Probability Theory

1. Lecture 1
Topics: Introduction to Statistics
and Data Analysis
SADAF SALEEM
Department of Computer Sciences
GIFT University, Pakistan
Course Title: Probability Theory
Course Code: MATH-313
Program: BS MATH
Semester: Spring 2023

2. Introduction
Statistics:
• The practice of collecting and analyzing the data
• Use :Areas involve the gathering of information or scientific data
Probability:
• The chance of an outcome or event to occur.
• Examples: Toss a die
Flip a coin
Draw a card
↓
Randomness Deterministic Random
Models
↓
↓
Output can be expressed
exactly with help of a
mathematical equation
Output cannot be
expressed exactly with
help of a mathematical
equation

3. Introduction
• Three different types of probability
• Theoretical Probability → based on some laws
• Empirical probability →based on historical and geological records
• Subjective Probability →based upon intuition
• Circle of Trust

4. Introduction to Statistics
and Data Analysis

5. Data:
• Data: it is Latin for “those that are given”
: Can be thought as the results of the observation
• For Example:
1. Statements given to a police officer or physician or psychologist in a interview is a data
2. Correct and incorrect answers by a student on a final exam
3. Time required by a runner to complete a marathon
4. Positions of aircrafts
5. The spectral composition of light emitted by a star

6. Types of data:
• Types of data:
Quantitative:
When the data studied can be reported numerically, the data is called a
quantitative data
Examples: age, weight, income
Qualitative:
When the characteristic being studied is nonnumeric, it is called a qualitative data
Examples: education, eye color, poverty

7. Types of data:
• Types of Quantitative Date:
Discrete Data : can take only a discrete set of integers or whole numbers
Examples: No of persons in a house
No of room in a house
No of death in an accident
Continuous Data : can take on any value fractional or integer
Examples: Height of a plant
Weight of student
Temperature of a room
Age of a person

8. Types of data:
Types of Data

9. Study of Quantitative Data

10. Study of Quantitative Data
• As show in the previous Chart data can be studied by two different Parts:
Modeling:
In this process data is represented by different plot and charts and tables.
In this form of study we will take an overview of the data
Analysis:
This is the part we summarizes the data in two or three number such that mean, median, mode
etc.
The quantitative data can be grouped or ungrouped.
The methods to apply theses statistical tool to grouped and ungrouped data are different.
We will study each part of analysis in deal in coming lectures

11. Data Modeling:
Frequency Table:
• Data can be arranged in the form of a table
• Table consist of 2 or some time 3 columns.
• First column represent the range of the data
• Second column represent the Frequency of data
• Frequency: total no of observation which are present in that range
Example: suppose you have the following data points
Solution:
Identify the minimum and the maximum data point.
Choose a range for the class,
There is no hard and fast rule to do that.
Tips: avoid any class have zero entry
:5-8 classes are recommended. This may very with the no of data points

12. Frequency Table
Example 1:
Solution:
Data range Frequency
-10 to -1 3
0-9 13
10-19 2
20-29 1
30-39 1

13. Frequency Table
Example 2:
Solution:
Data range Frequency
6.5-7.4 4
7.5-8.4 1
8.5-9.4 5
9.5-10.4 7
10.5-11.4 3

14. Frequency Table
The third column some is the relative frequency
Relative frequency= frequency of the class/ total number of the observations
Example 1 Example 2
Data range Frequency Relative
Frequency
-10 to -9 3 3/20
0-9 13 13/20
10-19 2 2/30
20-29 1 1/30
30-39 1 1/30
Data range Frequency Relative
frequency
6.5-7.4 4 4/20
7.5-8.4 1 1/20
8.5-9.4 5 5/20
9.5-10.4 7 7/20
10.5-11.4 3 3/20

15. Statistical Modeling
• The importance of characterizing or summarizing the nature of collections of data should be
obvious.
• Often a summary of a collection of data via a graphical display can provide insight regarding the
system from which the data were taken.
1. Dot Plot
2. Scatter Plot
3. Stem and Leaf Plot
4. Histogram
5. Box and Whisker Plot/ Box Plot

16. Dot Plot:
Find the Max. and Min. of the data
Length and Scaling of the line should appropriate
Example 1: 7,7,9,8,10
:19,20,21,20,22
:2,5,4,9,12,17,20
Example 2: Sketch the dot plot of the given data:
You can represent two different observation on the same line

17. Scatter Plot:
Sketch the Scatter Plot of the given data:
• What cotton % is more distinct from other?
• Does cotton % enhance the tensile strength?
• For which cotten % population mean is higher?

18. Stem and Leaf plot: case 1
Step 1: Split the observation into two parts
Step 2: Generate Table

19. Double Stem and Leaf plot:

20. Class Practice:
• Construct a stem and Leaf plot for following data
• Construct double Stem and leaf plot

21. Stem and Leaf plot: Case 2
• When data is in whole number
Step 1: observe max and min of data
Step 2: modify data
Step 3: break into stem and leaf
Step 4: draw table
Stem Leaf Frequency
0 1 3 7 8 9 5
1 0 3 7 4 4 5 6 7 8
2 0 1 2
Stem Leaf Frequency
0 1 3 2
0* 7 8 9 3
1 0 3 4 4 4
1* 5 6 7 7 4
2 0 1 2

22. Stem and Leaf plot: Case 2

23. Stem and Leaf plot: Case 3
When data is in whole number (big one)
• Prices of 100 cars from a dealer ($)
18800, 19100, 19600,18885, 19075….
When data is in mixed number
• 21.8 to 74.9
Stem and Leaf plot: Case 4

24. SADAF SALEEM
Department of Computer Sciences
GIFT University, Pakistan