MECH 373 Instrumentation and Measurements Lecture 13 Statistical Analysis of Experimental Data (Chapter 6) • Introduction • General Concepts and Definitions • Probability • Probability Distribution Function

Measures of Central Tendency (review) Mean (sample mean, population mean) Median If the measured data are arranged in ascending or descending order, the median is the value at the center of the set. Odd: middle one Even: average of the middle two Mode The mode is the value that occurs most often. If no number is repeated, then there is no mode for the list. Range The range is just the difference between the largest and smallest values.

Measures of Dispersion (review) Spread or variability of the data Deviation Mean deviation Standard deviation (Sample, Population) Variance

Example 2 (review) Find the mean, median, mode, range, standard deviation and variance of the following dataset. There is a total of 60 samples

Basics of Probability Probability Probability is a numerical value expressing the likelihood of occurrences of an event relative to all possibilities in a sample space. successful occurrences Probability of event A = total number of possible outcomes If A is certain to occur, P(A) = 1 If A is certain not to occur, P(A) = 0 If B is the complement of A, then P(B) = 1 - P(A) If A and B are mutually exclusive (the probability of simultaneous occurrence is zero), P(A or B) = P(A) + P(B) If A and B are independent, P(AB) = P(A)P(B) (occurrence of both A and B) P(AB) = P(A) + P(B) - P(AB) (occurrence of A or B or both)

Probability Distribution Functions • An important function of statistics is to use information from a sample to predict the behavior of a population. • We are often interested to know the probability that our next measurement will be within certain range. • One approach is to use the sample data directly (as shown in next slide). This approach is called use of an empirical distribution. • For particular situations, the distribution of random variable follows certain mathematical functions. • The sample data are used to compute parameters in these mathematical functions, and then these mathematical functions are used to predict behavior of the population. • For discrete random variables, these functions are called probability mass functions. • For continuous random variables, these functions are called probability density functions.

Probability Distribution Functions From: Dr. McNair

Probability Distribution Functions Probability Mass Function If a discrete random variable can have values x1,…,xn, then the probability of occurrence of a particular value of xi is P(xi), where P is the probability mass function for the variable x. Normalization Mean Variance

Probability Distribution Functions Probability Mass Function From: Dr. McNair

Probability Distribution Functions Probability Mass Function From: Dr. McNair

Probability Distribution Functions Probability Density Function Probability of occurrence in an interval xi and xi+dx Probability of occurrence in an interval [a,b] Mean of population – is also the expected value Variance of population

Probability Distribution Functions Probability Density Function From: Dr. McNair

Probability Distribution Functions Probability Density Function From: Dr. McNair

Probability Density Functions Example: Ball bearing life probability distribution function: f(x) = 0 for x < 10h; f(x) = 200/x3 for x ≥ 10h A, Expected bearing life (mean): E(x) = u = ∫∞10 x.(200/x3) dx = 20h B, Less than 20 hours. P(x<20) = ∫10-∞ 0 dx + ∫2010 200/x3 dx = 0.75 P(x ≥20) = 1 – 0.75 = 0.25 From: Dr. McNair

PDF with Engineering Applications A number of distribution functions are used in engineering applications. We will discuss briefly about some common distribution functions. ►Binomial (values either true/false) ►Poisson ►Normal (Gaussian) ►Student’s t ► χ2 (Chi-squared) ►Weibull ►Exponential ►Uniform

Probability Distribution Functions

Probability Distribution Functions

Probability Distribution Functions

Probability Distribution Function  Binomial Distribution • The binomial distribution is a distribution which describes discrete random variables that can have only two possible outcomes: “success” or “failure”. • This distribution has applications in production quality control, when the quality of a product is either acceptable or unacceptable. • The binomial distribution provides the probability (P) of finding exactly r successes in a total of n trials and is expressed as where, p is the probability of success which remains constant throughout the experiment, and is called as n combination r, which is the number of ways that we can choose r identical items from n items. • The expected number of successes in n trials for binomial distribution is • The standard deviation of the binomial distribution is

Probability Distribution Function  Binomial Distribution - Example Machines being tested. Success rate = 90% (10% failure). Probability of 15 successes (r) out of 20 machines (n)? P(15) = [20!/(5!.15!)]∙ 0.915 ∙ (1 – 0.9)(20-15) P(15) = 0.032 3.2% Chance having 5 out of a batch of 20 machines needs some repair.

Probability Distribution Function  Poisson Distribution • The Poisson distribution is used to estimate the number of random occurrences of an event in a specified interval of time or space if the average number of occurrences is already known. • The probability (P) of occurrence of x events is given by where, λ is the expected or mean number of occurrences during the interval of interest. • The expected value of x for the Poisson distribution, the same as the mean (μ), is given by • The standard deviation is given by • Probability that the number of occurrences is less than or equal to k is given by

Probability Distribution Function  Poisson Distribution - Example Welded joint pipes having an average of five defects per 10 linear meters of weld (0.5 defects per meter). A, Probability of a single defect in a weld of 0.5m long. λ = Average number of defects in 0.5 m = 0.5 * 0.5 = 0.25. P(1) = e-0.25 0.251 / 1! = 0.194 (19.4%). B, More than one defect in weld of 0.5m long. Zero defect P(0) = e-0.25 0.250 / 0! = 0.778. P (x > 1) = 1 – 0.778 – 0.194 = 0.028 (2.8%)

Probability Distribution Function  Normal Distribution • The Normal or Gaussian distribution is a simple distribution function that is useful for a large number of common problems involving continuous random variables. -1 0 1 2 3 4 5 Symmetric about μ Bell-shaped Mean μ : the peak of the density occurs Standard deviation σ: indicates the spread of the bell curve. m = 2

Standard Normal Distribution (mean=0, standard deviation=1) 68% 95% 99.7% 1s Z[-1,1] 2s Z [-2,2] 3s Z [-3,3]

Normal Distribution Example The distribution of heights of American women aged 18 to 24 is approximately normally distributed with mean 65.5 inches and standard deviation 2.5 inches. 68% of these American women have heights between 65.5 – 1(2.5) and 65.5 + 1(2.5) inches, or between 63 and 68 inches. 95% of these American women have heights between 65.5 - 2(2.5) and 65.5 + 2(2.5) inches, or between 60.5 and 70.5 inches. 99.7% of these American women have heights between 65.5 - 3(2.5) and 65.5 + 3(2.5) inches, or between 58 and 73 inches. 68% 99.7% 95%