1. Slide 13 - 1 Copyright © 2009 Pearson Education, Inc. Unit 9 Seminar Agenda Final Project and Due Dates Measures of Central Tendency Measures of Dispersion Celebrate!

2. Slide 13 - 2 Copyright © 2009 Pearson Education, Inc. Final Project and Due Dates Final project is due to the dropbox by Tuesday, September 6 th by 11:59 PM ET. Late final projects will be docked 5% per day late and will not be accepted after 4 days late. If you have any questions about the assignment requirements there is a page in the Unit 9 introduction with all of the info you should need, as well as some sample projects to give you an idea of what I'm expecting from the finished product. Late assignments: The last day to submit any late assignments or message board posts will be Sunday, September 11 th, 11:59 PM ET.

3. Slide 13 - 3 Copyright © 2009 Pearson Education, Inc. An average is a number that is representative of a group of data. The arithmetic mean, or simply the mean is symbolized by, when it is a sample of a population or by the Greek letter mu, , when it is the entire population. 9.1 Measures of Central Tendency

4. Slide 13 - 4 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.

5. Slide 13 - 5 Copyright © 2009 Pearson Education, Inc. Example-find the mean Find the mean amount of money parents spent on new school supplies and clothes if 5 parents randomly surveyed replied as follows: $327 $465 $672 $150 $230

6. Slide 13 - 6 Copyright © 2009 Pearson Education, Inc. Median The median is the value in the middle of a set of ranked data. Example: Determine the median of $327 $465 $672 $150 $230. Rank the data from smallest to largest. $150 $230 $327 $465 $672 middle value (median)

7. Slide 13 - 7 Copyright © 2009 Pearson Education, Inc. Example: Median (even data) Determine the median of the following set of data: 8, 15, 9, 3, 4, 7, 11, 12, 6, 4. Rank the data: 3 4 4 6 7 8 9 11 12 15 There are 10 pieces of data so the median will lie halfway between the two middle pieces the 7 and 8. The median is (7 + 8)/2 = 7.5 3 4 4 6 9 11 12 15 7 8 (median) middle value

8. Slide 13 - 8 Copyright © 2009 Pearson Education, Inc. Mode The mode is the piece of data that occurs most frequently. Example: Determine the mode of the data set: 3, 4, 4, 6, 7, 8, 9, 11, 12, 15. The mode is 4 since it occurs twice and the other values only occur once.

9. Slide 13 - 9 Copyright © 2009 Pearson Education, Inc. Example The weights of eight Labrador retrievers rounded to the nearest pound are 85, 92, 88, 75, 94, 88, 84, and 101. Determine the a) mean b) median c) mode

10. Slide 13 - 10 Copyright © 2009 Pearson Education, Inc. Example--dog weights 85, 92, 88, 75, 94, 88, 84, 101 a. Mean b.Median Rank the data: 75, 84, 85, 88, 88, 92, 94, 101 The median is 88. c.Mode-the number that occurs most frequently. The mode is 88.

11. Slide 13 - 11 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. Both measure how many data points are less than the given value. 1 st Quartile – 25% of the data values are less than the 1 st Quartile 99 th Percentile – 99% of the data values are less than the 99 th Percentile

12. Slide 13 - 12 Copyright © 2009 Pearson Education, Inc. To Find the Quartiles of a Set of Data 1.Order the data from smallest to largest. 2.Find the median, or 2 nd quartile, of the set of data. If there are an odd number of pieces of data, the median is the middle value. If there are an even number of pieces of data, the median will be halfway between the two middle pieces of data.

13. Slide 13 - 13 Copyright © 2009 Pearson Education, Inc. To Find the Quartiles of a Set of Data continued 3.The first quartile, Q 1, is the median of the lower half of the data; that is, Q 1, is the median of the data less than Q 2. 4.The third quartile, Q 3, is the median of the upper half of the data; that is, Q 3 is the median of the data greater than Q 2.

14. Slide 13 - 14 Copyright © 2009 Pearson Education, Inc. Example: Quartiles The weekly grocery bills for 23 families are as follows. Determine Q 1, Q 2, and Q 3. 170210270270280 33080170240270 22522521531050 751601307481 95172190

15. Slide 13 - 15 Copyright © 2009 Pearson Education, Inc. Example: Quartiles continued Order the data: 50 75 74 80 81 95130 160170170172190210215 225225240270270270280 310330 Q 2 is the median of the entire data set which is 190. Q 1 is the median of the numbers from 50 to 172 which is 95. Q 3 is the median of the numbers from 210 to 330 which is 270.

16. Slide 13 - 16 Copyright © 2009 Pearson Education, Inc. Measures of Dispersion Measures of dispersion are used to indicate the spread of the data. The range is the difference between the highest and lowest values; it indicates the total spread of the data. Range = highest value – lowest value

17. Slide 13 - 17 Copyright © 2009 Pearson Education, Inc. Example: Range Nine different employees were selected and the amount of their salary was recorded. Find the range of the salaries. $24,000$32,000 $26,500 $56,000 $48,000 $27,000 $28,500 $34,500 $56,750 Range = $56,750  $24,000 = $32,750

18. Slide 13 - 18 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.

19. Slide 13 - 19 Copyright © 2009 Pearson Education, Inc. To compute mean and standard deviation using a calculator In Windows choose Start > Accessories > Calculator Select View > Statistics Enter numbers then the Add key to create a list Use the x-bar key for mean Use the σ(n-1) key for sample standard deviation

20. Slide 13 - 20 Copyright © 2009 Pearson Education, Inc. To compute mean and standard deviation using the web/Excel http://www.easycalculation.com/statistics/statistics.php In Excel use the Average() function for mean. Use the Stdev() function for sample standard deviation.

21. Slide 13 - 21 Copyright © 2009 Pearson Education, Inc. To Find the Standard Deviation of a Set of Data 1. Find the mean of the set of data. 2. Make a chart having three columns: Data Data - Mean (Data - Mean) 2 3. List the data vertically under the column marked Data. 4. Subtract the mean from each piece of data and place the difference in the Data - Mean column.

22. Slide 13 - 22 Copyright © 2009 Pearson Education, Inc. To Find the Standard Deviation of a Set of Data continued 5.Square the values obtained in the Data - Mean column and record these values in the (Data - Mean) 2 column. 6.Determine the sum of the values in the (Data - Mean) 2 column. 7.Divide the sum obtained in step 6 by n - 1, where n is the number of pieces of data. 8.Determine the square root of the number obtained in step 7. This number is the standard deviation of the set of data.

23. Slide 13 - 23 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.

24. Slide 13 - 24 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

25. Slide 13 - 25 Copyright © 2009 Pearson Education, Inc. Example continued, mean = 514 The standard deviation is $290.35.