MECH 373
Instrumentation and Measurements
Lecture 13
Statistical Analysis of Experimental Data
(Chapter 6)
• Introduction
• General Concepts and Definitions
• Probability
• Probability Distribution Function
Lecture 13
Lecture Notes on MECH 373 – Instrumentation and Measurements
1
Measures of Central Tendency
(review)

Mean (sample mean, population mean)
n
x1  x2    xn
xi
x

n
i 1 n



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N
x1  x2    x N
x

 i
N
i 1 N
Median
•
If the measured data are arranged in ascending or descending order,
the median is the value at the center of the set.
•
Odd: middle one
•
Even: average of the middle two
Mode
•
The mode is the value that occurs most often. If no number is
repeated, then there is no mode for the list.
Range
•
The range is just the difference between the largest and smallest
values.
Lecture Notes on MECH 373 – Instrumentation and Measurements
2
Measures of Dispersion
(review)




Dispersion
Spread or variability of the data
Deviation
di  xi  x
Mean deviation
n
di
i 1
n
d 
Standard deviation (Sample, Population)
( xi  x ) 2
S 
i 1 ( n  1)
( xi   ) 2
 
N
i 1
n

Variance
S2
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N
2
Lecture Notes on MECH 373 – Instrumentation and Measurements
3
Example 2
(review)
Find the mean, median, mode, range, standard deviation and variance of the
following dataset.
n  60
Median
x1  x2    xn
 11030 C
n
xm  1104 0 C
Mode :
m  1104 0 C
Range :
r  1089 0 C  11150 C
Standard deviation
S  5.79 0 C
Variance :
S 2  33.49 0 C
Mean
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x
Lecture Notes on MECH 373 – Instrumentation and Measurements
4
Basics of Probability

Probability
Probability is a numerical value expressing the likelihood of
occurrences of an event relative to all possibilities in a sample
space.
successful occurrences
Probability of event A =
total number of possible outcomes
•
•
•
•
•
•
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If A is certain to occur, P(A) = 1
If A is certain not to occur, P(A) = 0
If B is the complement of A, then P(B) = 1 - P(A)
If A and B are mutually exclusive (the probability of simultaneous
occurrence is zero), P(A or B) = P(A) + P(B)
If A and B are independent, P(AB) = P(A)P(B) (occurrence of both A
and B)
P(AB) = P(A) + P(B) - P(AB) (occurrence of A or B or both)
Lecture Notes on MECH 373 – Instrumentation and Measurements
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Probability Distribution Functions
• An important function of statistics is to use information from a sample
to predict the behavior of a population.
• We are often interested to know the probability that our next
measurement will be within certain range.
• One approach is to use the sample data directly (as shown in next
slide). This approach is called use of an empirical distribution.
• For particular situations, the distribution of random variable follows
certain mathematical functions.
• The sample data are used to compute parameters in these
mathematical functions, and then these mathematical functions are used
to predict behavior of the population.
• For discrete random variables, these functions are called probability
mass functions.
• For continuous random variables, these functions are called probability
density functions.
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Lecture Notes on MECH 373 – Instrumentation and Measurements
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Probability Distribution Functions
From: Dr. McNair
Lecture 13
Lecture Notes on MECH 373 – Instrumentation and Measurements
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Probability Distribution Functions

Probability Mass Function
If a discrete random variable can have values x1,…,xn,
then the probability of occurrence of a particular value of
xi is P(xi), where P is the probability mass function for the
variable x.
N
 Normalization
 P( xi )  1
i 1

N
   xi P( xi )
Mean
i 1

N
Variance
   ( xi   ) 2 P( xi )
2
i 1
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Lecture Notes on MECH 373 – Instrumentation and Measurements
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Probability Distribution Functions

Probability Mass Function
From: Dr. McNair
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Lecture Notes on MECH 373 – Instrumentation and Measurements
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Probability Distribution Functions

Probability Mass Function
From: Dr. McNair
Lecture 13
Lecture Notes on MECH 373 – Instrumentation and Measurements
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Probability Distribution Functions

Probability Density Function

Probability of occurrence in an interval xi and xi+dx
P( xi  x  xi  dx)  f ( xi )dx

Probability of occurrence in an interval [a,b]
b
P(a  x  b)   f ( x)dx
a

Mean of population – is also the expected value

   xf ( x)dx


Variance of population

 2   ( x   ) 2 f ( x)dx

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Lecture Notes on MECH 373 – Instrumentation and Measurements
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Probability Distribution Functions
•
Probability Density Function
From: Dr. McNair
Lecture 13
Lecture Notes on MECH 373 – Instrumentation and Measurements
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Probability Distribution Functions
•
Probability Density Function
From: Dr. McNair
Lecture 13
Lecture Notes on MECH 373 – Instrumentation and Measurements
13
Probability Density Functions
•
•
•
•
•
•
•
•
Example:
Ball bearing life probability distribution
function:
f(x) = 0 for x < 10h; f(x) = 200/x3 for x ≥ 10h
A, Expected bearing life (mean):
E(x) = u = ∫∞10 x.(200/x3) dx = 20h
B, Less than 20 hours.
P(x<20) = ∫10-∞ 0 dx + ∫2010 200/x3 dx = 0.75
P(x ≥20) = 1 – 0.75 = 0.25
From: Dr. McNair
Lecture 13
Lecture Notes on MECH 373 – Instrumentation and Measurements
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PDF with Engineering Applications
A number of distribution functions are used in
engineering applications. We will discuss briefly about
some common distribution functions.
►Binomial (values either true/false)
►Poisson
►Normal (Gaussian)
►Student’s t
► χ2 (Chi-squared)
►Weibull
►Exponential
►Uniform
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Lecture Notes on MECH 373 – Instrumentation and Measurements
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Probability Distribution Functions
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Probability Distribution Functions
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Probability Distribution Functions
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Probability Distribution Function
 Binomial Distribution
• The binomial distribution is a distribution which describes discrete random
variables that can have only two possible outcomes: “success” or “failure”.
• This distribution has applications in production quality control, when the
quality of a product is either acceptable or unacceptable.
• The binomial distribution provides the probability (P) of finding exactly r
successes in a total of n trials and is expressed as
n r
P(r )    p (1  p) n r
r
where, p is the probability of success which remains constant throughout the
experiment, and  n 
n!
  
 r  r!(n  r )!
is called as n combination r, which is the number of ways that we can choose r
identical items from n items.
• The expected number of successes in n trials for binomial distribution is   np
• The standard deviation of the binomial distribution is   np(1  p)
Lecture 13
Lecture Notes on MECH 373 – Instrumentation and Measurements
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Probability Distribution Function
 Binomial Distribution - Example
• Machines being tested.
• Success rate = 90% (10% failure).
• Probability of 15 successes (r) out of 20
machines (n)?
• P(15) = [20!/(5!.15!)]∙ 0.915 ∙ (1 – 0.9)(20-15)
• P(15) = 0.032
• 3.2% Chance having 5 out of a batch of 20
machines needs some repair.
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Lecture Notes on MECH 373 – Instrumentation and Measurements
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Probability Distribution Function
 Poisson Distribution
• The Poisson distribution is used to estimate the number of random occurrences
of an event in a specified interval of time or space if the average number of
occurrences is already known.
• The probability (P) of occurrence of x events is given by
e   x
P( x) 
x!
where, λ is the expected or mean number of occurrences during the interval of interest.
• The expected value of x for the Poisson distribution, the same as the mean (μ),
is given by
E (x)    
• The standard deviation is given by
 
• Probability that the number of occurrences is less than or equal to k is given by
e  i
P( x  k )  
i!
i 0
k
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Lecture Notes on MECH 373 – Instrumentation and Measurements
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Probability Distribution Function
 Poisson Distribution - Example
• Welded joint pipes having an average of five defects per 10
linear meters of weld (0.5 defects per meter).
• A, Probability of a single defect in a weld of 0.5m long.
– λ = Average number of defects in 0.5 m = 0.5 * 0.5 = 0.25.
– P(1) = e-0.25 0.251 / 1! = 0.194 (19.4%).
• B, More than one defect in weld of 0.5m long.
– Zero defect P(0) = e-0.25 0.250 / 0! = 0.778.
– P (x > 1) = 1 – 0.778 – 0.194 = 0.028 (2.8%)
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Lecture Notes on MECH 373 – Instrumentation and Measurements
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Probability Distribution Function
 Normal Distribution
• The Normal or Gaussian distribution is a simple distribution function that is useful for
a large number of common problems involving continuous random variables.
1
 ( x  ) 2 / 2  2
f ( x) 
e
 2
• Symmetric about μ
• Bell-shaped
• Mean μ : the peak of
the density occurs
• Standard deviation σ:
indicates the spread of
the bell curve.
-1
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0
1
2
2
Lecture Notes on MECH 373 – Instrumentation and Measurements
3
4
5
23
Standard Normal Distribution
(mean=0, standard deviation=1)
z
x
1 z2 / 2
f ( z) 
e
2

x2
z2
x1
z1
P( x1  x  x2 )   f ( x)dx   f ( z )dz
P( x1  x  x2 )  P( z1  z  z 2 )  P(
1
Z[1,1]
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

x


x2  

)
3
Z [-3,3]
2
Z [-2,2]
68%
x1  
95%
Lecture Notes on MECH 373 – Instrumentation and Measurements
99.7%
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Normal Distribution Example
• The distribution of heights of American women aged 18 to 24 is
approximately normally distributed with mean 65.5 inches and
standard deviation 2.5 inches.
• 68% of these American women have heights between 65.5 –
1(2.5) and 65.5 + 1(2.5) inches, or between 63 and 68 inches.
• 95% of these American women have heights between 65.5 2(2.5) and 65.5 + 2(2.5) inches, or between 60.5 and 70.5 inches.
• 99.7% of these American women have heights between 65.5 3(2.5) and 65.5 + 3(2.5) inches, or between 58 and 73 inches.
68%
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95%
Lecture Notes on MECH 373 – Instrumentation and Measurements
99.7%
25