Chi-Square Test Made Easy

QA Programs for Uptake Probes and Well Counters

A series of data obtained by counting a single radioactive sample repeatedly will always demonstrate variation.  To determine if the variation is due to the natural randomness of radioactive decay or is due to inconsistency in instrument performance, a chi-square (x2) test is performed.

The chi-square test allows you to determine how close a series of counts would come to a true Poisson distribution (bell curve) and provides a measure of the precision of the counting instrument’s performance.

Since it is time consuming and inconvenient to count every patient’s thyroid or wipe sample multiple times in a row, you would want to be certain that the individual values given by the counting instrument can be used with confidence.

Although newer thyroid probes and well counters have programs that will calculate the chi-square, manual calculations with a calculator are simple if you use the step-by-step method demonstrated below.

How to determine the chi-square value:

1.  Obtain a series of a minimum of 10 sample measurements of at least 10,000 counts each. A 0.1uCi Cs-137 rod source can be used or this test can be performed without using any radioactive source. A preset stop time of 10 seconds for each measurement may be used when using a radioactive source.

2.  Calculate the mean ( mean) for the number of measurements (n).

3.  For each measurement (Ni), calculate (Ni – mean), and then square that number (Ni – mean)2 .

4.  Calculate the sum () of the (Ni – mean)2 values, then insert values into the x2 formula. See example chi-square calculation below.

5.  After calculating x2, P-values and degrees of freedom used are obtained using a chi-square table.  The P-value indicates the acceptability of the data variation based on a Poisson distribution. The degrees of freedom are equal to n-1 where n equals the number of measurements made. Refer to the chi-square table below.

For the purpose of testing a well-counter or an uptake probe, the x2 is acceptable if it falls between P-values of 0.9 to 0.10 which are 4.168 to 14.684.

A P-value of 0.5 would be perfect and shows the variation in data is exactly as expected according to the Poisson distribution.

A P-value that falls below 0.10 indicates that it is unlikely that the variation produced by the instrument would match a Poisson distribution.  The variation appears to be too great and instrument malfunction must be considered, such as fluctuations in high voltage supply to the instrument or as a fixed voltage drop.

A P-value that exceeds 0.90 is probably too good to account for randomness and is also cause for concern. It could indicate that electrical background noise is being counted and could contribute to the lack of variation.

If the chi-square value falls between a P- value of 0.90 and 0.10 it is acceptable to use a single or small number of values when counting a radioactive sample.  It can be assumed that a single value has an acceptably high probability of falling within the normal Poisson distribution that would be produced by obtaining a large number of counts from the sample.


Cherry, S., Sorenson, J., Phelps, M. (2003).  Physics in Nuclear Medicine (3rd ed.).  Philadelphia:  W.B. Saunders.

Wells, P. (1999). Practical Mathematics in Nuclear Medicine Technology (1999). Virginia: Society of Nuclear Medicine