Chapter 20 Statistical Considerations Lecture Slides The McGraw-Hill Companies 2012 Chapter Outline Shigleys Mechanical Engineering Design Random Variables A random experiment is an experiment in which a number of specimens are selected at random from a larger batch. A random variable is a variable quantity whose value depends on the outcome of a random experiment. A random variable is also called a stochastic variable. A sample space contains all the possible outcomes. An example sample space for a pair of dice, Shigleys Mechanical Engineering Design Fig. 201 Probability Distribution The probability of a random variable x taking on a specific value is called p = f (x). p is the number of times a specific x occurs divided by the total number of possible outcomes. A list of all possible values along with the corresponding probabilities is called a probability distribution. Define a random variable x as the sum of numbers obtained from tossing a pair of dice. The probability distribution for x, Shigleys Mechanical Engineering Design Table 201 Frequency Distribution Plotting the probability distribution makes it clear that the probability is a function of x. The probability function p = f (x) is often called the frequency function, or the probability density function (PDF). Shigleys Mechanical Engineering Design Fig. 202 Cumulative Probability Distribution The probability that x is less than or equal to a certain value of x i is found by summing the probability of all xs up to and including x i. This results in a cumulative probability distribution. The function F(x) is a cumulative density function (CDF) of x. Shigleys Mechanical Engineering Design Table 202 Cumulative Frequency Distribution A plot of the cumulative density function is called a cumulative frequency distribution. Shigleys Mechanical Engineering Design Fig. 203 Continuous Random Variables A discrete random variable x can only have discrete values. A continuous random variable x can have any value in a specified interval. For continuous random variables, the distribution plots are continuous curves. For a continuous probability density function F(x), When x goes to , Differentiation of Eq. (20-2) gives Shigleys Mechanical Engineering Design Arithmetic Mean, Variance, and Standard Deviation Sample mean of N elements Sample variance Sample standard deviation Alternate form Shigleys Mechanical Engineering Design Population Mean and Standard Deviation If the entire population is considered, and s x are replaced with x and respectively. The N 1 in the denominator is replaced by N. Shigleys Mechanical Engineering Design Discrete Frequency Histogram A discrete frequency histogram gives the number of occurrences, or class frequency f i, within a given range. Shigleys Mechanical Engineering Design Fig. 204 Mean and Standard Deviation with Class Frequency When the data are grouped by class frequency f i, the mean and standard deviation can be expressed as The cumulative density function that gives the probability of an occurrence at class mark x i or less is Where w i is the class width at x i. Shigleys Mechanical Engineering Design Notation Boldface characters indicate random variables that can be characterized by a mean and a standard deviation. The terms stochastic variable or variate are used to mean a random variable. A deterministic quantity is something that has a single specified value. A coefficient of variation is defined by A variate x can be expressed in the following two ways: where X represents a variate probability distribution function. Shigleys Mechanical Engineering Design Example 201 Shigleys Mechanical Engineering Design Table 203 Example 201 Shigleys Mechanical Engineering Design Example 202 Shigleys Mechanical Engineering Design Table 204 Example 202 Shigleys Mechanical Engineering Design Probability Distributions Some standard discrete and continuous probability distributions Gaussian, or normal Lognormal Uniform Weibull Shigleys Mechanical Engineering Design Gaussian (Normal) Distribution The Gaussian, or normal, distribution is defined as It can be expressed as Typical plots of the Gaussian distribution, with small or large standard deviation, look like the following Shigleys Mechanical Engineering Design Fig. 205 Transformation Variate z The deviation from the mean is expressed in units of standard deviation by the transform The transformation variate z is normally distributed, with a mean of zero and a standard deviation and variance of unity. That is, z = N (0, 1) Shigleys Mechanical Engineering Design Cumulative Distribution Function for Gaussian Distribution Integration of the Gaussian distribution to find the cumulative density function F(x) is accomplished numerically. To avoid the need for many tables for different values of mean and standard deviation, the z transform is used. The integral of the transform is tabulated in Table A10. A sketch of the standard normal distribution, showing the z transform is given below. The normal cumulative density function is labeled (z) Shigleys Mechanical Engineering Design Fig. 206 Example 203 Shigleys Mechanical Engineering Design Example 203 Shigleys Mechanical Engineering Design Fig. 207 Example 203 Shigleys Mechanical Engineering Design Lognormal Distribution In the lognormal distribution the logarithms of the variate have a normal distribution. Thus the variate is said to by lognormally distributed. The parent, or principal, distribution is expressed as The companion, or subsidiary, distribution is express through a transformation, Shigleys Mechanical Engineering Design Lognormal Distribution The probability density function (PDF) for x is derived from that for y. The PDF for the companion distribution is Shigleys Mechanical Engineering Design Example 204 Shigleys Mechanical Engineering Design Example 204 Shigleys Mechanical Engineering Design Table 205 Example 205 Shigleys Mechanical Engineering Design Example 205 Shigleys Mechanical Engineering Design Fig. 208 Uniform Distribution The uniform distribution is a closed-interval distribution that arises when the chance of an observation is the same as the chance for any other observation. The probability density function (PDF) is The cumulative density function, the integral of f(x), is Shigleys Mechanical Engineering Design Uniform Distribution The mean for the uniform distribution, The standard deviation, Shigleys Mechanical Engineering Design Weibull Distribution The Weibull distribution is a flexible, asymmetrical distribution with different values for mean and median. It contains within it a good approximation of the normal distribution and an exact representation of the exponential distribution. It is widely used to represent laboratory and field service data. Shigleys Mechanical Engineering Design Weibull Distribution The reliability given by a three-parameter Weibull distribution is For the special case when x 0 = 0, the two-parameter Weibull is If a specific reliability is given, solving Eq. (2024) for x, Shigleys Mechanical Engineering Design Weibull Distribution The characteristic variate represents a value of x below which lie 63.2% of the observations. The shape parameter b controls the skewness of the distribution. A good approximation to the normal distribution is obtained when 3.3 < b < 3.5 The distribution is exponential when b = 1 Shigleys Mechanical Engineering Design Fig. 209 Weibull Distribution To find the probability function, note that Shigleys Mechanical Engineering Design Weibull Distribution The mean and standard deviation are given by is the gamma function, tabulated in Table A34. The notation for a Weibull distribution is Shigleys Mechanical Engineering Design Example 206 Shigleys Mechanical Engineering Design Example 206 Shigleys Mechanical Engineering Design Propagation of Error In the equation for axial stress If F and A are random variables, then the stress is also a random variable, Errors inherent in F and A are said to be propagated to the stress variate . Shigleys Mechanical Engineering Design Propagation of Error If two variates are added, Similar relations for other arithmetic functions are given in Table 206. Shigleys Mechanical Engineering Design Means and Standard Deviations for Algebraic Operations Shigleys Mechanical Engineering Design Table 206 Example 20-7 Shigleys Mechanical Engineering Design Example 20-7 Shigleys Mechanical Engineering Design Linear Regression A process called regression is used to obtain a curve that best fits a set of data points. The process is called linear regression when the best-fitting straight line is found. Shigleys Mechanical Engineering Design Fig. 2010 Linear Regression Shigleys Mechanical Engineering Design Linear Regression The correlation coefficient r indicates the degree to which a set of data points correlates with a regression line. The standard deviations for the slope and intercept are given by Shigleys Mechanical Engineering Design Example 208 Shigleys Mechanical Engineering Design Example 208 Shigleys Mechanical Engineering Design Table 207 Example 208 Shigleys Mechanical Engineering Design Example 208 Shigleys Mechanical Engineering Design Fig. 2011