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Engineering StatisticsLecture Number: 01

COURSE CODE: GS – 301 COURSE INSTRUCTOR: ZAFFER

ELAHI

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Lecture Day/Timing:Monday 08:30 – 11:00 a.m. (Even Roll Nos.)Tuesday 08:30 – 11:00 a.m. (Environmental)

Wednesday 08:30 – 11:00 a.m. (Odd Roll Nos.) General Instructions:

All announcements regarding change in class will be made through class representative (C.R).

Ensure the punctuality/ regularity in the class. Late comer students will not be entertained (in any case). Put your assignment/s to the instructor office before the upcoming class otherwise it will

not entertained. No separate quiz will be conducted if the student/s miss the class quiz. (Due to any reason) Every one must have his/ her own scientific calculator in the class. Use of mobile phone

as a calculator is not allowed in the class, Exam or in the quiz. Switch off the mobile phone in the class.

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Serial No. Marks Distribution % weight

1 Quiz 10

2 Assignment 10

3 Mid Term Exam 20

4 Final Term Exam 60

Total 100

Grading Criteria

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Week Topic/s Covered

1 Introduction to Statistics

2 of Data

3 Measures of central tendency, Variance, standard deviation

4 Counting Principle, Probability & its Elementary Theorems

5 Conditional probability, Bay’s Theorem

6 Mathematical expectation and decision making

7 Regression Correlation & Rank Correlation

8 Regression analysis by least square methods incorporating linear, polynomial, exponential & power function.

9 Probability Distributions

10 Binomials Distribution

11 Poisson processes

12 Probability densities

13 Normal Distribution

14 Students t- Distribution, Chi-square Distribution

15 Sampling Distributions

16 The sampling distribution of the mean (known and unknown) & variance

COURSE CONTENTS

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APPLIED STATISTICS AND PROBABILITY FOR ENGINEERS, BY DOUGLAS C. MONTGOMERY

PROBABILITY FOR ENGINEERS BY IRWIN MILLER, JOHN E FREUND

STATISTICAL METHODS FOR ENGINEERING & SCIENTISTS, BY WALPOL & MEYERS

INTRODUCTION TO STATISTICS THEORY, BY SHER MUHAMMAD CHUADHRY

Recommended Books

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The word “statistics” which comes from the Latin word status, meaning a political state, originally meant information useful to the state, e.g., information about the size of populations and armed forces. But this word has now acquired different meanings.

1. The word statistics refers to “numerical facts systematically arranged”.

2. The word statistics is defined as a discipline that includes procedures and techniques used to collect, process and analyze numerical data to make inferences and to reach decisions in the face of uncertainty.

3. The word statistics are numerical quantities calculated from sample observations; a single quantity that has been calculated is called a statistic.

Introduction to Statistics

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Main Branches Of Statistics

Theoretical Statistics

Design Experimen

t

Descriptive Statistics

Statistical Inference

Applied Statistics

Branches Of Statistics

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Design of Experiment:

is a sequence of steps taken to collect appropriate data for objective analysis to draw valid inferences with respect to a

problem under investigation. Descriptive Statistics:

is that branch of statistics that deals with concepts and methods concerned with summarization and description of the important aspects of numerical data.

Statistical Inference:

is that branch of statistics that deals with drawing valid conclusions about the population parameters on the basis of sample data along with an associated degree of their reliability.

Theoretical Statistics is a branch of statistics that formulates statistical methods and general rules to provide a basis for investigation into

specific problem.

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A/S is a branch of statistics that makes use of statistical methods and general rules in the investigation of a specific problem.

This branch is applied in the field of: Biometry Psychometric Genetics Engineering Physics Chemistry Banking Operation research Econometrics etc.

Applied Statistics

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Population: is the totality of the observations made on all the objects

possessing some common characteristic.

For instance:• Height of college students• Wages of skilled labor in company• Wheat prices in different markets of Pakistan etc.

A population may be finite or infinite.

Size of Population: The number of observations in a finite population.

Notation: It is denoted by N

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Parameter:is a numerical characteristic of a population such as its mean

or standard deviation etc.

Sample:is a representative part of the population which is selected to

obtain information concerning the characteristics of the population.

Size of Sample:Number of observations in a sample is called the size of sample.

Notation: It is denoted by n.

Statistic: is a numerical characteristic of a sample such as its mean or

standard deviation “ s” etc.

""""

""x

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In statistics, an observation means any sort of numerically recording of

information, whether it is a physical measurement such as height or weight; a

classification such as heads or tails, or an answer to a question such as Yes or

No.

Variables:

Any characteristic, which varies either in quantity or quality from one

individual to the other, is called a variable.

Examples:• Height of individuals• Weight of persons• Family size• Number of petals of flowers etc.

Observations And Variables

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Constant:A characteristics is called a constant if its domain contains only

one value.

Quantitative Variable:A characteristic, which varies only in quantity from one

individual to another, is called quantitative variable. It is also called Variate.

Examples:Wages, Prices, barometric readings, height, weights, etc.

Qualitative Variable:A characteristic, which varies only in quality from one

individual to another, is called qualitative variable. It is also called Attribute.

Examples:Marital status, Educational status, deafness, blindness, beauty etc.

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A variable, which can take only specified values, is called

a Discrete variable. Such as , number of rooms in house, number of heads in tossing a coin 4-times, size of family etc.

A variable that theoretically can assume any value

(fraction or integer) between two specified limits a and b, is called Continuous variable. Such as height of plant, weight of commodity, speed of car, temperature at a place, etc.

Continuous and Discrete Variable

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There are three main types for presentation of data. Classification Tabulation Graphical Display

Presentation of Data

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Classification is the sorting of data into homogeneousclasses or groups according to their being alike or not. (OR)Process of dividing a set of data into classes or groupsin such a way that, Observation in same class are similar Observation in each class are dissimilar to the other

classes.

Classification

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It is the systematic presentation of classified data under suitable heading and placed in the forms of rows and columns.

This sort of logical arrangement makes the data easy to understand, facilitates comparison and provides effective way to convey information to reader.

Tabulation

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AreaPopulation Male Female

M U.M. T M U.M. T M U.M. T

Urban25688 21072 46740 10934 12149 23083 14739 8923 23657

Rural133056 91280 224336 62200 54416 116616 70856 36864 107720

District158724 112352 271076 73134 66565 139699 85590 45787 131377

Population of district by marital status residence and sex

M: MarriedU.M.: UnmarriedT: Total

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A visual representation of statistical data in the form of lines, area and other geometrical shapes is known as graphical representation.

Graphical Display is further divided into two types. These types are as follow:

Graph Diagram

Graphical Display

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The organization of data in a table which shows distribution of data into classes or groups together with the number of observation in each class is called frequency distribution.

The number of observation in each class is referred as frequency.

Frequency Distribution

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Class Limit Class Boundary Class Interval Class Mark ( Midpoint value)

Frequency distribution terms

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106, 107, 76, 82, 109, 107, 115, 93, 187, 95, 123, 125, 111, 92, 86,

70,110,126, 68, 130, 129, 139, 115, 128, 100, 186, 84, 99, 113, 204, 111,

141, 136, 123,

90, 115, 98, 110, 78, 185, 162, 178, 140, 152, 173, 146, 158, 194, 148, 90,

107, 181, 131, 75, 184, 104, 110, 80, 118, 82.

Range: 204 – 68 = 136, Class size: 136/7=19.47 = 20 = h (say)

No. of Classes: k = 1+ 3.3 log (n) = 7;n = Total no. of observation

Make a Classification of data into groups

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Class LimitsClass

Boundaries Class MarksTally

Marks Frequency

65-84 64.5-84.5 74.5 IIII IIII 9

85-104 84.5-104.5 94.5 IIII IIII 10

105-124 104.5-124.5 114.5IIII IIII IIII II

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125-144 124.5-144.5 134.5 IIII IIII 10

145-164 144.5-164.5 154.5 IIII I 6

165-184 164.5-184.5 174.5 IIII 4

185-204 184.5-204.5 194.5 IIII 5

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A graph is a representation of data by continuous curve

Diagram is any other form of visual representation.

Difference between Graph and Diagram

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Simple Bar Chart:A simple bar chart consist of horizontal or vertical bar of equal widths and lengths equal to value represented by frequency.

Diagrams

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Years 1965 1966 1967 1968 1969

Turnover (in Dollars) 35000 42000 43500 48000 48500

Example: Draw a simple bar diagram to represent the turnover of a company for 5 years

1965 1966 1967 1968 19690

10000

20000

30000

40000

50000

60000

Years

Years

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A multiple Bar chart shows two or more characteristics corresponding to value of a common variable in the form of grouped bars whose lengths are proportional to the value of the characteristics and each bar is colored or shaped differently.

Example:Draw Multiple bar diagram to show area and production of cotton from the following data

Multiple Bar Chart

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Year Area Production1965-66 2866 1588

1970-71 3233 2229

1975-76 3420 1937

1965-66 1970-71 1975-760

500

1000

1500

2000

2500

3000

3500

4000

Chart Title

Area Production

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A component bar chart is an effective technique in which each bar is divided

into two or more sections proportional in size to component part of total being

displayed by each bar.

Example: Draw a component Bar chart of Population city wise

Component Bar Chart

Cities Total Male Female

Peshawar 64 33 31

Rawalpindi 40 21 19

Sargodha 60 32 28

Lahore 65 35 30

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Component Bar Chart

Peshawar Rawalpindi Sargodha Lahore0

10

20

30

40

50

60

70

Chart Title

Male Female

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Pie Diagram

 

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Items Food Clothing House rent

Fuel Misc.

Expenditure 50 30 20 15 35

Angle of Sector 50/150*360= 120

30/150*360= 72

20/150*360= 48

15/150*360= 36

35/150*360= 84

Example: Represent the total expenditure and expenditure of various items of a family by the Pie diagram.

Food Clothing House Rent Fuel Misc

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Historigram:

A curve showing changes in the value of one or more item from one period to next period of time is known as Historigram.

Graphs

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Year 1929 1930 1931 1932 1933 1934 1935 1936No. of Cars

98 74 68 50 99 172 245 302

Example

1929 1929 1931 1932 1933 1934 1935 19360

50

100

150

200

250

300

350

Historigram

number of car

No o

f c

ars

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A histogram consist of a set of adjacent rect-angles whose bases are marked off by classboundaries or the X-axis and whose height areproportional to frequency associated withrespective classes.

Histogram

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A frequency polygon is a graphic form of a frequency distribution which is constructed by plotting the class-marks along x-axis and frequency along y-axis.

Frequency polygon

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When a frequency polygon or histogram constructed over class intervals made sufficiently small for a large number of observations , is smoothed, it approaches a continuous curve called a frequency curve.

Frequency curve

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Age (Years) 18-19 20-24 25-29 30-34 35-44 45-54

No. of Operators 9 188 160 123 84 15

Example: Construct a histogram for following frequency Distribution relating ages of telephone operators

Class Boundaries Class Interval Proportional Height17.5-19.5 2 4.5

19.5-24.5 5 37.6

24.5-29.5 5 32

29.5-34.5 5 24.6

34.5-44.5 10 8.6

44.5-54.5 15 1