Machinery's Handbook, 31st Edition
APPLYING STATISTICS
141
Example 1:
Tool Life Tests of CNMG-432 Indexable Inserts Made of Two Different Carbide Grades (A and B) Carbide Grade A
Carbide Grades A B
Treatment of Data
Characteristics of Normal Distribution
Test Number n i
Data, x i minutes
x i – X
( x i – X )
Number of data points, n
7 6
7 6
2
Number of degrees of freedom, n – 1 Sample mean (s.m.), Equation (1) Sample standard deviation, Equation (2) Coefficient of variation, Equation (3) Absolute error of the s.m., Equation (4) Relative error of the s.m., Equation (5)
1 2 3 4 5 6 7
15.0 19.0 16.9 16.6 16.6 16.1 19.4
–2.1 1.9 –0.2 –0.5 –0.5 –1.0 2.3
4.41 3.61 0.04 0.25 0.25 1.00 5.29
17.1 14.8 1.6 2.1 9.4% 14.2% 0.6 0.8 3.5% 5.4%
t -value at 95% confidence level at given degrees of freedom, d.f. = 6, Table 1 2.447 2.447
n = 7
X = 17.1 2 = 14.85
Population mean is greater than: Population mean is less than:
15.6 12.8 18.6 16.8
Σ x i = 119.6
Σ ( x i – X )
Carbide Grade B
Comparison of Grades A and B
Treatment of Data
Pooled standard deviation, Equation (11) t -value at 95% confidence level at given degrees of freedom, d.f. = 6 + 6 = 12, Table 1 Decision criterion, Equation (12) Difference between the two sample means
1.9
Test Number n i
Data, x i minutes
x i – X
( x i – X )
2.179
2
2.2 2.3
1 2 3 4 5 6 7
14.6 13.5 15.6 12.4 14.6 13.8 19.1
–0.2 –1.3 –2.4 –0.2 –1.0 4.3 0.8
0.04 1.69 0.64 5.76
Conclusion:
Sample means of the tool life of carbide grades A and B are statistically significant at the 95% confidence level, since the difference of the sample means (17.1 – 14.8 = 2.3 min) is greater than the decision criterion (2.2 min). Note: n i = i th test or data point, x i = i th value of the data point
0.04 1.00 18.49
n = 7
X = 14.8 2 = 27.66
Σ x i = 103.6
Σ ( x i – X )
Example 2:
Tensile Strength of Carbon Steel Specimens Versus Heat Treatment Carbon Steel Sample A Samples Test Number n i Data, x i MPa Treatment of Data Characteristics of Normal Distribution A B x i – X ( x i – X ) 2 Number of data points, n 5 5 1 522.0 6.9 47.61 Number of degrees of freedom, n – 1 4 4 2 511.0 –4.1 16.81 Sample mean (s.m.), Equation (1) 515.1 504.7 3 488.9 –26.2 686.44 Sample standard deviation, Equation (2) 24.8 11.1 4 553.7 38.6 1490.00 Coefficient of variation, Equation (3) 4.8% 2.2% 5 499.9 –15.1 228.01 Absolute error of the s.m., Equation (4) 11.1 5.0 n = 5 X = 515.1 Relative error of the s.m., Equation (5) 2.2% 1.0% Σ x i = 2575.5 Σ( x i – X ) 2 = 2468.9 t -value at 95% confidence level at given degrees of freedom, d.f. = 4, Table 1 2.776 2.776 Carbon Steel Sample B Population mean is greater than: 484.3 490.8 Test Number n i Data, x i MPa Treatment of Data Population mean is less than: 545.9 518.6 x i – X ( x i – X ) 2 Comparison of Samples A and B 1 517.1 12.4 153.76 Pooled standard deviation, Equation (11) 19.2 2 490.2 –14.5 210.25 t -value at 95% confidence level at given degrees of freedom, d.f. = 4 + 4 = 8, Table 1 2.306 3 499.1 –5.6 31.36 4 514.4 9.7 94.09 Decision criterion, Equation (12) 28.0 5 502.6 –2.1 4.41 Difference between the two sample means 10.4 n = 5 X = 504.7 Conclusion: Σ x i = 2523.4 Σ ( x i – X ) 2 = 493.87
Sample means of the tensile strength of samples A and B are statistically insignificant at the 95% confidence level, since the difference of the sample means (515.1 – 504.7 = 10.4 MPa) is less than the decision criterion (28.0 MPa).
Note: n i = i th test or data point, x i = i th value of the data point
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