ME Mechanical and Thermal Systems Lab Fall 2011 Chapter 3: Assessing and Presenting Experimental Data Professor: Sam Kassegne, PhD, PE

Lecture Covers Terminologies (error, precision, resolution, etc) Statistical Theory Based on Population Statistical Theory Based on Sample (student ‘t’, Chi squared, etc) Homework Problems

1) TERMINOLOGIES ERRORS Error = ε = X m – X true :  Difference between measured and true values  Can never be calculated exactly Bias/Systematic errors: errors that occur the same way each time a measurement is taken  Cannot be treated using statistics  Commonly a zero offset or scale error Precision/Random errors: different for each measurement but average to zero

COMBINED ERROR Bias error > typical precision errortypical precision error > bias error

CLASSIFICATION OF ERROR Bias or systematic error  Calibration errors  Certain consistently recurring human errors  Certain errors caused by defective equipment  Loading errors  Limitations of system resolution Precision or random error  Human error  Errors caused by disturbance of the equipment  Errors caused by fluctuating experimental conditions  Errors derived from insufficient measuring- system sensitivity

OTHER ERRORS Illegitimate error  Blunders and mistakes during experimentation  Computational errors in data reduction Errors that are sometimes bias and sometimes precision error  Errors from instrument backlash, friction, hysteresis  Errors from calibration drift, variation in test and environmental conditions

Introduction to UNCERTAINTY Uncertainty: Where: B x = bias uncertainty P x = precision uncertainty

ESTIMATING PRECISION UNCERTAINTY Distribution of error: probability that an error of given size will occur Population: finite or infinite group from which samples are drawn Sample: limited set of measurements Gaussian or Normal Distribution (bell curve) Probable difference/Confidence Interval:  Estimate of precision uncertainty of the measured sample

SAMPLE vs. POPULATION A sample has a defined number of members (n) A population may have:  (n) members or may be infinite A sample is randomly taken from a population of indefinite size

PROBABILITY DISTRIBUTIONS Histogram (Tossed Dice) Distribution Types:  Gaussian (standard)  Students t-distribution  χ 2 distribution (lower right)

2) THEORY BASED ON POPULATION Probability Density Function (PDF)  Describes random precision error of sampling  Probability = area under curve

GAUSSIAN DISTRIBUTION For Gaussian Distribution:  x = magnitude of particular measurement  μ = mean value of entire population  σ = standard deviation of entire population μ  X avg = (x 1 +x 2 +…+x n )/n for large n For a measurement (x) the deviation (d) d = x - μ

GAUSSIAN DISTRIBUTION (ctd.) σ = standard deviation Probability estimates Standard curve

Probability Question: 1)What is the area under the curve between z = and z=1.43? 2)What range of z will contain 90% of the data?

3) THEORY BASED ON SAMPLE Mean Standard Deviation

Useful Terms in Theory Based on Samples Central Limit Theorem - For large number of n for a given number of samples, the distribution of the mean values is Gaussian. Eq 1. Confidence Interval (c%) - used to indicate the reliability of an estimate. How likely the interval is to contain the parameter is determined by the confidence level. For a population with unknown mean  and known standard deviation , a confidence interval for the population mean, based on a simple random sample of size n is: Eq 2. Remember that: Eq 3 & Eq 4.

A) Student’s t-statistic Useful for estimating confidence interval for small samples (n << 30) Assumes the underlying population satisfies the Gaussian distribution Considers the distribution of a quantity ‘t’ for a specific confidence interval

Student’s t-statistic Probability for the t-statistic approaches Gaussian as n  infinity t-statistic eliminates probable outliers and what remains must center around the true value = n-1 is called the degree of freedom)

Student’s t-statistic Example 3.6 on pg 74  designates the area (hence probability) outside a specified value of t  Conversely,  the probability that a given value will lie inside of the limits t  is 1- . n = 14, = oz. S x = , = 13. From Table 3.6, t 0.025,13 = two sided interval = +/- t 0.025,13 Sx/sqrt(n) = +/  = / oz with CI of 95%

B) CHI SQUARED TEST * The χ 2 test measures ‘goodness of fit’ * Tests the probability that a sample of data is or is not normally distributed. total area under curve = unity. Curve starts at chi^2 = 0 The curve is not symmetrical. As number of data increases, it becomes similar to normal distribution.

CHI SQUARED TEST – measures the ‘goodness of fit’ The χ 2 goodness of fit test  O i = the number of observed data in a bin  E i = the number of expected data in a bin  N = the number of data bins, or “class intervals”   = degrees of freedom=1- number of bins Large discrepancy between Oi and Ei

CHI SQUARED EXAMPLE For a group of 50 trees on a set area of land, to determine if the trees prefer dry soil we examine the following: If 50% of the land is dry, 30% is moist, and 20% is wet, then if there were no preference, we would expect: 25 trees on dry soil; 15 on moist; and 10 on wet If actually: 40 on dry; 8 on moist; 2 on wet 40-25=15; 15 2 /25=9 | 8-15=7; 7 2 /15=3.2 | 2-10=8; 8 2 /10=6.4 χ 2 = ( )= 18.6 Examining the previous graph it is shown that it is extremely unlikely that there is no soil preference, as our Chi squared value falls far above the questionable probability (Note that bin-size, n = 3; nu = 2)