MECH 373
Instrumentation and Measurements
Lecture 15
Statistical Analysis of Experimental Data
(Chapter 6)
• Criterion for Rejecting Questionable Data
Points
• Introduction
• General Concepts and Definitions • Correlation of Experimental Data
• Least-Squares Linear Fit
• Probability
• Probability Distribution Function • Outliers in x-y Data Sets
• Linear regression using data
• Parameter Estimation
transformation
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Lecture Notes on MECH 373 – Instrumentation and Measurements
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Criterion for Rejecting Questionable
Data Points
• In some experiments, it happens that one or more measured data points
appears to be out of the line with the rest of the data.
• If some clear fault can be detected in measuring those specific values, they
should be discarded.
• But often the seemingly faulty data cannot be traced to any specific
problem.
• Data that lie outside the probability of normal variation can bias the
statistical analysis.
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Criterion for Rejecting Questionable
Data Points
• There are several statistical methods for detecting and rejecting these wild
or outlier data points.
• The simplest method for the outlier detection is called the three-sigma test.
• In this method, first calculate the mean and the standard deviation from the
data set and then label all the data points that lie outside the range of 99.7%
probability of occurrence. That is outside the range
x  3  x  x  3
• Another recommended method is called the modified Thompson τ
technique.
• In this method, if we have n measurements that have a mean x and
standard deviation S, the data can be arranged in ascending order.
• The extreme values (i.e. highest and lowest) are suspected outliers.
• For these suspected points, the deviation is calculated as
i  xi  x
• In the next step, for the given value of n, the value of τ is obtained from
Table 6.8.
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Criterion for Rejecting Questionable Data Points
• Compare the largest value of δi with the
product of τ and standard deviation S.
• If δ > τS, the data value (xi) can be rejected
as the outlier.
• This method rejects one data value at a
time. Thus, the process needs to be repeated
with the recomputed values of mean and
standard deviation of the remaining data,
and continued until no more of the data can
be eliminated.
• It should be noted that eliminating an
outlier is not an entirely positive event.
• The outlier may be resulted from a
problem with the measuring system, the
influence of extraneous variable or the
measurement apparatus.
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Example Rejecting Questionable Data Points
• Nine voltage measurements: 12.02, 12.05, 11.96, 11.99, 12.10, 12.03,
12.00, 11.95 and 12.16V.
• Vmean = 12.03V, S = 0.07.
• D1 = |Vlargest – Vmean| = |12.16 – 12.03| = 0.13.
• D2 = |Vsmallest – Vmean| = |11.95 – 12.03| = 0.08.
• Table 6.8, n = 9, τ = 1.777. S τ = 0.07x1.777 = 0.12.
• D1 > S τ , 12.16 rejected.
• Re-calculate Vmean and S with 8 measurements.
• Vmean = 12.01V, S = 0.05.
• Table 6.8, n = 8, τ = 1.749. S τ = 0.05x1.749 = 0.09.
• D1 = |Vlargest – Vmean| = |12.10 – 12.01| = 0.09.
• D2 = |Vsmallest – Vmean| = |11.95 – 12.01| = 0.06.
• No rejection.
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Correlation of Experimental Data
Correlation Coefficient
• Scatter due to random errors is a
common characteristic of virtually
all measurements.
• However, in some cases, the
scatter is so large that it is difficult
to detect a trend. For example, Fig.
6.11.
• Figure (a) shows a strong
relationship between x and y.
• Figure (b) does not show any
functional relationship between x
and y.
• Figure (c) shows some vague
relationship between x and y,
however, it could be a consequence
of pure chance.
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Correlation of Experimental Data
• A statistical parameter that can be used to determine whether the apparent trend
between two variables is real or simply be a consequence of a pure chance is
called correlation coefficient, rxy.
• The magnitude of rxy is used to determine whether there is a functional
relationship between the two variables x and y and if it is strong or week.
• Consider from an experiment, we obtained a set of n data pairs of variables x
and y, i.e. [(xi,yi), i = 1, 2, 3, …, n].
• The linear correlation coefficient, rxy can be computed as
where
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Correlation of Experimental Data
• The values of rxy lie in the range from -1 to +1.
• A value of rxy = +1 indicates a perfectly linear relationship between x and y
with a positive slope (i.e. increasing x results in increasing y).
• A value of rxy = -1 also indicates a perfectly linear relationship between x and y
but with a negative slope (i.e. increasing x results in decreasing y).
• A value of rxy = 0 indicates there is no linear relationship between x and y.
However, in real data, even when there is no correlation between the two
variables, the value of rxy is usually nonzero.
• To determine if the computed value of correlation coefficient indicates a
functional relationship between the two variables or the trend is purely due to
chance, the minimum values of the correlation coefficient (rt) for different
significance or confidence levels are used which are presented in the table below.
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Correlation of Experimental Data
• For the given n number of data pairs
and for a given confidence level, if
rxy  rt , the real linear relationship
exists between the two variables
otherwise, if rxy  rt , we cannot be
confident that a linear functional
relationship between the two variables
exists.
• It should be noted that rxy only
indicates if a linear relationship exists
between the two variables. If the
relationship between the two variables
is nonlinear e.g. polynomial,
exponential, etc, rxy will not be a good
indicator.
• Outliers could have a significant
effect on the correlation coefficient;
therefore, outliers should be removed
before computing the correlation
coefficient.
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Correlation of Experimental Data
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Correlation of Experimental Data
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Correlation of Experimental Data
Least-Squares Linear Fit
• It is a common requirement in
experimentation to correlate
experimental data by fitting
mathematical function such as, straight
lines or exponentials through the data.
• Straight lines are the most common
functions used for this purpose.
• Linear fits are often appropriate for
the data, and in other cases, the data
can be transformed to be
approximately linear.
• If we have n pairs of data (xi, yi), we
seek to fit a straight line through the
data of the form
Y = ax + b
• We would like to obtain values of the
constants a and b which provides the
best fit to the data.
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Least-Squares Linear Fit
• A systematic approach to obtain the best fit is called the method of leastsquares or linear regression.
• Regression is a well-defined mathematical formulation that is readily
automated.
• Consider data consists of pairs (xi, yi). For each value of xi, we can predict a
value of Yi according to the linear relationship Y = ax + b.
• For each value of xi, we have an error
ei  Yi  yi
and the square of the error is
ei2  Yi  yi   (axi  b  yi )2
2
• The sum of the squared errors for all the data points is then
E   Yi  yi    axi  b  yi 
2
2
• We now choose a and b to minimize E. Thus
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Least-Squares Linear Fit
E
 0   2axi  b  yi xi
a
E
 0   2axi  b  yi 
b
• These two equations can be solved for a and b:

n xi yi   xi  yi
a 
2
2
n xi  ( xi )


2
x


i  yi   xi  xi yi
 y  ax
2
b 
2
n xi  ( xi )

n
 xi
x  i 1
n
n
 yi
y  i 1
n
• The resulting equation, Y = ax + b, is called the least-squares best fit to the
data represented by (xi, yi).
• When a linear regression analysis has been performed, it is important to
determine how good the fit actually is.
• A good measure of the adequacy of the regression model is called the
coefficient of determination, given by
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Least-Squares Linear Fit
2


ax

b

y
 i
i
r 2  1
2


y

y
 i
2


Y

y
 i i
 1
2


y

y
 i
• For a good fit, r2 should be close to unity.
• Another measure to estimate how well the best-fit line represents the data is
called the standard error of estimate, given by
S yx 
( y
i
 Yi )^ 2
n2
• This is the standard deviation of the differences between the data points and
the best-fit line, and it has the same units as y.
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Least-Squares Linear Fit
Best fit line: a = 0.9977, b = 0.0295, Y = 0.9977x + 0.0295.
Standard error of estimate = 0.0278.
Coefficient of determination:
y mean = 1.27666.
r 2 = 0.999286.
High value close to unity – Good fit.
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Least-Squares Linear Fit
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Least-Squares Linear Fit
• Many computer software packages include features to perform the
regression analysis. For example, Excel, Matlab, etc.
• The advantage of using the software is that you can try polynomial fits of
different orders and select the suitable one.
• There are some important considerations about the least-squares method.
• Variation in the data is assumed to be normally distributed and due
to random causes.
• In deriving the relation Y = ax + b, it is assumed that random
variation exists in y, while x values are error free.
• Since the error has been minimized in the y direction, an erroneous
conclusion can be made if x is estimated based on a value for y. That is,
linear regression of x in terms of y (i.e. X = cy + d) cannot simply be
derived from Y = ax + b.
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Outliers in x-y Data Sets
• We discussed about how to detect outliers when static measurements are
made. That is, several measurements of a single variable.
• When a variable y is measured as a function of an independent variable x, in
most cases, there is only one value of y for each value of x.
• One way of identifying the outlier is to plot the data and the best fit line and
identify the outlier that has much larger deviation from the line than the other
data.
• A more sophisticated method of identifying outliers in x-y data set is by
computing the ratio of the residuals (ei) to the standard error of estimate (Syx),
which is called the standardized residual.
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Outliers in x-y Data Sets
• If the residuals are normally distributed, it is expected that 95% of the
standardized residuals would be in the range ±2. That is, within two standard
deviations from the best-fit line.
• If the standardized residual is much greater than 2 then it could be considered
as the outlier.
• Determination of outliers in x-y data sets is not a simple mechanistic process.
• The experimenter can use plots of data with the best-fit, plots of standardized
residuals, but ultimately it is a judgment call as to whether reject any data point.
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Example - Outliers in x-y Data Sets
• Water-turbine
experiment:
• Fit least squares straight
line. Check outlier.
Lecture 15
Speed rpm
Lecture Notes on MECH 373 – Instrumentation and Measurements
Torque (Nm)
100
4.89
201
4.77
298
3.79
402
3.76
500
2.84
601
4
699
2.05
799
1.61
21
Example - Outliers in x-y Data Sets
• Water turbine data:
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Example - Outliers in x-y Data Sets
• Torque = at 600rpm, ei/Syx > 2 – high
probability of being outlier.
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Linear Regression Using Data Transformation
• Non-linear relationship can be transformed to linear equation.
• e.g. y = aebx ; In y = bx + In(a)
• Example: Compression process in a piston cylinder – Temperature and
pressure relationship:
• T/To = (P/Po) (n-1)/n
• T = Absolute temperature = Tabs (K) = T + 460
• P = Absolute pressure = Gauge pressure + atmospheric pressure (14.7psi)
• To and Po = reference data.
• In (T) = a In (P) + b
• a = (n-1)/n, b = Constant
•In(T+460) = 0.1652 In(P) + 5.72222
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Linear Regression Using Data Transformation
• Non-linear relationship can be transformed to linear equation.
•.
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Linear Regression Using Data Transformation
• Non-linear relationship can be transformed to linear equation.
•
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