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Chapter VI – Statistical Analysis of Experimental Data CHAPTER VI Statistical Analysis of Experimental Data Measurements do not lead to a unique value. This is a result of the multitude of errors (mainly random errors) that can affect them. It is therefore essential to take into account these variabilities to use statistical methods to interpret the results obtained through an experiment. 6.1. Introduction An example of the use of statistical analysis of experimental data is to use a representation under the form of a histogram. Let us consider the following data representing the measurement of a temperature. Number of readings 1 1 2 4 8 9 12 4 5 5 4 3 2 Temperature (C) 1089 1092 1094 1095 1098 1100 1104 1105 1107 1108 1110 1112 1115 The data are first arranged into groups called bins. Here the size of a bin is 5C. The bins have to satisfy a couple of conditions: - The bins usually have the same size and cover the entire range of the data without overlap. Figure 6.1. Histogram. Instrumentation and Measurements \ LK\ 2009 47 Chapter VI – Statistical Analysis of Experimental Data The above ‘bell’ shaped curve of the histogram is typical of experimental data (although this is not a rule, see figure 6.2 for other types of histograms). Figure 6.2. Different distributions of data. a) symmetric; b) skewed; c) j-shaped; d) bimodale; e) uniform. - Discrete and random variables: Continuous random variables: It is a variable that can take any real value in a certain domain. Discrete variables: It a variable that can take a limited number of values. 6.2. General concepts and definitions Population: The population comprises the entire collection of objects, measurements, observations, and so on whose properties are under consideration and about which some generalizations are to be made. Sample: A sample is a representative subset of a population on which an experiment is performed and numerical data are obtained. Instrumentation and Measurements \ LK\ 2009 48 Chapter VI – Statistical Analysis of Experimental Data Sample space: The set of all possible outcomes of an experiment is called the sample space. Is can be a discrete sample space of a continuous sample space. Random variable: It is a variable that will change no matter how you try to precisely repeat the experiment. A random variable can be discrete or continuous. Distribution function: It is a mathematical relationship used to represent the values of the random variable. Parameter: It is an attribute of the entire population (exp. average, median, …) Event: It is the outcome of a random experiment. Statistic: It is an attribute of the sample (exp. average, median, …) Probability: It is the chance of occurrence of the event in an experiment. 6.2.2. Measures of central tendency n - Mean: x i 1 xi n N xi i 1 N - Median: it is the value at the center of a set, arranged in ascending or descending order. If the size of the set is even, the median represents the average of the two central peaks. And for a finite number of elements: - Mode: It represents the value of the variable that corresponds to the peak value of the probability of occurrence of an event. 6.2.3. Measures of dispersion - deviation: d i xi x - mean deviation: d n - di i 1 n standard deviation (for a population with a finite number of elements): N xi 2 N - Sample standard deviation: i 1 Instrumentation and Measurements \ LK\ 2009 49 Chapter VI – Statistical Analysis of Experimental Data S n xi x 2 i 1 n it is used to estimate the population standard deviation. - 2 for population The variance: variance 2 S for sample 6.3. Probability ‘Probability is a numerical value expressing the likelihood of occurrence of an event relative to all possibilities in a sample space.’ The probability of occurrence of an event A is defined as the number of successful occurrences (m) divided by the total number of possible outcomes (n) in a sample space, evaluated for n >> 1. probabilit y of event A m n The event can be represented by: 1) a continuous random variable (the probability is expressed as P(x)); 2) a discrete random variable (the probability is expressed as P(xi)). Here are some properties relative to probability: a- 0 Px or xi 1 b- If event A is the complement of event A, then: P( A ) 1 P( A) c- It the events A and B are mutually exclusive (A and B can not occur simultaneously): P( A B) P( A) P( B) d- It the events A and B are independent, the probability that both A and B will occur tighter is: P( AB) P( A) P( B) eThe probability of occurrence of A or B or both is: P( A B) P( A) P( B) P( AB) Example A distributor claims that the chance that any of the three major components of a computer (CPU, monitor, and keyboard) is defective is 3%. Calculate the chance that all three will be defective in a single computer? 6.3.1. Probability distribution functions An important function of statistics is to use information from a sample to predict the behavior of a population. For particular situations, experience has shown that the distribution of the random variable follow a certain mathematical pattern (function). Then, if the parameters of this function can be determined using the sample data, it will be possible to predict the properties of the parent population. Such functions are called: probability mass Instrumentation and Measurements \ LK\ 2009 50 Chapter VI – Statistical Analysis of Experimental Data functions for discrete random variables . For continuous random variables, these functions are called probability density functions. - Probability mass function: n P( x ) 1 ; i 1 i The mean of the population for a discrete random variable (also called the expected N value): xi P xi i 1 N 2 The variance of the population is given by: 2 xi Pxi i 1 - Probability density function: Pxi x xi dx f xi dx And then, to find the probability of x to occur between a and b values: b Pa x b f x dx a The mean of the population is: x f x dx The variance of the population is: 2 x f x dx 2 Example Consider the following probability distribution function for a continuous random variable: 3x 2 2 x3 f ( x) 35 0 elsewhere a- Show that this function satisfies the requirements of a probability distribution function. b- Calculate the expected mean value of x. c- Calculate the variance and the standard deviation of x. - Cumulative distribution function: Instrumentation and Measurements \ LK\ 2009 51 Chapter VI – Statistical Analysis of Experimental Data This type of distributions is used when you want to know the probability of event to be lower that a certain value (x). F rv x F ( x) f ( x)dx P(rv x) x i For discrete random variable: F rv xi P( x j ) j 1 Cumulative distribution function has the two following properties: P(a x b) F (b) F (a) P( x a) 1 F (a) Instrumentation and Measurements \ LK\ 2009 52