Machinery's Handbook, 31st Edition
APPLYING STATISTICS 139 the sample mean, X , absolute error of the sample mean, A x , and t -distribution (Student’s) value. A confidence interval for the population mean satisfies the inequality: (6) Applying Statistics X A x t – µ X A x t + ≤ ≤ Minimum Number of Test or Data Points.— The minimum number of data points needed to represent a sample size can be determined through the formulas for the coeffi - cient of variation V , Equation (3), the absolute error of the sample mean A x , Equation (4), and the relative error of the sample mean R x , Equation (5). According to Equation (4), the absolute error of the sample mean is: A x S n = ---- The other expression for the absolute error of the sample mean from Equation (5) is: (7) Because the values to the left of the equal sign in Equations (4) and (7) are equal, the values on the right are also equal and, therefore: (8) Solving for n in Equation (8) produces: (9) Because S / X is the coefficient of variation V , see Equation (3), then: (10) The coefficient of variation of the sample mean must be known or selected according to previously collected data of a similar kind, or, if necessary, preliminary tests should be conducted to estimate its value. Based on numerous studies of cutting tool performance and publications on mechanical properties of cutting tool materials, values of the coeffi cient of variation within 25 to 45% are considered typical. A relative error of the sample mean between 6 and 12% is also considered typical. The coefficient of variation and the relative error are used to estimate how many tests are required. For example, if V = 30% and R x = 8%, then the number of tests required is n = 30 2 /8 2 = 14. Comparing Products with Respect to Average Performance.— Lab and field tests are usually conducted to compare the average performance of two or more products. The term “average performance” is a quantitative value, which can be any mechanical, physical, or chemical characteristic of a product. For example, the average tool life of drills and indexable cutting inserts or the average hardness of cemented carbide grades, etc. ( Tool life usually refers to the usable life of the tool based on wear, a judgment that is made by the manufacturer for a specific product based on various factors.) The products may differ in manufacturing procedure (CVD or PVD coatings), in chemical composition (alloying elements and their amount), and in other parameters. Data collected during experiments must be statistically treated to determine whether products have the same performance characteristics or not. A x XR x = S n ---- XR x = n S XR x = -------- n V R x = --- and n V 2 ( R x ) 2 = - - -------
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