Analysis of Experimental Data; Introduction Some form of analysis must be performed on all experimental data. In this chapter, we will consider the analysis of data to determine errors and uncertainty, precession, and general validity of experimental data. Real errors are those factors which are vague to some extent and carry some uncertainty. Our task is to determine just how uncertain a particular observation may be and to device a consistent way specifying the uncertainty in analytical form.

Analysis of Experimental Data; Introduction Some of the types of errors that may cause uncertainty in an experimental data are: Gross blunders in instruments construction. Fixed, systematic, or bias errors that cause the same repeated readings to be mistaken. (Have unknown reasons!!!) Random errors that may be caused by personal fluctuations, friction influences, electrical fluctuations,... This type usually follow a certain statistical distribution

Analysis of Experimental Data; Introduction Errors can be broadly analyzed depending on the “common sense” or some “rules of thumb”. e.g. Assume the calculation of electrical power from P = EI, where E and I are measured as E = 100V  2V, I = 10A  0.2A the nominal value of P is 10010=1000W the worst possible variations: Pmax=(100+2)(10+0.2) = 1040.4W & Pmin= (100-2)(10-0.2) = 964.4W i.e., the uncertainty in the power is +4.04% & -3.96%

Uncertainty Analysis Yet, a more precise methods for estimating uncertainty are needed. Consider the following “Suppose a set of measurements are made. Each measurement may be expressed with the same odds. These measurements are then used to calculate some desired result of the experiment. We like to estimate the uncertainty in the calculated result. The result R is a given function of the independent variables x1,x2,…,xn. Thus: R = R(x1,x2,x3,…xn)”

Uncertainty Analysis Let R be the uncertainty in the result and 1, 2,…, n be the uncertainty in the independent variables, then: For product functions: R = [((R/x1)1)2 +((R/x2)2)2 +…+((R/x3)3)2 ]1/2 R = x1a1 x2a2 …xnan R/R = [(aixi/xi)2]1/2

Uncertainty Analysis R = [(aixi)2]1/2 For additive functions: R =  aixi Note that i has the same units of xi R = [(aixi)2]1/2

Uncertainty Analysis E.g. The resistance for a certain copper wire is given by: R = Ro[1 + (T-20)] Where Ro = 6±0.3% at 20ºC,  = 0.004 ±1%, and T of the wire is T =30±1ºC. Calculate the resistance of the wire and its uncertainty? Sol …… Read examples 3.2 and 3.3. C-1

Statistical Analysis of Experimental Data Definitions: * Mean: * Median: is the value that divides the data points into half * Standard deviation * is called the variance

Probability distributions It shows how the probability of success, p(x), in a certain event is distributed over the distance x. Two main categories; district and continuous. The binomial distribution is an example of a district probability distribution. It gives the number of successes n out of N possible independent events when each event has a probability of success p.

Probability distributions The probability that n events will succeed is given as: E.g. if a coined is flipped three times, calculate the probability of getting 0, 1, 2, or 3 heads in these tosses?

The Gaussian or normal error distribution It is a continuous probability distribution type. The most common type. If the measurement is designated by x, the Gaussian distribution gives the probability that the measurement will lie between x and x+dx, as:

The Gaussian or normal error distribution x P(x) 1 2 1<2

The Gaussian or normal error distribution The probability that a measurement will fall within a certain range x1 of the mean reading is let =(x-xm)/, then P becomes

The Gaussian or normal error distribution Values for the integral of the Gaussian function are given in table 3.2. Example

The Gaussian or normal error distribution-Confidence level The confidence interval expresses the probability that the mean value will lie within a certain number of  values. The z symbol is used to represent it. Thus: For small data samples; z is replaced by: Using the Gaussian function integral values, the confidence level (error) in percent can be found. (Table 3.4)

The Gaussian or normal error distribution-Confidence level The level of significance is: 1- the confidence level See example 3.11

The Gaussian or normal error distribution-Chauvenet’s Criterion It is a way to eliminate dubious data points. The Chauvenet’s criterion specifies that a reading may be rejected if the probability of obtaining the particular deviation from the mean is less than 1/2n. The attached table list values of the ratio of deviation (d=abs(x-xm) to standard deviation for various n according to this criterion n dmax/ 3 1.38 4 1.54 5 1.65 6 1.73 7 1.80 10 1.96 15 2.13 25 2.33 50 2.57 100 2.81 300 3.14 500 3.29 1000 3.48

Method of Least Squares We seek an equation of the form: y = ax + b where y is a dependent variable and x is an independent variable. The idea is to minimize the quantity: This is accomplished by setting the derivatives with respect to a and b equal to zero.

Method of Least Squares Performing this, the results are:

Method of Least Squares Designating the computed value of y as y`, we have y` = ax + b And the standard error of estimate y for the data is:

The Correlation Coefficient After building the y-x correlation, we want to know how good this correlation is. This is done by the correlation coefficient r which is defined as: