2
Lecture Four : Introductory Statistics
Monte Carlo Methods
Monte Carlo Methods
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Introductory Statistics-Stochastic Process-Lecture Slides, Slides of Stochastic Processes
Main topics for this course are Stochastic process, random variables, linear congruent generators, pdfs and cdfs, rejection method, metropolis methods, sampling techniques, random walks and genetic algorithm. This lecture includes: Introductory, Statistics, Population, Sample, Inductive, Deductive, Discrete, Continuous, Variable, Graph, Frequency, Distribution, Tabular
Typology: Slides
2011/2012
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Lecture Four :
Introductory Statistics
Monte Carlo Methods
Monte Carlo Methods
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Basic Ideas Population:
The whole group of objects to be analyzed is called
Population or universe. Sample:
When we examines a small part of the population then
it is termed as Sample.
Statistics is concerned with the scientific methods forcollecting, organizing, summarizing, presenting andanalyzing data.
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Graphs: bar graphs
africa
asia
europe
north america
oceania
south america
russia
0
5
10
15
20
25
30
Area (millions of square Kilometers)
Continent
area
africa
asia
europe
north america
oceania
south america
russia
Table showing area of various continents of the world.
This figure shows a bargraph in which bars arehorizontal instead ofvertical.
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Graphs: bar graphs
Continent
area
africa
asia
europe
north america
oceania
south america
russia
Table showing area of various continents of the world.
This figure shows a bargraph in vertical form.
africa
asia
europe n. america oceania s. america
russia
5 0 30 25 20 15 10
Area (millions of square kilometers)
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Graphs: bar graphs
time
Value
Table showing area of various continents of the world.
This figure shows a line graph.
0
2
4
6
8
10
0
80 60 40 20
140 120 100
number of tonnes
time (year)
scales
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Example: Radioactive decay Plot this data on a linear graph paper.
Time (min)
R(counts/s)
0
50
100
150
200
0 50 450 400 350 300 250 200 150 100
Count Rate (# per sec)
Time (min)
data values
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11
0
50
100
150
200
1
10
100
1000 Count Rate
Time (min)
data value
Time(min)
R
(counts/s)
4
392.
36
161.
68
65.
100
26.
132
10.
164
4.
196
1.
218
1
Example: Radioactive decay
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Inverse Square Law
r (cm)
Dose (μSv/h)
Exposure X
(mR/h)
The exposure and doserate decreases as thedistance between thesource and the personincreases. The governing relationis known as Inversesquare Law
2
r
A
X
Γ
=
&
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Frequency Distributions Raw Data:
It is the collected data that has not been organized yet.
Arrays:
An array is an arrangement of the raw numerical data in ascending
or descending order of magnitude. Range of the data
is the difference between maximum and minimum of data.
Weight
(kg)
Number of
students
Total:
Data is often distributed into classesor categories or bins. Number of classes
depend on user.
And to determine number ofindividuals in each class is calledclass frequency. The tabular for of data in form offrequency versus classes is calledfrequency distribution
shown here in
table.
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Frequency Distribution
Class Limits
Class
Boundaries
Class
Mark (X)
Number of students (f)
Total:
Class intervals and
Limits:
The class interval is forexample 60 –
in this table
for first class. The endnumbers are class limits. Class
Boundaries:
If weight
is recorded to nearestkilogram then measurementwill include 59.5 to 62.5 in thefirst class. These are realboundaries. The Size or class inreval:
It
is the difference between thelower and upper classboundaries. Class Mark:
It is the mid
point of the class interval. Forfirst class (60 + 62)/2 = 61
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Histogram and Frequency Polygon
60
62
64
66
68
70
72
5 0 45 40 35 30 25 20 15 10
Frequency
Weight, X, (kg)
Frequency Polygonconsists of a linegraph of the classfrequency (f) plottedagainst the classmark (X). This is shown here.
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Cumulative Frequency Distribution
60
62
64
66
68
70
72
74
76
0 80 60 40 20 100
Frequency
Weight, X, (kg)
Weight
(kg)
Number of
students
Less than 59.
Less than 62.
Less than 65.
Less than 68.
Less than 71.
Less than 74.
The total frequency of all values less than the upper class boundary of agiven class is called the cumulative frequency upto
that class.
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Median and Mode Median For a set of numbers arranged in order of magnitude, it is eithermiddle value or the AM of two middle values. Example:
The set of numbers 3,4,4,6,8,8,8 and 10 has median
6. Mode The mode of a set of numbers is the value that is repeatedmaximum times. Example:
The set 2, 2, 5, 7, 9, 9, 9, 10, 11, 12
and
18 has
mode 9.
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Mean Value Calculations
Weight
(kg)
Class Mark
(X)
Number of
students
( f )
f X
Total sum =
Example: Average weight ofthe students.
=
fX
f
= 6745 / 100
= 67.45 kg
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