# $\chi^2$ test to determine whether different observations come from the same quantity

Three experiments are performed to measure the same quantity $$T$$. The three results are

$$\begin{cases} T_A = 990\pm 3\\ T_B = 920\pm 30\\ T_C = 950\pm 12 \end{cases}$$

Using a $$\chi^2$$ test, and assuming Gaussian errors, verify the hypothesis that the three experiments are measuring the same quantity.

Can anyone explain how a $$\chi^2$$ test should be performed for this task? Normally I would have used a t-test. I know how the $$\chi^2$$ works to verify the goodness of a fit but I don't know how it could be used for this task. Thanks!

• Although one could apply a $\chi^2$ test in the sense of computing a suitable statistic, it almost surely would not follow a $\chi^2$ distribution, which means the computed p-value would be wrong. The reason is that the ratio $30/3$ reflecting the range of measurement errors is strong evidence that their variances differ. The best response from a statistical perspective is to refuse to conduct a $\chi^2$ test and to offer a better procedure in its stead.
– whuber
Apr 18, 2022 at 18:48
• What data have you summarized to get results such as $920\pm 30?$ More to the point, what does that notation mean. Is the result $951$ rare? not yet observed? unthinkable? Apr 18, 2022 at 20:33
• This is actually an exercise so I have no background for these measurements. The error is supposed Gaussian. Apr 19, 2022 at 10:39

If all 3 random variables are normally distributed with known mean and variance, then $$\tilde T_x = \frac{\bar T_x-\mu}{\sigma_x} \sim \mathcal{N}(0,1)$$ for $$x \in {A,B,C}$$ are standard normal distributed, where $$\bar T_x$$ is the sample mean. Notice how the same mean $$\mu$$ is assumed for all values (but not $$\sigma_x$$), which corresponds to the 3-equation null $$H_0: \mu_A = \mu; \mu_B = \mu; \mu_C = \mu;$$, i.e. are all experiments measuring the same mean? We know that the sum of $$n$$ squared standard normal variables is chi-squared distributed with $$n$$ degress of freedom, i.e. $$c = (\tilde T_A^2 + \tilde T_B^2 + \tilde T_C^2) \sim \chi^2_3$$. Now a right-tailed test can be performed on the test statistic $$c$$.
As far as the actual numbers go I would assume the values to the left of $$\pm$$ are the sample means $$\bar T_x$$ and the values to the right are the standard deviations $$\sigma_x$$. For $$\mu$$ I'm not sure, maybe the mean of all 3 sample means.
• What would this "right tailed test" based on $c$ actually test? What is your null hypothesis?