Machinery's Handbook, 31st Edition
STATISTICAL ANALYSIS 137 Such simple examples are not that useful in explaining manufacturing statistics , where any event can have many values—for instance, in the hardness of an alloy or duration of tool life. However, more complex statistical analysis starts with the same concepts: whether a company is inspecting for defects or for a range of values of physical charac- teristics, the analysis involves how data from an experiment is regarded and how it can be used to determine the probability of various outcomes. Statistical analysis depends on the type of statistical distributions that apply to various properties of the data being examined. Some common statistical distributions are: 1) normal; 2) log-normal; 3) exponential; 4) binomial; 5) Weibull; and 6) Poisson. Normal distribution is described here. Normal Distribution Analysis.— The normal distribution is the most widely used statis- tical distribution for modeling mechanical, physical, electrical, and chemical properties that scatter randomly about a well-defined mean value without either positive or negative bias. This curve is frequently called a bell curve. In the following discussion, the charac- teristics of the normal distribution curve are assumed. Statistical analysis of raw data is a crucially important scientific and engineering tool that summarizes the characteristics of samples (number of observations, trials, data points, etc.). If a sample of data is randomly selected from the population, its statisti- cal characteristics converge towards the statistical characteristics of the population as the sample size increases. Because economic constraints, such as testing time and cost, prevent a large number of repeat tests, it is important to understand how a sample of data represents an approximation of the real population of data. The following parameters must be calculated to evaluate the sample of data with respect to the population of data: X = sample mean S = sample standard deviation V = coefficient of variation t = critical value of t -distribution (or Student’s t -distribution) m = population mean σ = population standard deviation X ± t A x = confidence interval for the population mean Sample Mean (X): There are several types of average measures, the most common being the arithmetic mean , or sample mean . It is the value about which all data are “centered.” The sample mean X is calculated as: (1) A x = absolute error of the sample mean R x = relative error of the sample mean where x i = individual data point values, and n = number of data points. According to the central limit theorem , when a large number of samples is taken independently from a population, the sample means follow a normal distribution, regardless of how data in each sample are distributed. Sample Standard Deviation (S): A measure of the dispersion of data about its standard mean X . The sample standard deviation is calculated by the formula: i = 1 X 1 n -- n ∑ = x i
n ∑
2
x i X – ( )
(2)
i = 1
n – 1 = ----------------
S
where n – 1 = the number of degrees of freedom ( d. f. ) Degrees of Freedom (d.f.): The number of observations made in excess of the minimum needed to estimate a statistical parameter or quantity. For example, only one measure- ment is required to identify the width of an indexable insert’s flank wear that occurs while machining a workpiece. If the measurements are repeated seven times, then the sample variance of flank wear measurement has six degrees of freedom.
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