# The effect of measurement accuracy and rounding on hypothesis testing

I am checking the temperature I have at home with accuracy of one tenth of a grade (Celsius). The easily publicly available information is temperature in grades, with no decimals.

I am doing very basic hypothesis testing: My null hypothesis is that temperatures I measure do not have a systematic bias upwards or downwards from the public data, while my alternative hypothesis is that the temperatures I measure are systematically higher or systematically lower. I am doing a simple T-test and checking if the average of the differences is far from zero.

Does it make a difference whether I round my own measurements to integers, or, (essentially equivalently,) round the differences to integers? In particular, is there a bias in a particular direction, towards throwing away the null hypothesis or the opposite, if my data has more accuracy then what I am comparing it to?

**Notes**

This project breaks assumptions of hypothesis testing; at least independence of measurements and possibly the normal distribution of the differences.

The point of the question is not these, but rather the possible effect of rounding or different precision in the ground truth data and the measurements on the hypothesis test.

## 1 answer

If the standard deviation of your measurements is in the trillions, then rounding to integers will make little difference – probably no practical difference – since the rounding error is so tiny by comparison to the value. If the standard deviation is $1/2$ then you would be throwing away huge amounts of information and your results will be wrong.

Generally you shouldn't round more than you have to until the last step.

And notice that $(20+\tfrac13)\times 3 = 60 + \left(\tfrac13\times3\right) = \text{exactly } 61,$

but $20.33\times3 = 60.99,$

So $61$ is an exactly answer and $60.99$ is a rounded answer.

If you show $61$ and $60.99$ to a person whose grasp of arithmetic is at a naive level, and as which one is rounded, they'll get it wrong.

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