Answer to Question #5982 Submitted to "Ask the Experts"

Why is the range of acceptable values for chi-square so wide (0.98-0.02)? How is the acceptable range determined when dealing with measurements involving randomness of decay?

The probability range that is specified by the individual invoking the statistical test can be as wide or narrow as desired. Recall that the chi-square statistic, for a given number of observations (counts in the case of radioactivity measurements), is represented by a probability density function and has a shape similar to a normal distribution that is skewed to the right (that is, has a somewhat longer tail on the right than on the left). If a range from 0.02 to 0.98 is specified as acceptable, the investigator is implying that he/she will accept a value of chi-square that falls in a range that has, in each tail at both the low-value and high-value ends, no less than a 2 percent probability of occurring.

When we want to use the chi-square test to evaluate count data, or other data, we are testing whether the data points (counts) belong to an expected distribution—for example, a normal distribution. The actual value of chi-square is obtained from the following summation of squared deviations from the expectation (mean) value divided by the expectation value:

χ² = Σ[(x_i - x_m)2/x_m],

where x_i is one data point value, and x_m is the mean value.

For radioactivity count data, to which the Poisson distribution applies, in which the variance of a count represents the expectation value, the theoretical variance is the mean value that appears in the denominator. Thus, the chi-square statistic is a measure of the overall variance of the individual data points, (x_i - x_m)², compared to the expectation variance, x_m.

For a given distribution of data points, we calculate a single value for chi-square and ask the question "Is this value consistent with expectations that the data are represented by a normal distribution?" (The normal distribution is a good approximation to the Poisson distribution as long as we have a reasonable mean number of counts.)

In order to answer this question we must decide what criteria we will use to establish "consistency." Most often this is done by specifying a probability value, α, that represents the probability of obtaining a value of chi-square that falls outside of the range of acceptable values even though the actual data and variances are consistent with the assumed model.

As we noted above, the chi-square distribution is a two-tailed distribution; if the count data and their variances are fully consistent with the assumed (normal distribution) model, then the expectation value of chi-square is that value that has about a 50 percent chance of being obtained. At times we would expect to obtain a chi-square value that is less than the expectation value and at times we would expect a value greater than the expectation value. When we specify alpha, the usual procedure is to divide the value of alpha by two, assigning a probability of α/2 as representative of a portion of the area of each tail of the chi-square distribution curve. Thus, if we specify α = 0.04 and assign the value of 0.02 to each end of the chi-square distribution curve we are actually saying that we are willing to accept a 4 percent false chance that the distribution model for the data points (counts) and/or the expected variances we assumed are wrong. The chi-square probability values for this case are then α/2 = 0.02 (low value end) and 1 - α/2 = 0.98 (high value end). If the value of chi-square obtained falls within the range of values that correspond to these two values, then we accept the hypothesis that our data and variances are consistent with our model assumptions.

For example, if we obtained 25 data points (counts), the chi-square values corresponding to the probability values of 0.02 and 0.98 are 11.992 and 40.270, and if we obtained a chi-square value between these two values we would conclude that our data are consistent with our model assumptions; a chi-square value less than 11.992 or greater than 40.270 would lead us to reject our data and conclude that they are inconsistent with the assumed model.

The extent of separation between the probability values (0.02 and 0.98 in this case) then depends on the level of uncertainty we are willing to accept as to whether we have concluded improperly that our data and/or variances are inconsistent with our assumed model. The most common value of alpha used in routine health physics counting applications seems to be 0.05, with α/2 = 0.025, thus implying that health physicists are willing to draw false inferences about good data 5 percent of the time.

There may be instances when it is desirable to either narrow or to expand the range of probabilities. This depends, among other things, on the nature of the work being done, the importance of reducing either the number of false judgments made, and the flexibility and gambling instincts of the investigator. Good luck in your choices.

George Chabot, PhD, CHP