October 2011 / Volume 63 / Issue 10|
By Edmund Isakov, Ph.D.
Courtesy of Walter USA
Statistical analysis of metalcutting data can help shops select the right indexable insert geometry.
Statistical analysis of metalcutting data, such as tool life and cutting edge wear, is an important scientific and engineering method because it defines the relationship between the limited test data (“sample”) and large quantity of real data (“population”). Testing time and cost prevent conducting a large number of repeated tests (specimens). It is critical, then, to understand how a sample of data represents an approximation of the real population of data.
For example, Walter USA LLC, Waukesha, Wis., conducted field tests for finishing drive shafts made of AISI 1050 carbon steel using the following tool and cutting parameters.
Indexable insert: DNMG 432-xxx WPP10S (geometry A)
Feed rate: 0.012 ipr
DOC: 0.032 "
Cutting speed: 800 sfm
The field test results (Table 1) show the calculated average number of drive shafts (sample mean) machined with Walter’s indexable inserts is 28 percent larger than the number of drive shafts machined with a competitor’s inserts. However, to calculate only the sample mean is not enough to conclude that these values are statistically significant. Therefore, in addition to the sample mean, the following parameters are required to perform statistical analysis. These parameters must be calculated to determine how the sample size, or limited data, reflects the population size, or unlimited data.
Sample Standard Deviation
Sample standard deviation (σ) is a measure of the dispersion of data about its standard mean. Sample standard deviation is calculated with the formula:
Where xi is an individual data point, X is a sample mean and n - 1 is the number of degrees of freedom.
In our case, degrees of freedom can be defined as the number of tests conducted in excess of the minimum needed to estimate a statistical parameter or quantity. For example, one field test identified the quantity of the drive shafts machined with a certain cutting insert. If a field test was repeated four times, then the sample variance of the quantity of the drive shafts has three degrees of freedom, because three more tests were conducted to observe the difference in quantity per test.
To calculate sample standard deviation for each sample mean, the treatment of field test data was performed by the author’s calculator developed in Microsoft Excel (Tables 2 and 3).Absolute Error
The absolute error of the sample mean (α) is calculated by dividing the sample standard deviation (σ) by the square root of the number of data (n). The result is expressed in the same unit of measure as the sample standard deviation and sample mean:
Absolute error of the sample mean in Group 1 is:
Absolute error of the sample mean in Group 2 is:
The concept of t-Distribution was introduced in 1908 by William Sealy Gosset (1876 to 1937), who is best known by his pen name Student for his work on Student’s t-Distribution.
The critical value of t depends on the number of degrees of freedom and the probability of error. If a 95 percent two-sided confidence is used for statistical analysis, then the probability of error is ±5 percent, or ±2.5 percent per side. A 5 percent probability of error provides practical accuracy, which is commonly acceptable in various engineering calculations.
Critical values of t-Distribution should be used to locate such value in the column under the heading t0.025 with respect to the number of degrees of freedom (Table 4).
The calculated sample mean (Tables 2 and 3) is only the estimate of the true population mean. Therefore, it is important to define a confidence interval that determines a range within which the population mean lies. Such an interval depends on the sample mean (X), absolute error of the sample mean (α) and t-Distribution value.
A confidence interval for the population mean (µ) satisfies the inequality:
A 95 percent two-sided confidence interval is recommended for statistical analysis. In this case, the probability of error is ±5 percent, or ±2.5 percent per side. A 5 percent probability of error provides practical accuracy, which is commonly acceptable in various engineering calculations.
In our case, the number of degrees of freedom is three, and a chosen confidence level is 95 percent, therefore the t-Distribution value is 3.182 (Table 4). Based on the values of sample mean and absolute error of the sample mean (previously shown), the ranges of the population mean values are:
• 585 to 615 drive shafts can be machined with the geometry A inserts.
• 435 to 505 drive shafts can be machined with the competitor’s inserts.
Laboratory 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 characteristics of a product.
In our case, the average insert life in Groups 1 and 2 is expressed in how many drive shafts are machined. These groups are equivalent to the two products we want to compare and find out whether or not they represent one or two population means. Two parameters must be calculated to get the answer: weighted standard deviation and decision criterion.
Weighted standard deviation (σw) is calculated with the formula:
Where σ 1 and σ 2 are the sample standard deviations (Tables 2 and 3, respectively), and n1 and n2 are the number of field tests with inserts for Groups 1 and 2, respectively.
Decision criterion (DC) is calculated with the formula:
Where t0.025 is a value of t-Distribution at a 95 percent two-sided confidence interval, depending on degrees of freedom (Table 4), and σ w is a weighted standard deviation.
Due to space limitations, step-by-step calculations are omitted. The complete statistical treatment of data was performed using the previously mentioned calculator. The results are summarized in Table 5.
The sample means of the number of drive shafts finish machined are statistically significant, because the difference between them (130 pieces) is greater than the decision criterion (29.1). Therefore, these sample means represent two different population means.
Statistical treatment of the field tests data confirmed that tool life of the Walter’s DNMG 432-xxx WPP10S inserts (geometry A) is 28 percent longer than that of the competitor’s inserts. CTE
About the Author: Edmund Isakov, Ph.D., is a consultant, writer and frequent contributor to CTE. He is the author of the books “Engineering Formulas for Metalcutting” (Industrial Press, 2004), “Cutting Data of Turning of Steel” (Industrial Press, 2009) and the software “Advanced Metalcutting Calculators” (Industrial Press, 2005). He can be e-mailed at email@example.com or reached at (561) 369-4063. The author thanks the following people for providing the field tests data and image for this article: Terry Ashley, training manager, and Kurt Ludeking, product manager-turning, Walter USA LLC; and Gerd Kussmaul, global product manager-turning, Walter AG.
CUTTING TOOL ENGINEERING Magazine is protected under U.S. and international copyright laws. Before reproducing anything from this Web site, call the Copyright Clearance Center Inc. |
at (978) 750-8400.