Lecture 3 analysis of experimental data
Download
1 / 39

LECTURE 3: ANALYSIS OF EXPERIMENTAL DATA - PowerPoint PPT Presentation


  • 106 Views
  • Uploaded on

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

loader
I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.
capcha
Download Presentation

PowerPoint Slideshow about ' LECTURE 3: ANALYSIS OF EXPERIMENTAL DATA' - pallavi-gaurav


An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript
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


Solution:

Tabulate the data using this table:


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



From the result before we can find coeff of determination r2

by tabulating the following values

r2=


Next lecture
Next Lecture

Experimental Uncertainty Analysis

End of Lecture 3


ad