in the lower half of a data set. The middle quartile (or second quartile) is the overall median. The upper quartile (or third quartile) divides the lower three-fourths of a data set from the upper fourth. It is the median of the data values in the upper half of a data set.

7. 6-B Copyright © 2011 Pearson Education, Inc. Slide 6-7 The five-number summary for a data set consists of the following five numbers: low value lower quartile median upper quartile high value A boxplot shows the five-number summary visually, with a rectangular box enclosing the lower and upper quartiles, a line marking the median, and whiskers extending to the low and high values. The Five-Number Summary

8. 6-B Copyright © 2011 Pearson Education, Inc. Slide 6-8 Best BankBig Bank low value (min) = 4.1 lower quartile = 5.6 median = 7.2 upper quartile = 8.5 high value (max) = 11.0 low value (min) = 6.6 lower quartile = 6.7 median = 7.2 upper quartile = 7.7 high value (max) = 7.8 Five-number summary of the waiting times at each bank: The corresponding boxplot: The Five-Number Summary

9. 6-B Copyright © 2011 Pearson Education, Inc. Slide 6-9 Standard Deviation The standard deviation is the single number most commonly used to describe variation.

10. 6-B Copyright © 2011 Pearson Education, Inc. Slide 6-10 The standard deviation is calculated by completing the following steps: 1.Compute the mean of the data set. Then find the deviation from the mean for every data value. deviation from the mean = data value – mean 2.Find the squares of all the deviations from the mean. 3.Add all the squares of the deviations from the mean. 4.Divide this sum by the total number of data values minus 1. 5.The standard deviation is the square root of this quotient. Calculating the Standard Deviation

11. 6-B Copyright © 2011 Pearson Education, Inc. Slide 6-11 Standard Deviation Let A = {2, 8, 9, 12, 19} with a mean of 10. Find the sample standard deviation of the data set A.

12. 6-B Copyright © 2011 Pearson Education, Inc. Slide 6-12 Standard Deviation Let B = {1, 7, 8, 11, 18} with a mean of 9. Find the sample standard deviation of the data set B.

13. 6-B Copyright © 2011 Pearson Education, Inc. Slide 6-13 Standard Deviation Let B = {1, 7, 8, 11, 18} with a mean of 9. Find the sample standard deviation of the data set B.

14. 6-B Copyright © 2011 Pearson Education, Inc. Slide 6-14 Standard Deviation Let B = {1, 7, 8, 11, 18} with a mean of 9. Find the sample standard deviation of the data set B.

15. 6-B Copyright © 2011 Pearson Education, Inc. Slide 6-15 Standard Deviation Let B = {1, 7, 8, 11, 18} with a mean of 9. Find the sample standard deviation of the data set B.

16. 6-B Copyright © 2011 Pearson Education, Inc. Slide 6-16 Standard Deviation Let B = {1, 7, 8, 11, 18} with a mean of 9. Find the sample standard deviation of the data set B.

17. 6-B Copyright © 2011 Pearson Education, Inc. Slide 6-17 Standard Deviation Let B = {1, 7, 8, 11, 18} with a mean of 9. Find the sample standard deviation of the data set B.

18. 6-B Copyright © 2011 Pearson Education, Inc. Slide 6-18 Standard Deviation Let B = {1, 7, 8, 11, 18} with a mean of 9. Find the sample standard deviation of the data set B.

19. 6-B Copyright © 2011 Pearson Education, Inc. Slide 6-19 The Range Rule of Thumb The standard deviation is approximately related to the range of a data set by the range rule of thumb: If we know the standard deviation for a data set, we estimate the low and high values as follows: