1. 3.2 - 1 Copyright © 2010, 2007, 2004 Pearson Education, Inc. Lecture Slides Elementary Statistics Eleventh Edition and the Triola Statistics Series by Mario F. Triola

2. 3.2 - 2 Copyright © 2010, 2007, 2004 Pearson Education, Inc. Chapter 3 Statistics for Describing, Exploring, and Comparing Data 3-1 Review and Preview 3-2 Measures of Center 3-3 Measures of Variation 3-4 Measures of Relative Standing and Boxplots

3. 3.2 - 3 Copyright © 2010, 2007, 2004 Pearson Education, Inc. Section 3-2 Measures of Center

4. 3.2 - 4 Copyright © 2010, 2007, 2004 Pearson Education, Inc. Key Concept Characteristics of center. Measures of center, including mean and median, as tools for analyzing data. Not only determine the value of each measure of center, but also interpret those values.

5. 3.2 - 5 Copyright © 2010, 2007, 2004 Pearson Education, Inc. Basics Concepts of Measures of Center Part 1

6. 3.2 - 6 Copyright © 2010, 2007, 2004 Pearson Education, Inc. Measure of Center  Measure of Center the value at the center or middle of a data set

7. 3.2 - 7 Copyright © 2010, 2007, 2004 Pearson Education, Inc. Arithmetic Mean  Arithmetic Mean (Mean) the measure of center obtained by adding the values and dividing the total by the number of values What most people call an average.

8. 3.2 - 8 Copyright © 2010, 2007, 2004 Pearson Education, Inc. Notation  denotes the sum of a set of values. x is the variable usually used to represent the individual data values. n represents the number of data values in a sample. N represents the number of data values in a population.

9. 3.2 - 9 Copyright © 2010, 2007, 2004 Pearson Education, Inc. Notation µ is pronounced ‘mu’ and denotes the mean of all values in a population x = n  x x is pronounced ‘x-bar’ and denotes the mean of a set of sample values x N µ =  x x

10. 3.2 - 10 Copyright © 2010, 2007, 2004 Pearson Education, Inc.  Advantages Is relatively reliable, means of samples drawn from the same population don’t vary as much as other measures of center Takes every data value into account Mean  Disadvantage Is sensitive to every data value, one extreme value can affect it dramatically; is not a resistant measure of center

11. 3.2 - 11 EXAMPLE The given values are the numbers of Dutchess County car crashes for each month in a recent year. Find the mean. 27 8 17 11 15 25 16 14 14 14 13 18

12. 3.2 - 12 FINDING THE MEAN ON THE TI-83/84 1.Press STAT; select 1:Edit…. 2.Enter your data values in L1. (You may enter the values in any of the lists.) 3.Press 2ND, MODE (for QUIT). 4.Press STAT; arrow over to CALC. Select 1:1-Var Stats. 5.Enter L1 by pressing 2ND, 1. 6.Press ENTER.

13. 3.2 - 13 Copyright © 2010, 2007, 2004 Pearson Education, Inc. Median  Median the middle value when the original data values are arranged in order of increasing (or decreasing) magnitude  often denoted by x (pronounced ‘x-tilde’) ~  is not affected by an extreme value - is a resistant measure of the center

14. 3.2 - 14 Copyright © 2010, 2007, 2004 Pearson Education, Inc. Finding the Median 1.If the number of data values is odd, the median is the number located in the exact middle of the list. 2.If the number of data values is even, the median is found by computing the mean of the two middle numbers. First sort the values (arrange them in order), the follow one of these

15. 3.2 - 15 Copyright © 2010, 2007, 2004 Pearson Education, Inc. 5.40 1.10 0.420.73 0.48 1.10 0.66 0.42 0.48 0.660.73 1.10 1.10 5.40 (in order - odd number of values) exact middle MEDIAN is 0.73 5.40 1.10 0.420.73 0.48 1.10 0.42 0.48 0.731.10 1.10 5.40 0.73 + 1.10 2 (in order - even number of values – no exact middle shared by two numbers) MEDIAN is 0.915

16. 3.2 - 16 Copyright © 2010, 2007, 2004 Pearson Education, Inc. Mode  Mode the value that occurs with the greatest frequency  Data set can have one, more than one, or no mode Mode is the only measure of central tendency that can be used with nominal data Bimodal two data values occur with the same greatest frequency Multimodalmore than two data values occur with the same greatest frequency No Modeno data value is repeated

17. 3.2 - 17 Copyright © 2010, 2007, 2004 Pearson Education, Inc. a. 5.40 1.10 0.42 0.73 0.48 1.10 b. 27 27 27 55 55 55 88 88 99 c. 1 2 3 6 7 8 9 10 Mode - Examples  Mode is 1.10  Bimodal - 27 & 55  No Mode

18. 3.2 - 18 Copyright © 2010, 2007, 2004 Pearson Education, Inc.  Midrange the value midway between the maximum and minimum values in the original data set Definition Midrange = maximum value + minimum value 2

19. 3.2 - 19 Copyright © 2010, 2007, 2004 Pearson Education, Inc.  Sensitive to extremes because it uses only the maximum and minimum values, so rarely used Midrange  Redeeming Features (1)very easy to compute (2) reinforces that there are several ways to define the center (3)Avoids confusion with median

20. 3.2 - 20 EXAMPLE The given values are the numbers of Dutchess County car crashes for each month in a recent year. Find the midrange. 27 8 17 11 15 25 16 14 14 14 13 18

21. 3.2 - 21 ROUND-OFF RULE FOR MEASURES OF CENTER Carry one more decimal place than is present in the original data set.

22. 3.2 - 22 MEAN FROM A FREQUENCY DISTRIBUTION To compute the mean from a frequency distribution, we assume that all sample values are equal to the class midpoint. x = class midpoint f = frequency Σf = sum of frequencies = n

23. 3.2 - 23 EXAMPLE The following data represent the number of people aged 25– 64 covered by health insurance in 1998. Approximate the mean age. Age Number (in millions) 25–3438.5 35–4444.7 45–5435.2 55–6422.9 Source: US Census Bureau

24. 3.2 - 24 FINDING THE MEAN FROM A FREQUENCY TABLE ON TI-83/84 1.Enter the class midpoints in L1. 2.Enter the frequencies in L2. 3.Press STAT, arrow over to CALC, and select 1:1-Var Stats. 4.Press L1,L2 followed by ENTER. 5.The mean will be the first item.

25. 3.2 - 25 Copyright © 2010, 2007, 2004 Pearson Education, Inc. Think about the method used to collect the sample data. Critical Thinking Think about whether the results are reasonable.

26. 3.2 - 26 Copyright © 2010, 2007, 2004 Pearson Education, Inc. Weighted Mean x = w  (w x)  When data values are assigned different weights, we can compute a weighted mean.

27. 3.2 - 27 A=4.0, B=3.0, C=2.0, D=1.0, F=0.0 Use the number of credit hours as weights. Copyright © 2010, 2007, 2004 Pearson Education, Inc.

28. 3.2 - 28 Copyright © 2010, 2007, 2004 Pearson Education, Inc. Best Measure of Center

29. 3.2 - 29 Copyright © 2010, 2007, 2004 Pearson Education, Inc.  Symmetric distribution of data is symmetric if the left half of its histogram is roughly a mirror image of its right half  Skewed distribution of data is skewed if it is not symmetric and extends more to one side than the other Skewed and Symmetric

30. 3.2 - 30 Copyright © 2010, 2007, 2004 Pearson Education, Inc.  Skewed to the left (also called negatively skewed) have a longer left tail, mean and median are to the left of the mode  Skewed to the right (also called positively skewed) have a longer right tail, mean and median are to the right of the mode Skewed Left or Right

31. 3.2 - 31 Copyright © 2010, 2007, 2004 Pearson Education, Inc. The mean and median cannot always be used to identify the shape of the distribution. Shape of the Distribution

32. 3.2 - 32 Copyright © 2010, 2007, 2004 Pearson Education, Inc. Skewness

33. 3.2 - 33 Copyright © 2010, 2007, 2004 Pearson Education, Inc. Recap In this section we have discussed:  Types of measures of center Mean Median Mode  Mean from a frequency distribution  Weighted means  Best measures of center  Skewness