1. Chapter# 2 Frequency Distribution and Graph

2. When data are in original form, they are called raw data.
2-1 Organizing Data:
Frequency distribution:
Categorical distribution
Grouped distribution
Ungrouped Distribution

3. For example : row data (scores) 2 5 8 7 6 4
The researcher organized the raw data into a frequency distribution (is the organizing of raw data in table form, using classes and frequencies).

4. Class
(score) f 8 3 7 2 6 5 4 1

5. Types of Frequency Distributions
Categorical frequency distributions
Grouped Frequency Distributions
Ungrouped Frequency Distributions

6. Categorical frequency distributions: can be used for data that can be placed in specific categories (nominal- or ordinal).
Examples:
political affiliation, religious affiliation, blood type etc.

7. Example: 25 Army inductees were given a blood test to determine their blood type. The data set is :
A B B AB O
O O B AB B
B B O A O
A O O O AB
AB A O B A
*Find the frequency distribution:

8. P=(f/n)*100
Rf=f/n
CLASS
TALLY
FREQUENCY
(f)
Relative frequency
PERCENT A 5
0.2
20% B 7
0.28
28% O 9
0.36
36% AB
|||| 4
0.16
16%
n=25 1
100%
IIII
IIII II
IIII IIII

9. Grouped frequency distributions: can be used when the range of values in the data set is very large.
The data must be grouped into classes that are more than one unit in width.
Examples - the life of boat batteries in hours.

10. Class limits Class boundaries Tally Frequency
Upper boundary
Lower boundary
Upper class
Lower class
Class
limits
Class boundaries
Tally
Frequency
24-30
/// 3
31-37 / 1
38-44
//// 5
45-51
//// //// 9
52-58
//// / 6
59-65

11. Class limits: represent the smallest and largest data values that can be included in a class.
In the lifetimes of boat batteries example, the values 24 and 30 of the first class are the class limits
The lower class limit is 24 and the upper class limit is 30.

12. Lower boundary= lower limit - 0.5 Upper boundary= upper limit + 0.5
The class boundaries are used to separate the classes so that there are no gaps in the frequency distribution.
Lower boundary= lower limit - 0.5
Upper boundary= upper limit + 0.5

13. Class limits should have the same decimal place value as the data, but the class boundaries should have one additional place value and end in a 5.
For example: Class limit
find the class boundaries?

14. The class width for a class in a frequency distribution is found by subtracting the lower (or upper) class limit of one class minus the lower (or upper) class limit of the previous class
Class width=lower of second class limit-lower of first class limit Or
Class width=upper of second class limit-upper of first class limit

15. The class midpoint Xm is found by adding the lower and upper class limit (or boundary) and dividing by 2 .
Xm = Or
Xm =
For example :

16. *Example: *Find the boundaries for the following class limits: 44 - 37
22.2 – 23.0
*Find the class boundaries for the class 24:
*Find the class width for the following class limit:
45 – 52

17. Better way to construct a frequency distribution
Age
10-20
20-30
30-40
40-50
Better way to construct a frequency distribution
Age
10-20
21-31
32-42
43-53

18. Procedure for Constructing a Grouped Frequency Distribution
STEP 1:Determine the classes :
Find the highest and lowest value.
Find the range.
Select the number of classes desired.
Find the width by dividing the range by the number of classes and rounding up.
Select a starting point (usually the lowest value); add the width to get the lower limits.
Find the upper class limits.
Find the boundaries.
STEP 2: Tally the data.
STEP 3: Find the frequencies and find the cumulative frequency.

19. Example:
1- The following data represent the record high temperatures for each of the 50 states. Construct a grouped frequency distribution for the data using 7 classes

20. STEP 1 Determine the classes
STEP 1 Determine the classes. Find the class width by dividing the range by the number of classes 7. Range = High – Low = 134 – 100 = 34 Width= 𝑅𝑎𝑛𝑔𝑒 7 = 34 7 = 5
STEP 2 Tally the data.
STEP 3 Find the frequencies.

21. Class Limits
Class Boundaries
Frequency
Cumulative Frequency 2 8 18 13 7 1 2 10 28 41 48 49 50

22. Ungrouped Frequency distribution:
Example :The data shown here represent the number of miles per gallon that 30 selected four-wheel- drive sports utility vehicles obtained in city driving.

23. range of the data is small STEP 1 Determine the classes.
The rang of the data set is small .
Range = High – Low
= 19 – 12 = 7
So the class consisting of the single data value can be used. They are 12,13,14,15,16,17,18,19.
This type of distribution is called ungrouped frequency distribution
range of the data
is small

24. STEP 3 Find the frequencies.
STEP 2 Tally the data.
STEP 3 Find the frequencies.
Class Limits
Class Boundaries
Frequency
Cumulative Frequency 12 13 14 15 16 17 18 19 6 1 3 8 2 7 10 24 26 29 30

25. Histograms, Frequency Polygons, Ogives

26. Class Agenda:
The three most commonly used graphs in research are: 1. The histogram 2. The frequency polygon 3. The cumulative frequency graph (ogive)

27. Graphical representation: why?
Purpose of graphs in statistics is to convey the data to the viewers in pictorial form
• Easier for most people to understand the meaning of data in form of graphs
• They can also be used to discover a trend or pattern in a situation over a period of time
• Useful in getting the audience’s attention in a publication or a speaking presentation

28. Histogram
The histogram is a graph that displays the data by using contiguous vertical bars (unless the frequency of a class is 0) of various heights to represent the frequencies of the classes.
The class boundaries are represented on the horizontal axis

29. Example 2-4: Construct a histogram to represent the data for the record high temperatures for each of the 50 states (see Example 2–2 for the data).
Class Limits
Class Boundaries
Frequency 2 8 18 13 7 1

30. Histograms use class boundaries and
frequencies of the classes.

31. The class midpoints are represented on the horizontal axis.
Frequency polygons
The frequency polygon is a graph that displays the data
by using lines that connect points plotted for the frequencies at the midpoints of the classes. The frequencies are represented by the heights of the points.
The class midpoints are represented on the horizontal axis.

32. Example 2-5:Construct a frequency polygon to represent the data for the record high temperatures for each of the 50 states (see Example 2–2 for the data).
Class Limits
Class Midpoints
Frequency
130 – 134
102
107
112
117
122
127
132 2 8 18 13 7 1

33. Frequency polygons use class midpoints and frequencies of the classes.
A frequency polygon
is anchored on the
x-axis before the first
class and after the
last class.

34. A frequency polygon is used to represent:
Dependent variables
Quantitative variables
Qualitative variables

35. Cumulative Frequency Graphs Or Ogives
The ogive is a graph that represents the cumulative frequencies for the classes in a frequency distribution
The upper class boundaries are represented on the horizontal axis

36. Example 2-6:Construct an ogive to represent the data for the record high temperatures for each of the 50 states (see Example 2–2 for the data).
Class Limits
Class Boundaries
Frequency 2 8 18 13 7 1

37. Class Boundaries Cumulative Frequency Less than 99.5 Less than 104.5
Class Limits
Class Boundaries
Frequency 2 8 18 13 7 1
Class Boundaries
Cumulative Frequency
Less than 99.5
Less than 104.5
Less than 109.5
Less than 114.5
Less than 119.5
Less than 124.5
Less than 129.5
Less than 134.5 2 10 28 41 48 49 50

38. Ogives use upper class boundaries and cumulative frequencies of the classes.

40. Constructing Statistical Graphs
Procedure Table
Constructing Statistical Graphs
1: Draw and label the x and y axes.
2: Choose a suitable scale for the frequencies or cumulative frequencies, and label it on the y axis.
3: Represent the class boundaries for the histogram or ogive, or the midpoint for the frequency polygon, on the x axis.
4: Plot the points and then draw the bars or lines.

41. * …… is used starting zero and finishing the total number of observations
Grouped frequency
Cumulative frequency graphs

42. If proportions are used instead of frequencies, the graphs are called ………………
The sum of the relative frequencies will always be ……..
The sum of the percent will always be ……..
………. And ……….. And …………..graphs are used to represent continuous data.

43. What types of frequency distribution?
“ Cars Colors” what type of frequency distribution should be used?
The boundaries of 16.7 are …………..
The boundaries of ………….. And class width is ………..

44. Shapes of Distributions
Flat
J shaped : few data values on left side and increases as one moves to right
Reverse J shaped: opposite of the
j-shaped distribution

45. Positively skewed Negatively skewed

46. 2.3 Other Types of Graphs
Bar graph
2. Pareto chart
3. The Pie graph

47. Bar graph represents the data by using vertical bar or horizontal bar whose heights or lengths represent the frequencies of the data.
When the data are qualitative or categorical
,bar graphs can be used.

48. *Example 2-8 P(69): College Spending for First-Year Students:
Class
Frequency
Electronic
728$
Dorm decor
344$
Clothing
141$
Shoes
72$

50. A Pareto chart is used to represent a frequency distribution for a categorical variable (Qualitative), and the frequencies are displayed by the heights of vertical bars, which are arranged in order from highest to lowest.

51. *Example 2-9 P(70): Turnpike costs:
Class (State)
Freq.
Indiana
2.9
Oklahoma
4.3
Florida 6
Maine
3.8
Pennsylvania
5.8

52. Time series graph represents data that occur
over a specific period of time.
year
Freq.
2000 11
2001 10
2002 16
2003 15
2004 18

53. A pie graph is a circle that is divided into sections or wedges according to the percentage of frequencies in each category of the distribution.

54. Example 2-12:
Construct a pie graph showing the blood types of the army inductees described in example 2-1 .
Class
Frequency
percent A 5 B 7 O 9 AB 4
Total 25

55. A stem and leaf plots is a data plot that uses part of a data value as the stem and part of the data value as the leaf to form groups or classes.
The stem and leaf plot is a method of organizing data and is a combination of sorting and graphing.
It has the advantage over a grouped frequency distribution of retaining the actual data while showing them in graphical form.

56. Example 2-13: At an outpatient testing center, the number of cardiograms performed each day for 20 days is shown. Construct a stem and leaf plot for the data.

57. The class width is the same for any class.
Formulas (CH#2)
Range: R= highest value – lowest value
Class width:
c.w= L2-L1 Or
c.w= U2-U1
The class width is the same for any class.
Class Limits
24 – 30
31 – 37
c.w= R/ no. of classes

58. Xm = Xm = ** Total of frequency= n= number of observations
Formulas (CH#2)
** Class midpoint
Xm =
Xm =
** Total of frequency= n= number of observations
** Relative Frequency= (f/n)
** Sum of relative frequency= 1
** Percentage= (f/n) * 100
** Sum of percentages= 100
** Class Width should be: Odd number
** Degree= (f/n) * 360
** sum of degrees= 360
** Less than  Cumulative frequency

59. Review
CH # 2

60. The following data represent the fire in the weak for 4 city
The following data represent the fire in the weak for 4 city. Choose the correct frequency distribution to organize the data:
Class Limit
Frequency
4-7 2
7-11 3
11-14 6
14-17 8
Class Limit
Frequency
4-7 2
8-11 5
12-15 9
16-19
Class Limit
Frequency
4-7 2
7-10 3
10-13 1
13-16 5

61. What are the boundaries of 49.005 ounces ?
The class width for the class is: 5 6 33 28

62. The following table shows the frequency distribution of temperature (in degree centigrade) of 30 countries : Class Limit Frequency Use the above table to answer questions: a) The number of countries with temperature less than 44.5 is: b)The midpoint in the first class is: c) The percentage of values in second class is :

63. What graph should be used to show the relationship between the parts and the whole ?
Histogram
Pie graph
Pareto graph
Ogives
In a pie chart, if the percentage of married people is 40 %. Then their corresponding degree of the angle is:
144⁰
151.1⁰
216.8⁰
180⁰

64. In a stem and leaf plot, the leaf part for the data value 347 is 34 7
Which graph should be used to represent "types of cars sold in Jeddah"?
A) Ogive
B) Histogram
C) Pareto chart
D) Time series graph

65. ……. can be drown by using vertical or horizontal bars :
Histogram
Bar graph
Pareto chart
Pie graph
Other name for ogive is:
a) Histogram
b) Frequency Polygon
c) Cumulative frequency graph
d) Pareto Chart

66. The graph has ……….. Peak:
a)Two
b)Three
c) One
d) None
Types of graph is:
a) Frequency polygon
b) Ogive
c) Cumulative frequency
d) Curve

67. Data collected over period of time can be graphed using ………..
a) Pareto graph
b) Pie graph
c) Time serious
d) Curve
what the correct stem and leaf plot which represent the :following data

68. In a stem and leaf plot, the leaf part for the data value 347 is

70. Class
Frequency
50-52 5
53-55 8
56-58 12
59-61 13
62-64 11
The number whose height or less than 55.5 inches 5 13 8 25

71. The heights of vertical bars in the histogram represent the
a)Class width
b)Sample size
c)Frequencies of classes
d)Number of classes
Which graph should be used to represent the relationship between years of production and number of buses product?