8/12/2019 M350 Wk3 Stats 2 Student t
1/23
MECH 350 Instrumentation and Design ofExperiments
Assessing and presenting experimental dataMain topics
3 s principle and its applicationThe bound of uncertainty with confidence
level Assessing Small Sample Sizes
Department of Mechanical Engineering andTechnology
Wentworth Institute of Technology
8/12/2019 M350 Wk3 Stats 2 Student t
2/23
Gaussian Distributions3 sigma ( s) principle
If the random variable complies with thenormal distribution, the probability of the
8/12/2019 M350 Wk3 Stats 2 Student t
3/23
Physical meaning of 3 principle inmeasurement
If the measurement can be described by normaldistribution:
(1) Any measured value in the range ( m-3 s , m+3 s)has the probability of 99.74%. In other word,any real measured value should or must be inthis range because the probability is 99.74%.
%74.999974.0
21
exp2
1)33(
3
3-
s
m
s s m s m
s m
s m
x x P
8/12/2019 M350 Wk3 Stats 2 Student t
4/23
(2) The probability of a measured value which islarger than ( m+3 s) or less than ( m-3s) is 0.0036(0.36%). That is, the chance of a realmeasured value out of range ( m-3s , m+3 s) isonly 0.36%.
(3) If a real measured value out of range ( m-3 s ,m+3 s) does happen, this measured value willbe rejected because of:
a) Something is wrong with the instrument, ormeasurement system, or measurement setting, orthere was some mistake.
b) The measured parameter had significant changedue to machine setting.In one word, There was some mistake duringmeasurement or the measured parameter wasnot the same.
8/12/2019 M350 Wk3 Stats 2 Student t
5/23
Application of the 3 principle inmeasurement
The 3 s principle is a tool to pre-analyze yourmeasured data to check whether themeasurement system and the experimentalsetting is in normal status. If some data isout the value range defined by 3 s principle,you should stop experiment andmeasurement and find the possible causes
before you resume your experiment andmeasurement. Otherwise, you will wasteyour time because the measured data mightnot be the data you intend to obtain.
8/12/2019 M350 Wk3 Stats 2 Student t
6/23
Example: Followings were 20 measured sound levels (dBA) ofa type of machine:101.03; 100.98; 101.30; 100.98; 101.10;101.25; 100.68; 100.75; 101.36; 101.67; 100.56; 100.85;100.39; 101.25; 101.56; 101.23; 101.40; 100.95; 100.67;
100.56. The mean and standard deviation of these dataare 101.03 dBA and 0.35 dBA. Now, another 3 test data:98.10, 101.45 and 103.89 were obtained. Are these threemeasured data acceptable?
Solution:Based on 6 s principle, the measured data should be in therange (101.03-3X0.35=99.98, 101.03+3X0.35=102.08) withthe probability 99.74%. The test data 98.10 was too low and should be rejected
(the cause for this was to use the wrong type of filter) the test data 103.89 was too high and should berejected (the cause for this was to wrong RPM setting).
The test data 101.45 was normal and acceptable.
8/12/2019 M350 Wk3 Stats 2 Student t
7/23
Application of probability theory inmeasurement
Population vs. samplePopulation: The set ofdata (numerical or
otherwise)corresponding to theentire collection of unitsabout which information
is soughtSample: A subset of thepopulation data that areactually collected in the
course of a study.
8/12/2019 M350 Wk3 Stats 2 Student t
8/23
In any real-life situation, we always deal withsamples from a population and not the populationitself.
Typically, our objectives for measurement are touse measurement of sample to estimate :a) the mean and standard deviation of the
population.b) the bound of measured parameter with a
confidence level.Since we normally use the average of the
measurement data to represent the measuredparameter, it is good to determine the bound of themean with a confidence level.
c) the bound of the mean with a confidence level.
8/12/2019 M350 Wk3 Stats 2 Student t
9/23
The confidence intervals for measurement
If the measured parameter complies with normaldistribution and the sample number is big enough (>30),the actual value of the measured parameter inside therange ( m z a/2 s, m z a/2 s ) will have the confidence level a
as listed in following table. m-mean. s -standard deviationTable 1 The confidence intervals
Confidencelevel
85% 90% 95% 99%
Za/2 1.439 1.645 1.960 2.575
Bounds (m-1.439 s ,m+1.439 s )
(m-1.645 s ,m+1.645 s )
(m-1.960 s ,m+1.960 s )
(m-2.575 s ,m+2.575 s )
8/12/2019 M350 Wk3 Stats 2 Student t
10/23
Za/2 can be determined by use of table 3.2 in page 50.For example, probability level is 95%. Half of 0.95 is0.475. From table 3.2, the z value for 0.475 is 1.960.
8/12/2019 M350 Wk3 Stats 2 Student t
11/23
Example 1 During a 12-hour test of a steam generator, the inlet
pressure was measured 100 times. The inlet pressure
should be held constant at 4.00 MPA. The results of a 12-hour pressure test is shown in following table. (page 52).The histogram is shown in following.
Pressure, p , in
MPA
Number of results,
m(3.965,3.975)(3.975,3.985)(3.985,3.995)(3.995,4.005)(4.005,4.015)(4.015,4.025)(4.025,4.035)(4.035,4.045)
(4.045,4.055)
13
1225331762
1
8/12/2019 M350 Wk3 Stats 2 Student t
12/23
The mean and standard deviation are m=4.008 MPA,s =0.014 MPA. For proper performance, the pressureshould no deviate from setting value 4.00 MPA by morethan about 1%.
(1) Determine the interval containing 95% of pressurereadings.
(2) Based on the test measurement, what is the probabilityof pressure reading whose deviation from setting value
4.00MPA is less than 1%. Solution:(1) The probability level is 99%. From previous table, we
have Z a/2 =2.575. The interval containing 99% of
pressure reading is:(m 2.575 s MPA )=(4.008 2.575X0.014 MPA)=(4.008 0.036 MPA)So, the interval containing 99% of pressure reading is:
(3.972, 4.044) MPA
8/12/2019 M350 Wk3 Stats 2 Student t
13/23
The probability level for mean atlarge samples (n>=30)
If the bias errors have been reduced to minimum(proper calibration on the proper instrument withproper measurement procedure), it is reasonable
to assume that the mean of the measurement isvery close to the true mean value in idealcondition.
Now lets assess the uncertainty between themeasured mean and the true mean
8/12/2019 M350 Wk3 Stats 2 Student t
14/23
If we do 100 measurement at one time, wecalculate the arithmetic average and will getone value for the mean.
If we do another 100 measurement, we will getanother value of the mean, which will certainlydifferent from previous one.
If we repeat these kind of measurement, we wouldobtain a set of samples for the mean. (this isnot the set of sample of measurement.)
Actually, we dont need to do above in orderto assess the uncertainty between themeasured mean and the true mean. We needuse one of probabilistic theory (Central limittheorem).
8/12/2019 M350 Wk3 Stats 2 Student t
15/23
Let us set a new random variable forrepresenting the mean:
x is the arithmetic averagen-is the sample size of the measurements is the standard deviation of the measurement
Here z is the random variable of mean andstill complies with normal distribution.
n
x x z
xs
m
s
m
8/12/2019 M350 Wk3 Stats 2 Student t
16/23
8/12/2019 M350 Wk3 Stats 2 Student t
17/23
Example 2 (cont.)Determine a 99% confidence interval for the mean
pressure calculated in example 1. Solution :For the 100 pressure measurement, the mean and
standard deviation are 4.008 and 0.014 MPA. Son=100, x=4.008 MPA, S x=0.014 MPA
For probability level of 99% from table 1, z c/2 =2.575.
The intervals of mean pressure with probability level99% is:
0036.0008.4
100
014.0575.2008.42/
n
S z x x
c
8/12/2019 M350 Wk3 Stats 2 Student t
18/23
Compare and notice the conceptualdifference
(1) The interval containing 99% of pressurereadings is: (expected range of measuredpressures)(m 2.575 s MPA )=(4.008 2.575X0.014 MPA)=(4.008 0.036 MPA)
(2) The interval of mean pressure with probabilitylevel 99% is: (Range of the true mean)
0036.0008.4100
014.0575.2008.42/
n
S z x x
c
8/12/2019 M350 Wk3 Stats 2 Student t
19/23
Probability intervals for mean atsmall samples (n
8/12/2019 M350 Wk3 Stats 2 Student t
20/23
In figure v is degree of freedom. v=n-1. Theshape of t-distribution is similar to the normaldistribution. When v is big enough, the t-
distribution is the same as normal distribution.
8/12/2019 M350 Wk3 Stats 2 Student t
21/23
The two-sided confidence interval for meanx of a small sample (n
8/12/2019 M350 Wk3 Stats 2 Student t
22/23
Example 3The 12 sound pressure-level measurement has mean101.35 dBA and standard deviation 0.27 dBA. What is thetwo-sided 95% confidence interval for the true meansound pressure-level?Solution:The required probability level is 95%, so a =1-0.95=0.05.v=n-1=12-1=11. According to table 3-5 in page 58:
t a/2, v =t 0.025, 11 =2.201.Hence the two-sided 95% confidence interval for soundpressure level is:
53.10117.101 11
27.0201.235.101
11
27.0201.235.101
,2/,2/
m
m
m n
S
t xn
S
t x x
vc
x
vc
8/12/2019 M350 Wk3 Stats 2 Student t
23/23
Homework
Blackboard Assignments