222TME100
Design of Experiments
Feb-May 2025
A Slot
Dr. Mubarak Ali
Department of Mechanical Engineering
TKM College of Engineering
1
Reference -1
Applied Statistics and
Probability for Engineers,
Douglas C. Montgomery,
George C. Runger,
John Wiley & Sons, Inc.,
6e, 2014.
2
Reference -2
Introduction to Statistical
Quality Control,
Douglas C. Montgomery,
John Wiley & Sons, Inc.,
6e, 2009
3
Why should an Engineering Student study
Design of Experiments?
1.
2.
3.
4.
Because it is in our curriculum, you know
Well, it is an easy subject
To get good GPA
I don’t know
All the above are Erroneous Answers
Course Outcomes (COs)
CO1
CO2
CO3
CO4
CO5
CO6
Perform statistical analysis of data.
Conduct statistical hypothesis tests on mean and variance of
populations.
Design and analyze single factor experiments.
Design and analyze full and fractional factorial experiments.
Apply Response Surface Methodology to optimize the
response in an experiment.
Carry out an experimental project and analyze the results
using a statistical software
Module 1
6
Who is an Engineer?
• An engineer is someone who solves problems of
interest to society by the efficient application of
scientific principles.
How?
Engineers accomplish their objectives either
1. by refining/improving an existing product or process
2. by designing a new product or process that meets
customers’ needs
7
Statistics
• Deals with the Collection, Processing, Analysis,
Interpretation, Prediction and Presentation of data
Application
• To make decisions & solve problems,
• To design products and improving existing products
• To design, develop & improve production processes
or business processes
8
Statistical Thinking
• All of us think statistically everyday
• Example:
Fuel Mileage of your bike
Time taken to reach your institute
Expected arrival time of a train
Data
Data is any factual information or measurement that is
collected and analyzed to make a decision, change or
any calculation
9
Experiment
Experiment is a test usually performed to discover something about
the process or system.
10
Experiment 1
Investigations on Workpiece/Tool Interface
Temperature during Vertical Milling
11
Vertical Milling Machine
• Major portion of the energy dissipated
in milling is converted into heat.
• Temperature of cutting zone affects
strength, wear resistance of tool,
dimensional accuracy of sample,
properties and life of the tool and
workpiece.
• Temperature to be measured using
infrared thermometer (non contact)
• Main Factors affecting Temperature:
Feed and Speed
12
Factor Levels in Experiment 1
Factor A: Feed (in mm/min)
Lower Feed: -1
Higher Feed: +1
Factor B: Speed
Lower Speed: -1
Higher Speed: +1
Response Variable (output): Temperature of workpiece/tool interface
Use principle of design and analysis of experiment to conduct the
experiment at all combinations and to analyse the data
13
Categories of Statistics
Tabular Methods
Graphical Methods
Numerical Methods
Confidence Intervals,
Hypothesis Testing
Parameter Estimation
14
Descriptive Statistics
Tabular and
Graphical methods
15
Tabular and graphical methods for
qualitative data
1. Frequency Distribution
SOFT DRINK
FREQUENCY
Coke Classic
19
Diet Coke
8
Dr. Pepper
5
Pepsi
13
Sprite
5
Total
50
16
Contd.
2. Relative and Percent Frequency Distribution
Relative Frequency = Frequency of a class/n
Percent Frequency = Relative Frequency x 100
17
Contd.
3. Bar Charts (or
Pareto Diagram)
4. Pie Charts
18
Tabular and graphical methods for
Quantitative data
1. Frequency Distribution
Data show the time in days required to complete year end audits
for 20 clients of an accounting firm
Year-end audit time in days
12
14
19
18
15
15
18
17
20
27
22
23
22
21
33
28
14
18
16
13
Audit Time(Days)
Frequency
10–14
4
15–19
8
20–24
5
25–29
2
30–34
1
Total
20
19
Contd.
2. Dot Plot
a
3. Histogram
20
Contd.
4. Cumulative Distributions
Ogive
graph
21
Contd.
5. Stem and Leaf Plot
NUMBER OF QUESTIONS ANSWERED CORRECTLY ON AN APTITUDE TEST
22
Contd.
6. Scatter Diagram and Trend line
SAMPLE DATA FOR THE STEREO AND SOUND EQUIPMENT STORE
23
Contd.
Types of Relationship Depicted by Scatter Diagram
24
Descriptive Statistics
Numerical methods
25
Numerical Methods of Descriptive Statistics
26
Measures of Central Tendency
1. Mean: It is the arithmetic average of all values:
Mean = Sum of all values /Total number of values
2. Median: It is the central value of the data points,
when arranged in ascending or descending order
With an odd number of observations, the median is the middle
value. An even number of observations has no single middle
value so in this case, we take average of the values for the
middle two observations
3. Mode: It is the most frequently occurring value .
Based on the nature of data, any one of these is used.
27
Measures of Dispersion
1. Range: It represents the gap between the highest
and lowest value in the group.
Range = Maximum value – Minimum Value
2. InterQuartile Range:
Percentile is a measure indicating the value below
which a given percentage of observations in a group
of observations fall.
28
contd.
InterQuartile Range (IQR):-A measure of variability that
overcomes the dependency on extreme values
29
Find IQR
No of questions answered correctly in an aptitude exam
Solution
Step1: Arrange all numbers in ascending order
68,69,72,73,73,75,76,76,80,81,
81,82,83,84,85,86,91,92,92,92,
94,95,95,96,97,98,98,100,100,102,
104,106,106,106,107,108,112,113,115,115,
118,119,119,124,126, 127,128,132,134,141
Step2: divide the data into two half sets (marked by diff colors)
Step3: Find medians entire data set, and also for each half sets)
Minimum = 68
Maximum = 141
Q2 = 97.5 (Median of entire data set)
Q1 = 83 (Median of First half set)
Q3 = 113 (Median of Second half set)
IQR = Q3-Q1 = 113-83=30
31
Contd.
3. Variance:- The variance is a measure of variability that
utilizes all the data.
• The difference between each xi and the mean μ is called a deviation
about the mean.
• If the data are for a population, the average of the squared deviations
is called the population variance.
If sum of the squared deviations about the sample mean is divided by
n-1, the resulting sample variance provides an unbiased estimate of the
population variance
Note S2 =
32
Contd.
The sum of squared deviations about the mean is (xi-x)2 =256. Hence, with n-1= 4,
the sample variance is
33
Contd.
4. Standard Deviation:- The standard deviation is defined to
be the positive square root of the variance.
Hence the Sample Standard Deviation=8, for the latest example
34
Contd.
• For class size data, sample mean =44 and a sample standard
deviation =8. The coefficient of variation is [ x100]%
=18.2%. In words, the coefficient of variation tells us that the
sample standard deviation is 18.2% of the value of the sample
mean.
s=2
s=4
Sample 2 has greater
variability than Sample 1
s=2
Standard Deviation only reflects
scatter about the average
35
Contd.
6. Five Number Summary- The following five numbers are
used to summarize the data: 1.Smallest value 2.First quartile (Q1)
3.Median (Q2) 4. Third quartile (Q3) 5. Largest value
7. Box Plot:- Graphical display that simultaneously displays
several important features of the data, such as central tendency,
spread, departure from symmetry, and identification of observations
that lie unusually far from the bulk of the data (“outliers”).
36
Inferential Statistics
Using the information in a sample for drawing
conclusions about population
Terminology used in Inferential Statistics
Population: Set of all units of interest
Sample: Subset of population
Random sample: Sample collected in such a way
that every unit in population is equally likely to be
selected to ensure good overall representation of
population
Random sampling: The activity/procedure to extract
random samples from a population.
37
Inferential Statistics – Contd…
Confidence Interval:
• Charpy V-notch method for notched bar impact testing of metallic
materials.
• Test a sample of 10 specimens of a particular material. Use the sample
average to estimate the true mean impact strength μ.
• True mean impact strength is unlikely to be exactly equal to your
estimate.
• Reporting the results of your test as a single number is sometimes
unprofessional.
• Our estimate could be very close, or it could be considerably far from
the true mean.
• A way to avoid this problem - report the estimate in terms of a range of
possible values called a Confidence Interval.
• A confidence interval always specifies a confidence level, usually 90%,
95%, or 99%, which is a measure of the reliability of the procedure.
Inferential Statistics – Contd…
Hypothesis Testing:
• Suppose that two different reaction temperatures T1 and T2 can be used
in a chemical process.
• The engineer believes that T1 will result in higher yields than T2.
Hypothesis testing is the framework for solving problems of this type.
• In this Problem, the engineer would be interested in formulating
hypotheses that allow him or her to demonstrate that the mean yield
using T1 is higher than the mean yield using T2.
• There is no emphasis on estimating yields; instead, the focus is on
drawing conclusions about a hypothesis that is relevant to the
engineering decision.