1. Stat 101Dr SaMeH1 Statistics (Stat 101) Associate Professor of Environmental Eng. Civil Engineering Department Engineering College Almajma’ah University smohamed1@ksu.edu.sa http://faculty.ksu.edu.sa/SaMeH 2010/2011 Sameh Saadeldin Ahmed

2. 2.2 Organizing and Graphing Qualitative Data Stat 101Dr SaMeH2 2.2.1 Frequency Distribution Bar Graphs Pie Chart. 2.2.2 Relative Frequency & Percentage 2.2.3 Graphical Presentation of Qualitative

3. 2.3 Organizing and Graphing Quantitative Data Stat 101Dr SaMeH3 2.3.1 Frequency Distribution 2.3.1 Frequency Distribution  2.3.2 Constructing Freq. Distribution Tables  2.3.3 Relative Freq. & Percentage Distribut.  2.3.4 Graphing Grouped Data.

4. 2.3.2Constructing Frequency Distribution Tables To construct the frequency distribution table, you have to make the following three steps:  Number of Classes  Class Width  Lower Limit of the First Class or the Starting Point. Stat 101Dr SaMeH4

5. Number of Classes Usually the number of classes for a frequency distribution table varies from 5 to 20. The decision of the number of classes is arbitrarily made by the data organizer. Stat 101Dr SaMeH5

6. Class Width It is preferable to have the same width for all classes. To do so, find the difference between the largest and smallest values in the data. Then, the approximate width of a class is obtained by dividing this difference by the number of desired classes. Stat 101Dr SaMeH6 Calculating of Class Width Approximate class width = [Largest value - Smallest value] / Number of classes

7. Lower Limit of the First Class or the Starting Point Any convenient number that is equal to or less than the smallest value in the data set can be used as the lower limit of the first class. Stat 101Dr SaMeH7

8. 2.3.3Relative Frequency and Percentage Distributions Stat 101Dr SaMeH8 Calculating Relative Frequency and Percentage Relative frequency of a class = Frequency of that class / Sum of all frequencies = f / ∑f Percentage = (Relative frequency) x 100

9. Stat 101Dr SaMeH9 Exercise Calculate the relative frequencies and percentage for the following table: Total goals scoredF 124 – 1456 146 – 16713 168 – 1894 190 – 2114 212 - 2333

10. 2.3.4Graphing Grouped Data Stat 101Dr SaMeH10 Histogram A histogram is a graph in which classes are marked on the horizontal axis and the frequencies, relative frequencies, or percentages are marked on the vertical axis. The frequencies, relative frequencies, or percentages are represented by the heights of the bars. In a histogram the bars are drawn adjacent to each other. Grouped quantitative data can be displayed in a histogram or a polygon.

11. Stat 101Dr SaMeH11 A histogram for the frequency distribution

12. Stat 101Dr SaMeH12 A histogram for the relative frequency A histogram for the percentages

13. Stat 101Dr SaMeH13 Polygons A graph formed by joining the midpoint of the tops of successive bars in a histogram with straight lines is called a polygon.

14. 2.3.5More on Classes and Frequency Distribution Less than Method for Writing Classes Single-Valued Classes Stat 101Dr SaMeH14

15. Stat 101Dr SaMeH15 Example 2.6: less than..... The following data give the average travel time from home to work (in minutes) for 50 cities. 22.418.223.719.826.723.423.522.524.326.7 24.219.727.021.717.617.722.523.721.229.2 26.122.721.621.923.216.016.122.324.428.7 19.931.222.615.422.119.621.423.821.9 15.622.723.620.821.125.424.925.520.117.1 1.Construct a frequency distribution table. 2.Calculate the relative frequencies and percentages for all classes. 3.Plot the histograms and polygons for the frequencies & percentages.

16. Stat 101Dr SaMeH16 Solution The min. value is 15.4 and the max. value is 31.2 Suppose we decide to group these data using six classes of equal width. Then, Approximate width of each class = [31.2 – 15.4] / 6 = 2.63 We round this number to a more convenient number to 3 Let us start the first class at 15, the classes are written as 15 to less than 18, and so on. Sturge’s formula to decide on the no. of classes: c = 1 + 3.3 log n

17. Stat 101Dr SaMeH17 Average travel time to work (minutes) Frequency (f) Relative Frequency Percentage 15 to less than 1870.1414 18 to less than 2170.1414 21 to less than 24230.4646 24 to less than 2790.1818 27 to less than 3030.066 30 to less than 3310.022 SUM501.00100%

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19. Stat 101Dr SaMeH19 Example 2.7: Single value...... In this case we use classes that are made of single values and not of intervals. It is useful in cases of discrete data with only a few possible values. See the following example. The governorate of Almajmaa’h city wanted to know the distribution of the computer sets owned by families in the city. A sample of 40 randomly selected houses from the city produced the following data on the number of computers owned. 5112011251 1113302511 1234212212 1221114212 1121141311

20. Stat 101Dr SaMeH20 Construct the frequency distribution table for these data using single-valued classes. Solution: The observations in this data set assumes only 6 distinct values: 0,1,2,3,4 and 5. Each of these values is used as a class in the frequency distribution. Computers OwnedFrequency f Relative Frequency Percentage 020.055 1180.4545 2110.27527.5 340.1010 430.0757.50 520.055 SUM401.00100%

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22. 2.4 Cumulative Frequency Distribution Stat 101Dr SaMeH22 2.1 Raw Data 2.2 Organizing Qualitative Data 2.3 Organizing Quantitative Data 2.4 Cumulative Frequency Distribution 2.5 Seam-and –Leaf Display

23. 2.4Cumulative Frequency Distribution 2.4 Cumulative Frequency Distribution Stat 101Dr SaMeH23 Cumulative Frequency Distribution A cumulative frequency distribution gives the total number of values that fall below the upper boundary of each class. In the cumulative frequency distribution table, each class has the same lower limit but a different upper limit. Example 2.8 illustrates the procedure to prepare cumulative frequency distribution.

24. Stat 101Dr SaMeH24 Example 2.8: Using the frequency distribution of the following table, prepare a cumulative frequency distribution for the total goals. Total goals scoredF 124 – 1456 146 – 16713 168 – 1894 190 – 2114 212 - 2333

25. Stat 101Dr SaMeH25 Solution: Class LimitsClass BoundariesCumulative frequency 124 – 145123.5 to less than 145.56 124 – 167123.5 to less than 167.56 + 13 = 19 124 – 189123.5 to less than 189.56 + 13 + 4 = 23 124 – 211123.5 to less than 211.56 + 13 + 4 + 4 = 27 124 - 233123.5 to less than 233.56 + 13 + 4 + 4 + 3 = 30 From the above table we can determine the number of observations that fall below the upper limit or boundary of each class. For example, 23 football teams scored a total of 189 goals or fewer.

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27. Stat 101Dr SaMeH27 Calculating the Relative Frequency and Cumulative Percentage A cumulative frequency distribution gives the total number of values that fall below the upper boundary of each class. Cumulative frequency of a class Cumulative relative frequency = --------------------------------------- Total observations in the data set

28. Stat 101Dr SaMeH28 The following table contains both the cumulative relative frequencies and the cumulative percentages for the data given in example 8. Class LimitsCumulative Relative Frequency Cumulative Percentage 124 – 145 6/30 = 0.20020.0 124 – 16719/30 = 0.63363.3 124 – 18923/30 = 0.76776.7 124 – 21127/30 = 0.90090.0 124 - 23330/30 = 1.000100.0

29. Ogives Ogive An ogive is a curve drawn for the cumulative frequency distribution by joining with straight lines the dots marked above the upper boundaries of classes at heights equal to the cumulative frequencies of respective classes. Stat 101Dr SaMeH29 When plotted on a diagram, the cumulative frequencies give that is called ogive. The next figure gives an ogive for the cumulative frequency distribution of ex. 8.

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31. End of Part 3 Get ready for a quiz (2)…… next lecture!! Stat 101Dr SaMeH31