Machinery's Handbook, 31st Edition
140 APPLYING STATISTICS Statistical treatment of data obtained from experiments with two products includes the following steps: a) Calculation of the sample means X 1 and X 2 using Equation (1) b) Calculation of the sample standard deviations S 1 and S 2 using Equation (2) c) Calculation of a weighted, or pooled standard deviation using the following formula: (11) where n 1 and n 2 represent the number of data points for products 1 and 2 respectively. d) Selection of a confidence level. A 95% two-sided confidence level is recommended. At this confidence level, the probability of error is ± 2.5% per side. The values of t -distribu tion versus degrees of freedom ( d.f. ) are provided in Table 1 , and for a 95% confidence level are located in the column under the heading “ t 0.025 ” with respect to given degrees of freedom ( d. f. = n 1 + n 2 – 2). e) Calculation of decision criterion ( d.c. ) using the following formula: (12) f) Comparison of the value of the decision criterion with the difference of the sample means: X 1 – X 2 if X 1 > X 2 , or X 2 – X 1 if X 2 > X 1 . A product’s average performance is statistically significant if the difference in the two sample means is greater than the decision criterion d.c. X 1 X 2 – d.c. > or X 2 X 1 – d.c. > Table 1. Critical Values of t -Distribution d . c . t 0.025 S × p n 1 n 2 + n 1 n 2 --------- = S p n 1 – 1 ( ) S 1 2 + n 1 – 1 ( ) n 2 – 1 ( ) + = ------------------------------ n 2 – 1 ( ) S 2 2
t
d.f.
d.f.
d.f.
d.f.
t 0.100 t 0.050 t 0.025 t 0.010 t 0.005
t 0.100 t 0.050 t 0.025 t 0.010 t 0.005
1 3.078 6.314 12.706 31.821 63.657 1 2 1.886 2.920 4.303 6.965 9.925 2 3 1.638 2.353 3.182 4.541 5.841 3 4 1.533 2.132 2.776 3.747 4.604 4 5 1.476 2.015 2.571 3.365 4.032 5 6 1.440 1.943 2.447 3.143 3.707 6 7 1.415 1.895 2.365 2.998 3.499 7 8 1.397 1.860 2.306 2.896 3.355 8 9 1.383 1.833 2.262 2.821 3.250 9 10 1.372 1.812 2.228 2.764 3.169 10 11 1.363 1.796 2.201 2.718 3.106 11 12 1.356 1.782 2.179 2.681 3.055 12 13 1.350 1.771 2.160 2.650 3.012 13 14 1.345 1.761 2.145 2.624 2.977 14 15 1.341 1.753 2.131 2.602 2.947 15
16 1.337 1.746 2.120 2.583 2.921 16 17 1.333 1.740 2.110 2.567 2.898 17 18 1.330 1.734 2.101 2.552 2.878 18 19 1.328 1.729 2.093 2.539 2.861 19 20 1.325 1.725 2.086 2.528 2.845 20 21 1.323 1.721 2.080 2.518 2.831 21 22 1.321 1.717 2.074 2.508 2.819 22 23 1.319 1.714 2.069 2.500 2.807 23 24 1.318 1.711 2.064 2.492 2.797 24 25 1.316 1.708 2.060 2.485 2.787 25 26 1.315 1.706 2.056 2.479 2.779 26 27 1.314 1.703 2.052 2.473 2.771 27 28 1.313 1.701 2.048 2.467 2.763 28 29 1.311 1.699 2.045 2.462 2.756 29 1.282 1.645 1.960 2.326 2.576 Inf.
Inf.
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