Probability and Statistics

Charts and Frequency Distributions

When the variable of interest is qualitative, the statistical table is a list of the categories being considered along with measure of how often each value occurred. The measure can be presented in the following way:1. The frequency or number of measurements in

each category2. The relative frequency, or proportion of

measurements in each category3. The percentage of measurement in each

category

1. 4000 freshmen were admitted in Don Bosco Technical College in Mandaluyong for the school year 2010-2011. The Students were enrolled in the following program:

Example

Program Number of Students

Architecture 320

Computer Engineering 440

Computer Science 720

Electronics & Communication

1080

Entrepreneurship 800

Information Technology 400

Mechanical Engineering 240

Total 4000

Bar Graph Uses the height of the bar to display the amount in a

particular category.

Pie Chart Displays how the total quantity is distributed among the

categories.

Program Frequency

Relative

Percent

Architecture 320 0.08 8%

Computer Engineering 440 0.11 11%

Computer Science 720 0.18 18%

Electronics & Communication 1080 0.27 27%

Entrepreneurship 800 0.20 20%

Information Technology 400 0.10 10%

Mechanical Engineering 240 0.06 6%

Total 4000 1.00 100%

When a quantitative variable is recorded over time at equally spaced intervals, the data set forms a time series. Time series data are most effectively presented on a line chart.

Example. Table that shows the daily production of Gardenia Bread

Day 1 2 3 4 5 6 7

Number of Loaves(In Thousands)

200 190 230 170 240 250 245

Line Graph

A stem and leaf plot presents a graphical display of the data using the actual numerical values of each data point.

Steps in constructing:1. Divide each measurement into two parts: stem

and leaf2. List the stem in column, with a vertical line to the

right.3. For each measurement, record the leaf portion in

the same row as its corresponding stem.4. Order the leaves from lowest to highest in each

stem.

Stem and Leaf Plot

Daily sales of desktop computers of JRC Computer Company for 40 days.

Example

34 40 31 33 20 25 51 62

45 30 38 45 61 42 30 28

35 31 28 42 39 40 52 43

36 46 48 51 52 47 42 39

40 31 29 33 47 36 45 21

2 8 9 0 8 1

3 4 5 6 0 1 1 1 8 3 3 9 6 0 9

4 5 0 0 6 8 5 2 7 2 0 7 2 5 3

5 1 2 1 2

6 1 2

Solution

2 0 1 8 8 9

3 0 0 1 1 1 3 3 4 5 6 6 8 9 9

4 0 0 0 2 2 2 3 5 5 5 6 7 7 8

5 1 1 2 2

6 1 2

REORDERING

Steps in Constructing a Frequency Distribution Table

1. Determine the number of classes by using Sturges’ Formula:

K = 1 + 3.322 log n = approximate number of

classes n = number of observations

2. Determine the approximate class size. Whenever possible, all classes should be of the same size. The following steps can be used to determine the class size:* Solve for the range, R = max- min*Compute for C’ = R / K*Round-off C’ to a convenient number(nearest whole number)

Frequency Distribution

Steps in Constructing a Frequency Distribution Table

3. Determine the lowest class limit. The first class must include the smallest value in the data set.

4. Determine all class limits by adding the class size , C, to the limit of the previous class.

5. Tally the frequencies for each class. Sum the frequencies and check against the total number of observations.

Frequency Distribution

Construct a frequency distribution from the final grades of Stat 101 Students given below:

Example

82 82 83 79 72 71 84 59 77 50 87

83 82 63 75 50 85 76 79 68 69 62

79 69 74 53 73 71 50 76 57 81 62

72 88 84 80 68 50 74 84 71 73 68

71 80 72 60 81 89 94 80 84 81 50

84 76 75 82 76 53 91 69 60 89 79

59 62 79 82 72 81 60 84 68 66 94

77 78 87 75 86 82 74 73 72 84 51

50 69 75 70 77 87 86 77 75 96 66

87 73 84 68 85 62 87 92 69 52 65

1. Construct a stem and leaf plot.

Solution

5 0 0 0 0 0 0 1 2 3 3 7 9 9

6 0 0 0 2 2 2 2 3 5 6 6 8 8 8 8 8 9 9 9 9 9

7 0 1 1 1 1 2 2 2 2 2 3 3 3 3 4 4 4 5 5 5 5 5 6 6 6 6 7 7 7 7 8 9 9 9 9 9

8 0 0 0 1 1 1 1 2 2 2 2 2 2 3 3 4 4 4 4 4 4 4 5 5 6 6 7 7 7 7 7 7 8 9 9

9 1 2 4 4 6

The Complete Frequency Distribution Table

Class Frequency

LCB UCB RF RFP <CF >CF

Graphical Representation of a Frequency Distribution

1. Frequency Histogram – a bar graph that displays the classes on the horizontal axis and frequencies of the classes on the vertical axis; the vertical lines of the bars are erected at the class boundaries and the height of the bars correspond to the class frequency.

2. Relative Frequency Histogram – a graph that displays the classes on the horizontal axis and the relative frequencies on the vertical axis.

Graphical Representation of a Frequency Distribution

3. Frequency Polygon – a line chart that is constructed by plotting the frequencies at the class marks and connecting the plotted points by means of straight lines; the polygon is closed by considering an additional class at each end and the ends of the lines are brought down to the horizontal axis at the midpoints of the additional classes.

4. Ogives – graphs of the cumulative frequency distribution.a. < ogive – the <CF is plotted against the UCBb. > ogive – the >CF is plotted against the LCB

The Categorical frequency distribution is used for data that can be placed in specific categories, such as nominal or ordinal-level data.

Categorical Frequency Distribution

Twenty five students were given the following grades. The data set is:

C A B A DF B A C AC D F B BA B D F CB C C B D

Example

a. Make a table with A, B, C, D, and F as classes.

b. Tally the data and count the tallies.c. Find the percentage of values in each class

by using the formula

d. Find the total frequency and percent .

Solution

%100% 1 xb

b

o

Class Frequency Percent

A 5 20

B 7 28

C 6 24

D 4 16

F 3 12

TOTAL 25 100

Table of frequency and percent