PCM-2 Manual.pdf - Voss Associates

Chi-Squared Distribution Function
The Chi-Squared Distribution Function is applied post facto to counting experiments to evaluate
goodness-of-fit of the data to a theoretical distribution. Stated otherwise, observed results are
compared to expected (theoretical) results and a determinatIon is made as to whether or not the
difference is reasonable. A poor fit would suggest instrument malfunction indicated by spurious
counts. Chi-squared is calculated as indicated by Eq. 32.
Eq. 32
Where:Xe= .
The experimental mean, or average value of the counts collected in "N"
measurements.
N = The number of counting imervals involved.
x;=coum data collected in the i lb counting interval
As the number counting intervals that are evaluated increases. the goodness-of-fit is expected to
improve. Therefore, to be useful, the Chi-squared value must be considered along with the number
of data points. The parameter X 2 / II, known as "Reduced Chi-squared" is used to enter a table, such as
Table 3, or a plot, such as Figure 5, of Chi-squared values. The value II is (n - 1), or, the number of
degrees of freedom in the experiment.
• • •• • • II • • • " 788
0.000039 0.000157 0.0009ll2 0.0038 0.0158 0.102 0..055 1.32 2.71 3.84 5.02 8.63
0.005000 0.0101 0.0253 0.0515 0.106 0.288 0.895 1.39 2.31 3.00 389 481 5.30
0.0239 0.0383 0.0720 0.1173 0.195 0.403 0.790 t.37 2.08 2.80 312 3.77 4.27
0.0518 0.0743 0.121 0.178 0.285 0.480 0.840 1.35 1.95 2.37 2.78 3.33 3.73
0.0824 0111 0.188 0.230 0.322 0534 0.870 1.33 , 85 2.22 2.58 3.02 334
0.113 0.145 0207 0.273 0.367 0575 0.992 1.31 1.77 210 2.40 2.90 3.08
0.141 0.177 0.241 0.310 0.404 0.807 0907 129 1.71 2.01 2.29 2.84 2.90
0168 0.208 0273 0.341 0.438 0.834 0918 1.28 1.68 194 219 251 2.75
0.192 0232 0.300 0.370 0.463 0.858 0.927 127 1.63 1.89 211 2.41 2.62
0.218 0256 0.325 0.394 0.487 0.874 0.934 1.25 160 1.83 2.05 2.32 252
0 ..236 0.277 0.347 0.415 0.507 0689 Q.938 1.25 1.57 1.79 1.99 2.25 2.44
0.258 0.298 0367 0.438 0.525 0.703 0.942 1.23 1.54 1.75 1.94 2.18 238
0.275 0.318 0.385 0.453 0.542 0.7f5 0.948 1.23 1.52 172 1.90 2.13 2.29
0.291 0.333 0.402 0.489 0.558 0.729 0.950 1.22 1.51 1.69 188 2.08 2.24
0.307 0349 0.417 0.484 0570 0.733 0.953 121 1.49 1.87 1.83 2.04 219
0.321 0383 0.432 0.498 0.582 0.744 0.958 1.21 147 1.64 1.80 200 2.14
0.335 0.377 0.445 0.510 0.594 0.753 0.959 1.21 1.48 182 1.78 196 210
0.346 0.389 0.457 0.522 0806 0.781 0.961 1.20 1.44 161 1.75 1.93 2.07
0.360 0.402 0.469 0532 0.616 0.768 0963 1.t9 t.43 1.58 1.73 191 2.03
0.372 0413 0.480 0.545 0.620 0.775 0.965 119 1.42 1.57 1.71 188 200
0.382 0.424 0.490 0552 0.629 0.776 0.967 119 1.41 156 1.69 185 1 97
0393 0.434 0.500 0.559 0.638 0.762 0.968 118 1.40 154 1.87 183 1.95
0.403 0.443 0509 0.570 0843 0.767 0.970 1.t8 139 1.53 1.66 1.61 1 92
0.412 0.454 0517 0.575 0.854 0.792 0.971 1.t6 1.38 1.52 164 1.79 190
0.420 0.460 0.524 0.584 0.660 0.796 0.972 1.17 1.38 1.51 162 177 188
0.431 0.489 0.531 0592 0.665 0.800 0.973 1.17 t.37 1.50 1.61 1.75 186
0.437 0476 0.541 0.600 0833 0.904 0974 1.17 1.36 1.49 160 I 74 1 64
0.448 0468 0.548 0.904 0639 0.811 0975 116 '35 1.48 159 I 73 1.82
0.452 0493 0.552 0.810 0.683 0.814 0.978 1 18 1,35 1.47 159 1.71 160
0.460 0500 0.580 0.817 0.687 0.817 0.977 1.t6 134 1.46 157 1 70 1.79
Table 3, Chi-squared Distribution
.. I
PCM2.MAN