ASE 369K Measurements and Instrumentation Probability and Statistics for Analysis of Experimental Data Consider a variable u that is a real valued and in the range ∞≤≤∞−u . The probability distribution function associated with is defined as the probability that )(xF uxu : ≤ (1) (xuF(x) ≤= Prob) and must be determined through many observations (measurements of u). It is easy to see that must have the following properties: )(xF x)(xF 1 ∞≤≤∞−≤≤=∞=−∞≤≤aaFFFbaF(b)F(a)for1)(01)(0)(for(2) An example of a distribution function with these properties is shown in the sketch above. If the distribution function is smooth, it can be differentiated; this results in the definition of the probability density fuction, : )(xf dxxdFf(x))(= (3) As a consequence of the properties of the distribution function listed in Eq. (2), we get the following properties for the : )(xf (4) ∫[∫∫1)()()(Prob)()()(0==≤≤=−=≥∞∞−∞−dxxfxFdxxfbuaaFbFdxxff(x)xbax)(xfxµ] An example of a probability density function with the above properties is shown in the sketch above. is the probability of obtaining the vaue )(xfx. For any probability density function, is the expected value (the value of x that occurs most frequently) and is the variance. These quantities are defined by the following equations: xµ2xσ Professor K. Ravi-Chandar 1ASE 369K Measurements and Instrumentation (5) ()∫∫∞∞−∞∞−−==dxxfxdxxxfxxx)()(22µσµ We can explore these concepts through the example of a specific probability density function, , corresponding to a normal or Gaussian distribution: )(xfG ()22221)(xxxGexfσµπ−−= (6) This is perhaps the most important distribution function since it occurs in most cases. However, there are clear exceptions where this distribution does not hold. The interpretation of the expected value and the variance for the Gaussian distribution is the following: ()68.0)(Prob ==+<≤−∫+−xxxxdxxfxxxxxσµσµσµσµ (7) ()95.0)(22Prob22==+<≤−∫+−xxxxdxxfxxxxxσµσµσµσµ (8) Statistics based on sampled data: In a laboratory measurement aimed at determining a quantity x, one may make N measurements, where N is a finite number – probably 5 to 10 or in special cases ~1000! Now the statistical analysis indicated above must be performed based on this sample and not on the whole “population”; we do not know the probability density function completely, but get only )x(fN samples. Therefore, the expected value and variance are estimated as follows: ()∑∑==−−==NiixNiiNxxSNxx12211 (9) The integrals in Eq. (2) have been replaced by summation. Also, to distinguish between the true expected value and variance and their estimates based on N samples, we use x and to denote the estimates. 2xSx is the mean value and is the standard deviation. They are interpreted in the same manner as the expected value and variance: xS ()68.0Prob=+≤≤−xxSxxSx (10) Professor K. Ravi-Chandar 2ASE 369K Measurements and Instrumentation ()95.022Prob=−≤≤−xxSxxSx (11) So, the precision in the measurement with a 95% confidence interval may be written as follows: xSxx 2±= (12) i.e., there is a 95% confidence that any subsequent measurement will be within the interval given in Eq. (12). Imagine repeating the N measurements many times over; in each trial one gets an estimate of the mean value. These mean values are themselves expected to obey a Gaussian distribution. Based on these estimates of the mean, a standard deviation of the mean values can be determined: NSSxx= (13) The interpretation of the standard deviation of the mean value is the following: there is a 95% confidence that the best estimate for the quantity x will be within the interval NSxxxbest2±= (14) As N increases, the 95% confidence interval is tighter around the mean value. The standard deviation of the mean is taken to be the precision uncertainty in the measurement of the variable x (with an implicit 68% confidence). Now, let us look at an example. Suggested Thought Experiments [You don’t have to actually do these experiments to determine the answers]. 1. Consider the familiar exercise of throwing darts. Throw the dart 100 times, aiming for the bulls-eye (although this is not always the case in a game). For each trial, measure the distance of the dart from the center. Can you interpret your measurements in terms of a Gaussian distribution? What does the mean value indicate? Standard deviation? How would the measurements differ for different individuals? 2. Make a survey of heights of individuals in any group – your class, dorm, etc. What do you think the probability density function for height will look like? Wfactors might influence this distribution? [if you are a clothing or a shoe manufacturer or retailer, you need to know this!] hat 3. On a lazy Saturday afternoon, you are passing time by sitting near a roadway and measuring the speed of cars as they go by. In one hour, let us say you observed 300 cars. What do you expect your observations would reveal? Professor K. Ravi-Chandar 3ASE 369K Measurements and Instrumentation Professor K. Ravi-Chandar 4 Example 1 Sixteen measurements were taken of some variable x. These measurements are tabulated below. What is the sample mean? What is the sample standard deviation? Determine the 68% and 95% confidence intervals of a single measurement of x, and of the sample mean. 1 2 3 4 5 6 7 8 73.83 69.81 83.30 70.58 66.75 85.42 76.14 91.89 9 10 11 12 13 14 15 16 92.28 80.55 79.14 51.61 72.52 83.88 79.90 81.55 Solution: ()52.2 ,01.101 ,45.77121===−−===∑∑==NSSNxxSNxxxxNiixNii Confidence Interval For a single measurement For the mean value 68% ⇒+≤≤−xxSxxSx ⇒≤≤46.8744.67 x ⇒+≤≤−xxSxxSx 97.7992.74 ≤≤x 95% ⇒+≤≤−xxSxxSx 22 ⇒≤≤42.9443.57 x ⇒+≤≤−xxSxxSx 2250.8240.72 ≤≤xASE 369K Measurements and Instrumentation Propagation of Uncertainties Let us now look at how precision uncertainties in measured variables affect precision uncertainties in calculated results. The definitions in Eq. (5) involve linear operators. Therefore, if we consider two functions x and with expected values and and variances and , then the expected value and variance of the function yxµyµy2xσ2yσxβα+ (where ()βα, are two constants) are (15)