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MEASUREMENT AND INSTRUMENTATION BMCC 3743. LECTURE 3: ANALYSIS OF EXPERIMENTAL DATA. Mochamad Safarudin Faculty of Mechanical Engineering, UTeM 2010. Contents. Introduction Measures of dispersion Parameter estimation Criterion for rejection questionable data points

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lecture 3 analysis of experimental data
MEASUREMENT AND INSTRUMENTATION

BMCC 3743

LECTURE 3: ANALYSIS OF EXPERIMENTAL DATA

Mochamad Safarudin

Faculty of Mechanical Engineering, UTeM

2010

contents
Contents
  • Introduction
  • Measures of dispersion
  • Parameter estimation
  • Criterion for rejection questionable data points
  • Correlation of experimental data
introduction
Introduction
  • Needed in all measurements with random inputs, e.g. random broadband sound/noise
    • Tyre/road noise, rain drops, waterfall
  • Some important terms are:
    • Random variable (continuous or discrete), histogram, bins, population, sample, distribution function, parameter, event, statistic, probability.
terminology
Terminology
  • Population : the entire collection of objects, measurements, observations and so on whose properties are under consideration
  • Sample: a representative subset of a population on which an experiment is performed and numerical data are obtained
contents1
Contents
  • Introduction
  • Measures of dispersion
  • Parameter estimation
  • Criterion for rejection questionable data points
  • Correlation of experimental data
measures of dispersion
Measures of dispersion

=>Measures of data spreading or variability

  • Deviation (error) is defined as
  • Mean deviation is defined as
  • Population standard deviation is defined as
measures of dispersion1
Measures of dispersion
  • Sample standard deviation is defined as
    • is used when data of a sample are used to estimate population std dev.
  • Variance is defined as
exercise
Exercise
  • Find the mean, median, standard deviation and variance of this measurement:

1089, 1092, 1094, 1095, 1098, 1100, 1104, 1105, 1107, 1108, 1110, 1112, 1115

answer to exercise
Answer to exercise
  • Mean = 1103 (1102.2)
  • Median = 1104
  • Std deviation = 5.79 (7.89)
  • Variance = 33.49 (62.18)
contents2
Contents
  • Introduction
  • Measures of dispersion
  • Parameter estimation
  • Criterion for rejection questionable data points
  • Correlation of experimental data
parameter estimation
Parameter estimation

Generally,

  • Estimation of population mean, is sample mean, .
  • Estimation of population standard deviation, is sample standard deviation, S.
interval estimation of the population mean
Interval estimation of the population mean
  • Confidence interval is the interval between

to , where is an uncertainty.

  • Confidence level is the probability for the population mean to fall within specified interval:
interval estimation of the population mean1
Interval estimation of the population mean
  • Normally referred in terms of , also called level of significance, where

confidence level

  • If n is sufficiently large (> 30), we can apply the central limit theorem to find the estimation of the population mean.
central limit theorem
Central limit theorem
  • If original population is normal, then distribution for the sample means’ is normal (Gaussian)
  • If original population is not normal and n is large, then distribution for sample means’ is normal
  • If original population is not normal and n is small, then sample means’ follow a normal distribution only approximately.
normal gaussian distribution
Normal (Gaussian) distribution
  • When n is large,

where

  • Rearranged to get
  • Or with confidence level
student s t distribution
Student’s t distribution
  • When n is small,

where

  • Rearranged to get
  • Or with confidence level

t table

interval estimation of the population variance
Interval estimation of the population variance
  • Similarly as before, but now using chi-squared distribution, , (always positive)

where

interval estimation of the population variance1
Interval estimation of the population variance
  • Hence, the confidence interval on the population variance is

Chi squared table

contents3
Contents
  • Introduction
  • Measures of dispersion
  • Parameterestimation
  • Criterion for rejection questionable data points
  • Correlation of experimental data
criterion for rejection questionable data points
Criterion for rejection questionable data points
  • To eliminate data which has low probability of occurrence => use Thompson test.
  • Example: Data consists of nine values,

Dn = 12.02, 12.05, 11.96, 11.99, 12.10, 12.03, 12.00, 11.95 and 12.16.

  • = 12.03, S = 0.07
  • So, calculate deviation:
criterion for rejection questionable data points1
Criterion for rejection questionable data points
  • From Thompson’s table, when n = 9, then
  • Comparing with

where then D9 = 12.16 should be discarded.

  • Recalculate S and to obtain 0.05 and 12.01 respectively.
  • Hence forn = 8, and

so remaining data stay.

Thompson’s t table

contents4
Contents
  • Introduction
  • Measures of dispersion
  • Parameterestimation
  • Criterion for rejection questionable data points
  • Correlation of experimental data
correlation of experimental data
Correlation of experimental data
  • Correlation coefficient
  • Least-square linear fit
  • Linear regression using data transformation
a correlation coefficient
A) Correlation coefficient
  • Case I: Strong, linear relationship between x and y
  • Case II: Weak/no relationship
  • Case III: Pure chance

=> Use correlation coefficient, rxy to determine Case III

linear correlation coefficient
Linear correlation coefficient
  • Given as

where

  • +1 means positive slope (perfectly linear relationship)
  • -1 means negative slope (perfectly linear relationship)
  • 0 means no linear correlation
linear correlation coefficient1
Linear correlation coefficient
  • In practice, we use special Table (using critical values of rt) to determine Case III.
  • If from experimental value of |rxy|is equal or more than rt as given in the Table, then linear relationship exists.
  • If from experimental value of |rxy|is less than rt as given in the Table, then only pure chance => no linear relationship exists.
b least square linear fit
B) Least-square linear fit

To get best straight line on the plot:

  • Simple approach: ruler & eyes
  • More systematic approach: least squares
    • Variation in the data is assumed to be normally distributed and due to random causes
    • To get Y = ax + b, it is assumed that Y values are randomly vary and x values have no error.
least square best fit
Least-square best fit
  • For each value of xi, error for Y values are
  • Then, the sum of squared errors is
least square best fit1
Least-square best fit
  • Minimising this equation and solving it for a & b, we get
least square best fit2
Least-square best fit
  • Substitute a & b values into Y = ax + b, which is then called the least-squares best fit.
  • To measure how well the best-fit line represents the data, we calculate the standard error of estimate, given by

where Sy,x is the standard deviation of the differences between data points and the best-fit line. Its unit is the same as y.

coefficient of determination
Coefficient of determination
  • …Is another good measure to determine how well the best-fit line represents the data, using
  • For a good fit, must be close to unity.
c linear regression using data transformation
C) Linear regression using data transformation
  • For some special cases, such as
  • Applying natural logarithm at both sides, gives

where ln(a) is a constant, so ln(y) is linearly related to x.

example
Example
  • Thermocouples are usually approximately linear devices in a limited range of temperature. A manufacturer of a brand of thermocouple has obtained the following data for a pair of thermocouple wires:

Determine the linear correlation between T and V

slide35
Solution:

Tabulate the data using this table:

slide36
Another example

The following measurements were obtained in the calibration of

a pressure transducer:

  • Determine the best fit
  • straight line
  • Find the coefficient of
  • determination for the
  • best fit
next lecture
Next Lecture

Experimental Uncertainty Analysis

End of Lecture 3

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