Machinery's Handbook, 31st Edition
142
APPLYING STATISTICS
Example 3: Tool Life Tests of 6.0-mm Diameter Drills with Different Web Thickness Drills with 2.0 mm Web, Group A Drill Groups Test Number n i Data, x i minutes Treatment of Data Characteristics of Normal Distribution A B 2 Number of data points, n 14 16 ( x i – X ) x i – X
122.3236 Number of degrees of freedom, n – 1 61.7796 Sample mean (s.m.), Equation (1) Coefficient of variation, Equation (3) 12.5316 Absolute error of the s.m., Equation (4) 17.4724 5.4756 Relative error of the s.m., Equation (5)
13
15
1 2 3 4 5 6 7 8 9
15.68 18.88 19.20 22.56 23.20 24.40 24.64 26.56 27.20 30.24 32.16 33.60 36.80 39.20
–11.06 –7.86 –7.54 –4.18 –3.54 –2.34 –2.10 0.18
26.74 15.01
56.8516 Sample standard deviation, Equation (2) 6.94 7.30
26.0% 48.6% 1.85 1.83 6.9% 12.2%
4.4100 0.0324 0.2116
2.160 d.f.=13
2.131 d.f. =15
t -value at 95% confidence level at given degrees of freedom, Table 1
0.46 3.50 5.42 6.86
10 11 12 13 14
12.2500 Population mean is greater than: 29.3764 Population mean is less than:
22.74 11.11 30.74 18.91
47.0596
Comparison of Grades A and B
10.06 12.46
101.2036 Pooled standard deviation, Equation (11)
7.14
155.2516
t -value at 95% confidence level at given degrees of freedom, d.f. = 13 + 15 = 28, Table 1
2.048
X = 26.74
n = 14
5.35 11.73
Σ x i = 374.32
Σ( x i – X )
2 = 626.2296 Decision criterion, Equation (12)
Difference between the two sample means
Drills with 0.9 mm Web, Group B
Treatment of Data
Treatment of Data
Test Number n i
Data, x i minutes
Data, x i minutes
Test Number n i
x i – X
( x i – X )
i – X
( x i – X )
x
2
2
1 2 3 4 5 6
5.04 6.48 7.12 7.20 9.44 11.36
–9.97 –8.53 –7.89 –7.81 –5.57 –3.65
99.4009 72.7609 62.2521 60.9961 31.0249 13.3225
7 8 9
12.16 14.24 15.68 16.32 17.84 18.00 21.28 23.04 24.60 30.40
–2.85 –0.77 0.67 1.31
8.1225 0.5929 0.4489 1.7161 8.0089 8.9401 39.3129 64.4809 91.9681 236.8521
10 11 12 13 14 15 16
2.83 2.99 6.27 8.03
Conclusion:
Sample means of the tool life of the drills in Group A and B are statistically significant at the 95% confidence level, since the difference of the sample means (26.74 – 15.01 = 11.73 min) is greater than the decision criterion (5.35 min Difference). Note: n i = i th test or data point, x i = i th value of the data point
9.59 15.39
X = 15.01 2 = 800.2008
n = 16
( x i – X )
x i = 240.20
Regarding Machinability and Hardness: In cutting steels, the allowable cutting speed for a given tool life between grindings is, as a general rule, inversely proportional to the hardness of a given steel. To illustrate, tests in turning an alloy steel with a high-speed steel tool showed a cutting speed of 70 feet per minute (21.3 meters per minute) when the hardness of the steel was 180 BHN (Brinell Hardness Number); the cutting speed had to be reduced to about 35 feet per minute (10.7 meters per minute) when the hardness was increased to 360 BHN, the life between tool grindings for these tests being 20 minutes in each case. The machinability of other steels of the same hardness might vary. For exam- ple, the tests just referred to showed more or less variation in the cutting speeds for steels of the same hardness, but having different compositions or properties. Thus, while there is a constant relationship between the hardness of a steel and its tensile strength, there is not the same constant relationship between steel hardness and machinability as applied to different steels.
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