1. Probability & Statistics
Chapter 02
Summarizing and Graphing Data

2. Describing Data- 1. Frequency distribution
Frequency distribution: A grouping of data into categories showing the number of observations in each mutually exclusive category.

3. The steps for organizing data into a frequency distribution
Set up grouping called classes. Decide on the size of the class interval.
Tally the raw data or ungrouped data into classes.
Count the number of tallies in each class. The number of observation in the is called the class frequency.

4. Class: A class is one of the categories into which data can be classified.
Class interval for a frequency distribution is obtained by subtracting the lower limit of a class from the lower limit of the next class or subtracting the higher limit of a class from the higher limit of the next class

5. Class limits: represent the smallest and largest data values that can be included in a class
The class boundaries are used to separate the classes so that there are no gaps in the frequency distribution

6. Class mark (midpoint): A point that divides a class into two equal parts. This is the average between the upper and lower class limits.

7. EXAMPLE 1
Prof. Munasinghe is the head of the Department of Industrial Management (IM) and wishes to determine the amount of studying IM students do. He selects a random sample of 30 students and determines the number of hours each student studies per week: 15.0, 23.7, 19.7, 15.4, 18.3, 23.0, 14.2, 20.8, 13.5, 20.7, 17.4, 18.6, 12.9, 20.3, 13.7, 21.4, 18.3, 29.8, 17.1, 18.9, 10.3, 26.1, 15.7, 14.0, 17.8, 33.8, 23.2, 12.9, 27.1, 16.6,
Organize the data into a frequency distribution.

8. Guidelines for Constructing a Frequency Distribution
To determine the number of classes “2 to the k rule” can be used select the smallest integer (k) such that
2k  n
n is the number of observations
There should be between 5 and 20 classes.

9. Guidelines for Constructing a Frequency Distribution
Determine the class interval based on the number of suggested classes by using the formula:
The class intervals used in the frequency distribution should be equal.
Use the computed suggested class interval to construct the frequency distribution. Note: this is a suggested class interval; if the computed class interval is 97, it may be better to use 100.

10. Guidelines for Constructing a Frequency Distribution
Avoid open ended classes
Do not have overlapping classes
Count the number of values in each class.

11. EXAMPLE 1
15.0, 23.7, 19.7, 15.4, 18.3, 23.0, 14.2, 20.8, 13.5, 20.7, 17.4, 18.6, 12.9, 20.3, 13.7, 21.4, 18.3, 29.8, 17.1, 18.9, 10.3, 26.1, 15.7, 14.0, 17.8, 33.8, 23.2, 12.9, 27.1, 16.6
Number of possible classes: 2k  n need to find a value for k where 2k  30 then k = 5 number of possible classes = 5
Class interval: i = (highest value – lowest value) / number of classes i = (33.8 – 10.3) / 5 = 4.7 better to use i = 5

12. EXAMPLE 1 Classes Tally Frequency 10.3 – 15.2 |||| ||| 8 15.3 – 20.2
|||| |||| | 11
20.3 – 25.2
|||| || 7
25.3 – 30.2
||| 3
30.3 – 35.2 | 1

13. Relative Frequency Distribution
Shows the percentage of the observation in each class
The relative frequency of a class is obtained by dividing the class frequency by the total frequency.
This allows us to make statements regarding the number of observations in a single class relative to the entire sample

14. Relative Frequency Distribution for Exercise 1
Classes
Frequency
Relative Frequency
10.3 – 15.2 8
8/30 = 26.7%
15.3 – 20.2 11
11/30 = 36.7%
20.3 – 25.2 7
7/30 = 23.3%
25.3 – 30.2 3
3/30 = 10%
30.3 – 35.2 1
1/30 = 3.3%

15. Cumulative Relative Frequency Distribution
The cumulative relative frequency distribution express the cumulative frequency of each class relative to the entire sample

16. Cumulative Relative Frequency Distribution for Exercise 1
Classes
Frequency
Cumulative Frequency
Cumulative Relative Frequency
10.3 – 15.2 8
8/30 = 26.7%
15.3 – 20.2 11 19
19/30 = 63.3%
20.3 – 25.2 7 26
26/30 = 86.7%
25.3 – 30.2 3 29
29/30 = 96.7%
30.3 – 35.2 1 30
30/30 = 100%

17. Contingency Table
Contingency table indicates the number of observations for both variables that fall jointly in each category.
Example: Distribution of product complaint
Complaint Origin
Reason for complaint
Electrical
Mechanical
Appearance
During guarantee period
18%
13%
32%
After guarantee period
12%
22% 3%

18. Graphic Presentation
There are three commonly used graphic forms of a frequency distribution
histograms,
frequency polygons,
cumulative frequency distribution (ogive).

19. Histogram A graph in which
the classes are marked on the horizontal axis
the class frequencies on the vertical axis.
the class frequencies are represented by the heights of the bars
the bars are drawn adjacent to each other.

20. Histogram for Exercise 1
Class Midpoint
Frequency
12.75 8
17.75 11
22.75 7
27.75 3
32.75 1

21. Frequency Polygon
A frequency polygon consists of line segments connecting the points formed by the class midpoint and the class frequency.

22. Frequency Polygon for Exercise 1
Class Midpoint
Frequency
7.75
12.75 8
17.75 11
22.75 7
27.75 3
32.75 1
37.75

23. Cumulative Frequency distribution
A cumulative frequency distribution (ogive) is used to determine how many or what proportion of the data values are below or above a certain value.

24. Ogive for Exercise 1 Class Midpoint Frequency
Less than cumulative frequency
More than cumulative frequency
7.75 30
12.75 8
17.75 11 19 22
22.75 7 26
27.75 3 29 4
32.75 1
37.75

25. Ogive for Exercise 1

26. Number of unemployed (in ‘000)
2-17
Bar Chart
A bar chart can be used to depict any of the levels of measurement (nominal, ordinal, interval, or ratio).
City
Number of unemployed (in ‘000)
Gampaha
730
Matale
540
Chilaw
670
Colombo
890
Galle
820
Kandy
EXAMPLE : Construct a bar chart for the number of unemployed people per 100,000 population for selected cities of 1995.

27. 2-18
EXAMPLE continued
a bar chart for the number of unemployed people per 100,000 population for selected cities of 1995

28. Pie Chart
A pie chart is especially useful in displaying a relative frequency distribution. A circle is divided proportionally to the relative frequency and portions of the circle are allocated for the different groups.
Example: Draw a pie chart based on the following information.
Type of shoe
# of runners
Nike 92
Adidas 49
Reebok 37
Asics 13

29. EXAMPLE continued Type of shoe # of runners Nike 92 Adidas 49 Reebok37
Asics 13

30. Number of unemployed (in ‘000)
Line Chart
A graph of data that is mapped by a series of lines. Line charts show changes in data or categories of data over time and can be used to document trends
City
Number of unemployed (in ‘000)
Gampaha
730
Matale
540
Chilaw
670
Colombo
890
Galle
820
Kandy
EXAMPLE : Construct a bar chart for the number of unemployed people per 100,000 population for selected cities of 1995.

31. EXAMPLE continued

32. Stem-and-Leaf Displays
Stem-and-Leaf Display: A statistical technique for displaying a set of data. Each numerical value is divided into two parts: the leading digits become the stem and the trailing digits become the leaf.
Note: An advantage of the stem-and-leaf display over a frequency distribution is we do not lose the identity of each observation.

33. EXAMPLE
Saman achieved the following scores on his twelve accounting quizzes this semester: 86, 79, 92, 84, 69, 88, 91, 83, 96, 78, 82, 85. Construct a stem-and-leaf chart for the data.
stem
leaf 6 9 7
8 9 8
1 2 6