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Basics of EngineeringBasics of EngineeringBasics of EngineeringBasics of Engineering
MeasurementsMeasurementsMeasurementsMeasurements
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Dr. Nasser Mohamed ShelilDr. Nasser Mohamed Shelil Dr. Nasser Mohamed ShelilDr. Nasser Mohamed Shelil
B.Sc. & M.Sc. , Suez Canal University; PhD, Cardiff University/UK
Assistant Professor, Mechanical Engineering Dept., College of Applied Engineering, King Saud University
Applied Mechanical Engineering Program Basics of Engineering MeasurementsBasics of Engineering MeasurementsBasics of Engineering MeasurementsBasics of Engineering Measurements
Chapter 2
Chapter 2:
Calibration & Uncertainty Analysis
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Basic Terminology of Measurement
The International Vocabulary of Basic and General Terms in Metrology , using International Organization for Standardization (ISO) norms, has defined measurement as "a set of operations having the object of determining the value of a quantity". In other words, a measurement is the evaluation of a quantity made after comparing it to a quantity of the same type which we use as a "unit".
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Chapter 2
Basic Terminology of Measurement
the science and "grammar" of measurement is defined as “the field of knowledge concerned with measurement”. Standardized measurement units mean that scientific and economic figures can be understood, reproduced, and converted with a high degree of certitude.
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- The relationship between the value of the input to the measurement system and the system’s indicated output value is established during calibration of the measurement system.
- The known value used for the calibration is called the standard.
- The quantity to be measured being the measurand, which we call m, the sensor must convert m into an electrical variable called s. The expression s = F(m) is established by calibration. By using a standard or unit of measurement, we discover for these values of m (m1, m2 … mi ) electrical signals sent by the sensor (s1, s2 ... si ) and we trace the curve s(m), called the sensor calibration curve.
Calibration
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Chapter 2
Accuracy & Precision
- Accuracy of a system can be estimated during calibration. If the input value of calibration is known exactly, then it can called the true value. The accuracy of a measurement system refers to its ability to indicate a true value exactly.
- Accuracy : It is the ability of instrument to tell the truth
- Accuracy is related to absolute error, ε :
ε = true value – indicated value from which the percent accuracy is found by :
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Accuracy & Precision
- Precision: or repeatability of a measuring system refers to the ability of the system to indicate a particular value upon repeated but independent applications of a specific value input. Precision of a measurement describes the units used to measure something.
- Precision : It is the ability of the instrument to give the same output for the same input under the same conditions
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Applied Mechanical Engineering Program Basics of Engineering MeasurementsBasics of Engineering MeasurementsBasics of Engineering MeasurementsBasics of Engineering Measurements
Chapter 2
Accuracy & Precision
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Error Classifications
1. Systematic, Fixed or Bias Errors:
- Insidious in nature, exist unnoticed unless deliberately searched.
- Repeated readings to be in error by the same amount.
- Not susceptible to statistical analysis.
- Calibration errors
- Certain consistently recurring human error
- Technique error
- Uncorrected loading error
- Limitations of system resolution
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Chapter 2
Error Classifications
2. Precision or Random Errors:
- Distinguished by their lack of consistency. Usually (not always) follow a certain statistical distribution.
- In many instances very difficult to distinguish from bias errors.
- Error stemming from environmental variations
- Certain type of human error
- Error resulting from variations in definition.
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3. Illegitimate Errors
Illegitimate Errors are simply mistakes on the part of experimenter.
- Can be eliminated through the exercise of care and repetition of the measurement. - Blunders and mistakes - Computational errors - Chaotic errors.
Error Classifications
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Effects of precision and bias errors on calibration readings
Bias & Precision Errors
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Chapter 2
Uncertainty
The uncertainty is a numerical estimate of the possible range of the error in a measurement. In any measurement, the error is not known exactly since the true value is rarely known exactly. that the error is within certain bounds, a plus or minus range of the indicated reading
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Chapter .. StatisticsStatisticsStatisticsStatistics
ΣΣΣΣ denotes the addition of a set of values
x is the variable usually used to represent the individual
data values
n represents the number of data values in a sample
N represents the number of data values in a population
Notation
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Chapter .. StatisticsStatisticsStatisticsStatistics
Mean (Average): the number obtained by adding the values and
dividing the total by the number of values.
Median: the middle value when the original data values are
arranged in order of increasing (or decreasing) magnitude.
Variance: It is the expectation of the squared deviation of a
random variable from its mean
Standard Deviation: a measure of variation of the scores about
the mean (average deviation from the mean)
Definitions
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Chapter .. StatisticsStatisticsStatisticsStatistics
Sample and Population Mean
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Chapter .. StatisticsStatisticsStatisticsStatistics
Sample and Population Variance
Sample and Population Standard Deviations
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Chapter 2 UncertaintyUncertainty UncertaintyUncertainty
The estimate of the error is called the uncertainty. It includes both bias and precision errors. We need to identify all the potential significant errors for the instrument(s). All measurements should be given in three parts Mean value Uncertainty Confidence Interval on which that uncertainty is based ( typically 95% C.I. ) Uncertainty can be expressed in either absolute terms (i.e., 5 Volts ±0.5 Volts) or in percentage terms (i.e., 5 Volts ±10%) (relative uncertainty = DV / V x 100) We will use a 95 % confidence interval throughout this course
Uncertainty Analysis
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Calculation of bias Uncertainty
Manufacturers’ Specifications If you can’t do better, you may take it from the manufacturer’s specs. Accuracy - %FS, %reading, offset, or some combination (e.g., 0.1% reading + 0.15 counts) Unless you can identify otherwise, assume that these are at a 95% confidence interval Independent Calibration May be deduced from the calibration process
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Chapter 2
Calculation of precision Uncertainty
Use Statistics to Estimate Random Uncertainty Mean: the sum of measurement values divided by the number of measurements.
Deviation: the difference between a single result and the mean of many results.
Standard Deviation: the smaller standard deviation is the more precise data Large sample size
Small sample size (n<30) Slightly larger value
x =
N
xi
i = 1
N ∑
d i = xi − x
σ =
1 n
(x i − x)
2 ∑
1 2
σs = 1 n− 1
∑( x i − x)^2
1 2
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Distribution of errors on repeated measurements.
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Chapter 2
Student t-distribution (small sample sizes)
The t-distribution was formulated by W.S. Gosset, a scientist in the Guinness brewery in Ireland, who published his formulation in 1908 under the pen name (pseudonym) “Student.”
The t-distribution looks very much like the Gaussian distribution, bell shaped, symmetric and centered about the mean. The primary difference is that it has stronger tails, indicating a lower probability of being within an interval. The variability depends on the sample size, n.
With a confidence interval of c%
Where αααα =1-c and νννν =n-1 (Degrees of Freedom)
Don’t apply blindly - you may have better information about the population than you think.
n
X x t
n
x t s s
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Chapter 2
Confidence Interval
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