Lecture 05 - Statistical Analysis of Experimental Data.pptx
- 1. Measuring Instruments and Calibration Prof Dr Ossama Abouelatta, Department of Mechanical Engineering , Faculty of Engineering, Albaha University Albaha University Faculty of Engineering Mechanical Engineering Department MEASURING INSTRUMENTS AND CALIBRATION Lecture (5) Statistical Analysis of Experimental Data By: Ossama Abouelatta o_abouelatta@yahoo.com Mechanical Engineering Department Faculty of Engineering Albaha University 2013
- 2. Measuring Instruments and Calibration Prof Dr Ossama Abouelatta, Department of Mechanical Engineering , Faculty of Engineering, Albaha University Lecture (5): Statistical Analysis of Experimental Data (2) AIMS This lecture aims: to identify general concepts and definitions statistical analysis of experimental data. to determine criterion for rejecting questionable data points. to calculate correlation coefficient of experimental data. to apply computer software for statistical analysis of experimental data.
- 3. Measuring Instruments and Calibration Prof Dr Ossama Abouelatta, Department of Mechanical Engineering , Faculty of Engineering, Albaha University Lecture (5): Statistical Analysis of Experimental Data (3) OUTLINE Introduction General Concepts and Definitions Definitions Measures of Central Tendency Measures of Dispersion Criterion for Rejecting Questionable Data Points Correlation of Experimental Data Correlation Coefficient Least-Squares Linear Fit Outliers in x-y Data Sets Linear Regression Using Data Transformation Multiple and Polynomial Regression Applying Computer Software for Statistical Analysis of Experimental Data
- 4. Measuring Instruments and Calibration Prof Dr Ossama Abouelatta, Department of Mechanical Engineering , Faculty of Engineering, Albaha University Lecture (5): Statistical Analysis of Experimental Data (4) INTRODUCTION Measurement processes usually introduce a certain amount of variability or randomness into the results, and this randomness can affect the conclusions drawn from experiments. In this lecture, we discuss important statistical methods that can be used to plan experiments and interpret experimental data.
- 5. Measuring Instruments and Calibration Prof Dr Ossama Abouelatta, Department of Mechanical Engineering , Faculty of Engineering, Albaha University Lecture (5): Statistical Analysis of Experimental Data (5) GENERAL CONCEPTS Measures of central tendency The most common parameter used to describe the central tendency is the mean. This is the everyday concept of average and for a sample is defined by: Deviation of each measurement Dispersion is the spread or variability of the data. The deviation of each measurement is defined as: Deviation of measurement The mean deviation is defined as:
- 6. Measuring Instruments and Calibration Prof Dr Ossama Abouelatta, Department of Mechanical Engineering , Faculty of Engineering, Albaha University Lecture (5): Statistical Analysis of Experimental Data (6) GENERAL CONCEPTS Population standard deviation For a population with a finite number of elements, the population standard deviation is defined as: Sample standard deviation The sample standard deviation is defined as: Variance The variance is defined as:
- 7. Measuring Instruments and Calibration Prof Dr Ossama Abouelatta, Department of Mechanical Engineering , Faculty of Engineering, Albaha University Lecture (5): Statistical Analysis of Experimental Data (7) EXAMPLE (1) Table 1 shows the results of a set of 60 measurements of air temperature in the duct. These temperature data are observed values of a random variable. Number of readings Temperature (C) 1 1089 Average (x)= 1102.2 1 1092 Median (xm)= 1104.0 2 1094 Standard deviation (S) = 8.2 4 1095 Variance (S 2 ) = 67.4 8 1098 Mode (m) = 1104.0 9 1100 12 1104 4 1105 5 1107 5 1108 4 1110 3 1112 2 1115 Table 1
- 8. Measuring Instruments and Calibration Prof Dr Ossama Abouelatta, Department of Mechanical Engineering , Faculty of Engineering, Albaha University Lecture (5): Statistical Analysis of Experimental Data (8) =AVERAGE(B2:B14) =MEDIAN(B2:B14) =STDEV(B2:B14) =VAR(B2:B14) =CORREL(A2:A7,B2:B7) =MAX(B2:B14) =MIN(B2:B14)
- 9. Measuring Instruments and Calibration Prof Dr Ossama Abouelatta, Department of Mechanical Engineering , Faculty of Engineering, Albaha University Lecture (5): Statistical Analysis of Experimental Data (9) EXAMPLE (2) Table 2 shows a nine voltage measurements in a circuit have produced the following values: 12.02, 12.05, 11.96, 11.99, 12.10, 12.03, 12.00, 11.95, 12.16 V. Determine whether any of the values can be rejected. No. Volt Average (x) = 12.03 1 12.02 Standard deviation (S) = 0.07 2 12.05 d1 = (Vlargest - Vmean) = 0.13 3 11.96 d2 = (Vmean - Vsmallest) = 0.08 4 11 .99 From Table: n = 9 then, t = 1.777 5 12.10 St= S1.777 = 0.12 6 12.03 7 12.00 8 11.95 9 12.16 Table 2 SOLUTION:
- 10. Measuring Instruments and Calibration Prof Dr Ossama Abouelatta, Department of Mechanical Engineering , Faculty of Engineering, Albaha University Lecture (5): Statistical Analysis of Experimental Data (10) EXAMPLE (2) Table 2 shows a nine voltage measurements in a circuit have produced the following values: 12.02, 12.05, 11.96, 11.99, 12.10, 12.03, 12.00, 11.95, 12.16 V. Determine whether any of the values can be rejected. No. Volt Average (x) = 12.02 1 12.02 Standard deviation (S) = 0.05 2 12.05 d1 = (Vlargest - Vmean) = 0.08 3 11.96 d2 = (Vmean - Vsmallest) = 0.07 4 11 .99 From Table: n = 8 then, t = 1.749 5 12.10 St= S1.749= 0.09 6 12.03 7 12.00 8 11.95 Table 2 SOLUTION:
- 11. Measuring Instruments and Calibration Prof Dr Ossama Abouelatta, Department of Mechanical Engineering , Faculty of Engineering, Albaha University Lecture (5): Statistical Analysis of Experimental Data (11) CORRELATION OF EXPERIMENTAL DATA Correlation Coefficient The correlation coefficient, rxy, is a number whose magnitude can be used to determine whether there in fact exists a functional relationship between two measured variables x and y.
- 12. Measuring Instruments and Calibration Prof Dr Ossama Abouelatta, Department of Mechanical Engineering , Faculty of Engineering, Albaha University Lecture (5): Statistical Analysis of Experimental Data (12) EXAMPLE (3) It is thought that lap times for a race car depend on the ambient temperature. The following data for the same car with the same driver were measured at different races: Does a linear relationship exist between these two variables? Ambient temperature (o F) Lap time (S) Correlation coefficient = 0.4015 40 65.30 47 66.50 55 67.30 62 67.80 66 67.00 88 66.60 65.00 65.50 66.00 66.50 67.00 67.50 68.00 30 50 70 90 Lab time (s) Ambient temperature (oF) SOLUTION:
- 13. Measuring Instruments and Calibration Prof Dr Ossama Abouelatta, Department of Mechanical Engineering , Faculty of Engineering, Albaha University Lecture (5): Statistical Analysis of Experimental Data (13) CORRELATION OF EXPERIMENTAL DATA Least-Squares Linear Fit It is a common requirement in experimentation to correlate experimental data by fitting mathematical functions such as straight lines or exponentials through the data. One of the most common functions used for this purpose is the straight line. Linear fits are often appropriate for the data, and in other cases the data can be transformed to be approximately linear. Fitting a straight line through data.
- 14. Measuring Instruments and Calibration Prof Dr Ossama Abouelatta, Department of Mechanical Engineering , Faculty of Engineering, Albaha University Lecture (5): Statistical Analysis of Experimental Data (14) EXAMPLE (4) The following table represents the output (volts) of a linear variable differential transformer (LVDT; an electric output device used for measuring displacement) for five length inputs: Determine the best linear fit to these data, draw the data on a (V, L) plot, and calculate the standard error of estimate as well as the coefficient of determination. SOLUTION: L (cm) V (V) Correlation coefficient = 0.9996 0.0 0.05 0.5 0.52 1.0 1.03 1.5 1.50 2.0 2.00 2.5 2.56 y = 0.9977x + 0.0295 R² = 0.9993 0.0 0.5 1.0 1.5 2.0 2.5 3.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 V (V) L (cm)
- 15. Measuring Instruments and Calibration Prof Dr Ossama Abouelatta, Department of Mechanical Engineering , Faculty of Engineering, Albaha University Lecture (5): Statistical Analysis of Experimental Data (15) CORRELATION OF EXPERIMENTAL DATA Outliers in x-y Data Sets Thompson t technique was presented as a method for eliminating bad data when several measurements are made 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 measured for each value of x. As a result, at each x there are insufficient data to determine a meaningful standard deviation. One way to identify outliers is to plot the data and the best-fit line as shown in upper figure. A point such as A, which shows a much larger deviation from the line than the other data, might be identified as an outlier. x-y data showing an outlier. Plot of standardized residuals.
- 16. Measuring Instruments and Calibration Prof Dr Ossama Abouelatta, Department of Mechanical Engineering , Faculty of Engineering, Albaha University Lecture (5): Statistical Analysis of Experimental Data (16) EXAMPLE (5) The shown data were taken from a small water-turbine experiment. Fit a least-squares straight line to these data, and determine whether any of the torque values appear to be outliers. SOLUTION: rpm Torque (N.m) Torque (N.m) e/Sxy 100.0 4.89 5.00 0.19 201.0 4.77 4.55 -0.38 298.0 3.79 4.13 0.58 402.0 3.76 3.67 -0.16 500.0 2.84 3.24 0.69 601.0 4.00 2.79 -2.10 699.0 2.05 2.36 0.54 799.0 1.61 1.92 0.54 Standard error Sxy = 0.5758 y = -0.0044x + 5.437 R² = 0.802 0 1 2 3 4 5 6 0 200 400 600 800 Torque (N.m) rpm -4 -2 0 2 0 200 400 600 800 Torque (N.m) rpm e/Sxy
- 17. Measuring Instruments and Calibration Prof Dr Ossama Abouelatta, Department of Mechanical Engineering , Faculty of Engineering, Albaha University Lecture (5): Statistical Analysis of Experimental Data (17) CORRELATION OF EXPERIMENTAL DATA Multiple and Polynomial Regression Regression analysis is much more general than the methods presented so far. Best-fit functions can be determined for situations with more than one independent variable (multiple regression) or for polynomials of the independent variable (polynomial regression). The calculations for these methods can be quite tedious, but they are standard features of statistical-analysis programs, and many features are available in common spreadsheet programs. In multiple regression, we seek a function of the form:
- 18. Measuring Instruments and Calibration Prof Dr Ossama Abouelatta, Department of Mechanical Engineering , Faculty of Engineering, Albaha University Lecture (5): Statistical Analysis of Experimental Data (18) EXAMPLE (6) Consider the listed x-y data shown in the table. Perform a polynomial regression for a third-order polynomial on these data. SOLUTION: x y 0 4.997 1 6.165 2 6.950 3 8.218 4 9.405 5 10.404 6 10.425 7 10.440 8 9.393 9 7.854 10 5.168 y = -0.203x2 + 2.2159x + 4.1559 R² = 0.9123 0 2 4 6 8 10 12 0 2 4 6 8 10 12
- 19. Measuring Instruments and Calibration Prof Dr Ossama Abouelatta, Department of Mechanical Engineering , Faculty of Engineering, Albaha University Lecture (5): Statistical Analysis of Experimental Data (19) EXAMPLE (7) The following data represent the density of an oil mixture as a function of the temperature and mass fraction of three different component oils. Find the coefficients for a multiple regression of the form: SOLUTION: T(K) m1 m2 m3 rmixt 300 0.00 1.00 0.00 879.6 320 0.00 0.50 0.50 870.6 340 0.00 0.00 1.00 863.6 360 0.50 0.00 0.50 846.4 380 0.50 0.25 0.25 830.8 400 0.50 0.50 0.00 819.1 420 1.00 0.00 0.00 796.0 440 1.00 0.00 0.00 778.2
- 20. Measuring Instruments and Calibration Prof Dr Ossama Abouelatta, Department of Mechanical Engineering , Faculty of Engineering, Albaha University Lecture (5): Statistical Analysis of Experimental Data (20) THANK YOU Ossama Abouelatta Mechanical Engineering Department Faculty of Engineering Albaha University Albaha, KSA email: o_abouelatta@yahoo.com