1 Review Sections 2.1, 2.2, 1.3, 1.4, 1.5, 1.6 in text

2 Quantitative Data (Graphical)

3 This is numerical data We may describe quantitative data using the same methods as qualitative by breaking our numerical data into classes. That is 20-30, , 40-50,

4 Quantitative Data (Graphical) This is numerical data We may describe quantitative data using the same methods as qualitative by breaking our numerical data into classes. That is 20-30, , 40-50, Histograms, stem and leaf plots and dot plots are other common methods of displaying quantitative data.

5 Histograms A histogram is a bar graph where you use intervals for your data class. The following histogram summarizes the NBA payroll. You should note that the are adjacent to one another.

6 NBA Payroll

7 Stem and Leaf, and Dot Plots Notice in the histogram on the previous page we lose some information. That is we don’t know exactly what each team is paying in salary just how many are paying in the range of million dollars.

8 Stem and Leaf, and Dot Plots Notice in the histogram on the previous page we lose some information. That is we don’t know exactly what each team is paying in salary just how many are paying in the range of million dollars. A stem and leaf plot is a graphical device which uses numbers so that no information is lost.

9 Stem and Leaf, and Dot Plots A stem and leaf plot is a graphical device which uses numbers so that no information is lost. The technique separates each data point into two numbers, the stem (the leading digit) and the leaves.

10 Stem and Leaf, and Dot Plots The technique separates each data point into two numbers, the stem (the leading digit) and the leaves. In a dot plot we start with a number line of all possible values for the data. Each data point is represented with a dot above the appropriate number. If a number appears more than once in your data you build a tower of dots above that point.

11 Example Here is a list of exam scores: 88, 82, 89, 70, 85, 63, 100, 86, 67, 39, 90, 96, 76, 34, 81, 64, 75, 84, 89, 96 Construct a histogram (with interval size 10 starting at 24), a stem and leaf diagram and a dot plot.

12 Histogram of Exam Scores

13 Stem and Leaf Plot of Exam Scores 88, 82, 89, 70, 85, 63, 100, 86, 67, 39, 90, 96, 76, 34, 81, 64, 75, 84, 89, 96

14 Stem and Leaf Plot of Exam Scores 88, 82, 89, 70, 85, 63, 100, 86, 67, 39, 90, 96, 76, 34, 81, 64, 75, 84, 89,

15 Stem and Leaf Plot of Exam Scores 88, 82, 89, 70, 85, 63, 100, 86, 67, 39, 90, 96, 76, 34, 81, 64, 75, 84, 89,

16 Stem and Leaf Plot of Exam Scores 88, 82, 89, 70, 85, 63, 100, 86, 67, 39, 90, 96, 76, 34, 81, 64, 75, 84, 89,

17 Stem and Leaf Plot of Exam Scores 88, 82, 89, 70, 85, 63, 100, 86, 67, 39, 90, 96, 76, 34, 81, 64, 75, 84, 89,

18 Stem and Leaf Plot of Exam Scores 88, 82, 89, 70, 85, 63, 100, 86, 67, 39, 90, 96, 76, 34, 81, 64, 75, 84, 89,

19 Stem and Leaf Plot of Exam Scores 88, 82, 89, 70, 85, 63, 100, 86, 67, 39, 90, 96, 76, 34, 81, 64, 75, 84, 89,

20 Stem and Leaf Plot of Exam Scores 88, 82, 89, 70, 85, 63, 100, 86, 67, 39, 90, 96, 76, 34, 81, 64, 75, 84, 89,

21 Dot Plot of Exam Scores 88, 82, 89, 70, 85, 63, 100, 86, 67, 39, 90, 96, 76, 34, 81, 64, 75, 84, 89, 96

22 Dot Plot of Exam Scores 88, 82, 89, 70, 85, 63, 100, 86, 67, 39, 90, 96, 76, 34, 81, 64, 75, 84, 89,

23 88, 82, 89, 70, 85, 63, 100, 86, 67, 39, 90, 96, 76, 34, 81, 64, 75, 84, 89, 96 Dot Plot of Exam Scores

24 88, 82, 89, 70, 85, 63, 100, 86, 67, 39, 90, 96, 76, 34, 81, 64, 75, 84, 89, 96 Dot Plot of Exam Scores

25 88, 82, 89, 70, 85, 63, 100, 86, 67, 39, 90, 96, 76, 34, 81, 64, 75, 84, 89, 96 Dot Plot of Exam Scores

26 88, 82, 89, 70, 85, 63, 100, 86, 67, 39, 90, 96, 76, 34, 81, 64, 75, 84, 89, 96 Dot Plot of Exam Scores

27 88, 82, 89, 70, 85, 63, 100, 86, 67, 39, 90, 96, 76, 34, 81, 64, 75, 84, 89, 96 Dot Plot of Exam Scores

28 Summation Notation Here is a typical (small) data set: 27132

29 Summation Notation Here is a typical (small) data set: So we can talk about a general data set we let:

30 Summation Notation So we can talk about a general data set we let: In general for a sample of n points of data we call them, in order:

31 Summation Notation In general for a sample of n points of data we call them, in order: When we wish to sum (add them up), we use the notation: This is called summation notation.

32 Example

33 Example This says to add up the x i changing i from: 1 to 5

34 Example

35 Example

36 Example

37 Example

38 Example

39 Example

40 Example

41 Example

42 Example

43 Example

44 Example

45 Example

46 Example

47 Example

48 Summation Notation In statistics, sometimes the i is not included in the sum since it is implied that we are summing over all points in our data set. That is you may see the following:

49 Descriptive Statistics Qualitative Variables –Graphical Methods Quantitative Variables –Graphical Methods

50 Descriptive Statistics Qualitative Variables –Graphical Methods Quantitative Variables –Graphical Methods –Numerical Methods

51 Numerical descriptive measures Two types of measures we look for: 1)Ones which tell us about the central tendency of measurements 2)Ones which tell us about the variability or spread of the data.

52 Numerical Measures of Central Tendency Three Measures a) Mean b) Median c) Mode Problem

53 Mean The mean of a data set is the average or expected value of the readings in the data. Problem: I wish to talk about the mean of the population and the mean of the sample separately. Therefore we need to introduce two different notations.

54 Mean Sample: the size of the sample is usually denoted with n, and the mean of the sample (sample mean) is denoted with Population: the size of the population is usually denoted N and the population mean is denoted µ.

55 Mean The mean is given by

56 Example Given the sample: Find the mean.

57 Example Given the sample: Find the mean.

58 Example Given the sample: Find the mean.

59 Example Given the sample: Find the mean.

60 However, given the sample: we find the mean is quite different from Example

61 However, given the sample: we find the mean is quite different from This is not a good indication of the center of the sample. Example

62 Mean Usually the sample mean is used to estimate the population mean µ. The accuracy of this estimate tends to be effected by: –The size of the sample –Variability or spread of the data

63 Median The median of a quantitative data set is the middle number in the set. For example in the following data the median is 10.

64 Median The sample median is denoted M. If n is even, take the average of the two middle numbers.

65 Examples Find the median in the following two data sets:

66 Examples Find the median in the following two data sets: In both cases we found M=3.5. The median is sometimes a better estimate of the population mean µ than the sample mean because it puts less emphasis on outliers.

67 What the median and mean tell you A data set is skewed if one tail of the distribution has more extreme observations than the other. SkewDistribution/

68 What the median and mean tell you This data set is skewed to the right. Notice the mean is to the right of the median.

69 What the median and mean tell you Skewed to the right: The mean is bigger than the median.

70 What the median and mean tell you This data set is skewed to the left. Notice the mean is to the left of the median.

71 What the median and mean tell you Skewed to the left: The mean is less than the median.

72 What the median and mean tell you When the mean and median are equal, the data is symmetric

73 Mode The mode is the measurement which occurs most frequently

74 Mode The mode is the measurement which occurs most frequently a)mode= 4 b)mode= 4, 1

75 Mode When dealing with histograms or qualitative data, the measurement with the highest frequency is called the modal class.

76 Homework Read Sections 2.1 to 2.7 Find and start assignment 1 Familiarize yourself with your calculator.