Pictorial and Tabular Methods

Transcription

1 Example(Example 1.2 p5): The article Effects of Aggregates and Microfillers on the Flexural Properties of Concrete reported on a study of strength properties of high performance concrete obtained by using superplasticizers and certain binders. The accompanying data on flexural strength (in MPa) appeared in the article cited: We are interested in the average value of flexural strength for all beams that could be made in this way.

2

3 The decimal point is at the

4 The decimal point is at the 5 9

5 The decimal point is at the

6 The decimal point is at the

7 The decimal point is at the

8 The decimal point is at the

9 The decimal point is at the

10 The decimal point is at the

11 The decimal point is at the identification of a typical value

12 The decimal point is at the identification of a typical value presence of any gaps in the data

13 The decimal point is at the identification of a typical value presence of any gaps in the data extent of symmetry in the distribution of values

14 The decimal point is at the identification of a typical value presence of any gaps in the data extent of symmetry in the distribution of values number and location of peaks

15 The decimal point is at the identification of a typical value presence of any gaps in the data extent of symmetry in the distribution of values number and location of peaks presence of any outlying values

16

17 Stem-and-Leaf Displays

18 Stem-and-Leaf Displays 1. Select one or more leading digits for the stem values. The trailing digits become the leaves.

19 Stem-and-Leaf Displays 1. Select one or more leading digits for the stem values. The trailing digits become the leaves. 2. List possible stem values in a vertical column.

20 Stem-and-Leaf Displays 1. Select one or more leading digits for the stem values. The trailing digits become the leaves. 2. List possible stem values in a vertical column. 3. Record the leaf for every observation beside the corresponding stem value.

21 Stem-and-Leaf Displays 1. Select one or more leading digits for the stem values. The trailing digits become the leaves. 2. List possible stem values in a vertical column. 3. Record the leaf for every observation beside the corresponding stem value. 4. Indicate the units for stems and leaves someplace in the display.

22 Remark:

23 Remark: 1. Each data in the population must consist of at least two digits.

24 Remark: 1. Each data in the population must consist of at least two digits. e.g. the stem-and-leaf display is not suitable for the data set 1,2,1,4,1,5,2,6,1,3,2,3

25 Remark: 1. Each data in the population must consist of at least two digits. e.g. the stem-and-leaf display is not suitable for the data set 1,2,1,4,1,5,2,6,1,3,2,3 2. Ordering the leaves from smallest to largest is not necessary

26 The decimal point is at the The decimal point is at the

27 Dotplots:

28 Dotplots: e.g. The dotplot for the previous example:

29 Dotplots: e.g. The dotplot for the previous example: In a dotplot, each data is represented by a dot above the corresponding location on a horizontal measurement scale. When a value occurs more than once, there is a dot for each occurrence, and these dots are stacked vertically.

30 Histograms

31 Histograms e.g. The histogram for the previous example:

32 Discrete & Continuous Variables:

33 Discrete & Continuous Variables: A numerical variable is discrete if its set of possible values is either finite or can be listed in an infinite sequence.

34 Discrete & Continuous Variables: A numerical variable is discrete if its set of possible values is either finite or can be listed in an infinite sequence. e.g. x = number of students in this classroom who drove to school today

35 Discrete & Continuous Variables: A numerical variable is discrete if its set of possible values is either finite or can be listed in an infinite sequence. e.g. x = number of students in this classroom who drove to school today Usually arising from counting A numerical variable is continuous if its possible values consist of an entire interval on the number line.

36 Discrete & Continuous Variables: A numerical variable is discrete if its set of possible values is either finite or can be listed in an infinite sequence. e.g. x = number of students in this classroom who drove to school today Usually arising from counting A numerical variable is continuous if its possible values consist of an entire interval on the number line. e.g y = maximum hours a GE lamp can last

37 Discrete & Continuous Variables: A numerical variable is discrete if its set of possible values is either finite or can be listed in an infinite sequence. e.g. x = number of students in this classroom who drove to school today Usually arising from counting A numerical variable is continuous if its possible values consist of an entire interval on the number line. e.g y = maximum hours a GE lamp can last Usually arising from measuring

38 Frequency: the frequency of any particular data value is the number of times that value occurs in the data set.

39 Frequency: the frequency of any particular data value is the number of times that value occurs in the data set. Relative Frequency: the relative frequency of a value is the fraction of proportion of times the value occurs

40 Frequency: the frequency of any particular data value is the number of times that value occurs in the data set. Relative Frequency: the relative frequency of a value is the fraction of proportion of times the value occurs relative frequency = number of times the value occur number of observations in the data set

41 Frequency: the frequency of any particular data value is the number of times that value occurs in the data set. Relative Frequency: the relative frequency of a value is the fraction of proportion of times the value occurs relative frequency = number of times the value occur number of observations in the data set e.g. frequency of value 6.8: 2 relative frequency of the value 6.8: 2 27 = 0.074

42 Frequency: the frequency of any particular data value is the number of times that value occurs in the data set. Relative Frequency: the relative frequency of a value is the fraction of proportion of times the value occurs relative frequency = number of times the value occur number of observations in the data set e.g. frequency of value 6.8: 2 relative frequency of the value 6.8: 2 27 = Frequency Distribution: a tabulation of the frequencies and/or relative frequencies.

43 Constructing a Histogram for a Data Set:

44 Constructing a Histogram for a Data Set: 1. Divide the data set into a suitable number of class interval or classes;

45 Constructing a Histogram for a Data Set: 1. Divide the data set into a suitable number of class interval or classes; 2. Determine the frequency and relative frequency for each class;

46 Constructing a Histogram for a Data Set: 1. Divide the data set into a suitable number of class interval or classes; 2. Determine the frequency and relative frequency for each class; 3. Mark the class boundaries on a horizontal measurement axis;

47 Constructing a Histogram for a Data Set: 1. Divide the data set into a suitable number of class interval or classes; 2. Determine the frequency and relative frequency for each class; 3. Mark the class boundaries on a horizontal measurement axis; 4. Above each class interval, draw a rectangle whose height is the corresponding relative frequency(or frequency)

48 Determine frequency and relative frequency for each class: classes frequency relative frequency

49

50 Remark:

51 Remark: 1. For discrete data, we usually don t have to determine the class intervals.

52 Remark: 1. For discrete data, we usually don t have to determine the class intervals. 2. There is no hard-and-fast rules for the choice of class intervals. A reasonable rule of thumb is number of classes = number of observation

53 Remark: 1. For discrete data, we usually don t have to determine the class intervals. 2. There is no hard-and-fast rules for the choice of class intervals. A reasonable rule of thumb is number of classes = number of observation 3. Equal-width classes may not be a sensible choice if a data set stretches out to one side or the other.

54 Remark: 1. For discrete data, we usually don t have to determine the class intervals. 2. There is no hard-and-fast rules for the choice of class intervals. A reasonable rule of thumb is number of classes = number of observation 3. Equal-width classes may not be a sensible choice if a data set stretches out to one side or the other. e.g.

55 Remark: 3. Equal-width classes may not be a sensible choice if a data set stretches out to one side or the other. e.g.

56 Remark: 3. Equal-width classes may not be a sensible choice if a data set stretches out to one side or the other. e.g. Use a few wider intervals near extreme observations and narrower intervals in the region of high concentration.

57 Remark: 3. Equal-width classes may not be a sensible choice if a data set stretches out to one side or the other. e.g. Use a few wider intervals near extreme observations and narrower intervals in the region of high concentration. rectangle height = relative frequency of the class class width

58 Shapes of Histograms: