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Figure 6.10. Results of chi square tests for different sizes of error bars (ai), but for the same data points xi, yi. (a) Moderate error bars. The chi square is acceptable because the average deviation is on the order of 1 standard deviation. (b) Small error bars. The deviations measured in units of ai are very large, leading to an unacceptably low x2 probability that fluctuations in another trial would exceed these. (c) Large error bars leading to an unacceptably high probability.

Figure 6.10. Results of chi square tests for different sizes of error bars (ai), but for the same data points xi, yi. (a) Moderate error bars. The chi square is acceptable because the average deviation is on the order of 1 standard deviation. (b) Small error bars. The deviations measured in units of ai are very large, leading to an unacceptably low x2 probability that fluctuations in another trial would exceed these. (c) Large error bars leading to an unacceptably high probability.

What can we conclude in this case? Most likely something is wrong with the error bars; our model of the fluctuations could be wrong. For example, the cars could be passing at regular intervals in some sort of procession such as a parade, rather than at random times. This would lead to smaller fluctuations in the number of car passages in successive time intervals; Poisson statistics would not apply. (The remaining small error bars might represent some small jitter in the regularity of car arrival times.) Smaller error bars would serve to raise the x2 value and lower the probability. This could then make the data consistent with the trial function.

We do not derive here the x2 statistic that yields the desired probabilities, but it can be tabulated or plotted as in Fig. 11. One enters the figure with the value of X 2 and the number of degrees of freedom f as defined above. The curves give the probability that another measurement would yield a greater x2. Any probability between 0.1 and 0.9 can be considered to be consistent with the trial function curve, and values less than 0.02 and greater than 0.98 raise serious doubt about the appropriateness of the trial function.

Some insight into the behavior of x 2 follows from its definition (21). For normal fluctuations about the expected values, the deviation in the numerator of one term

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