# How does rounding affect subsequent calculations?

When we are doing calculations in mathematics, we often express exact values, like $$\sqrt 2$$ or $$\arctan (1)$$, as decimals and round these to a finite number of decimal places/significant figures before using these approximations in subsequent calculations. We might also round off a very long but finite decimal and use it in subsequent calculations. My question is, what is the minimum number of decimal places/significant figures which all of my intermediate values must be rounded to if I want my final answer to be accurate to a particular number of decimal places/significant figures?

Example: I’m doing a $$\chi^2$$ test. I want to find $$\chi^2$$ exact to $$1$$ decimal place. The exact $$\chi^2$$ contributions are: $$(0.56,0.32,1.76,1.99,0.72,0.88)$$. If I sum these, the exact value of $$\chi^2$$ is $$6.23$$, which becomes $$6.2$$ rounded to $$1$$ decimal place. Now if I round the contributions to $$1$$ decimal place, so they become $$(0.6,0.3,1.8,2.0,0.7,0.9)$$, before I sum them, I get the sum to be $$6.3$$, which is not accurate to $$1$$ decimal place, as we have shown it should be $$6.2$$.

How can I be sure that my intermediate values are rounded with enough remaining decimal places/significant figures that my final answer is accurate to a desired number of decimal places/significant figures? Not just for this particular example, but for a general calculation.

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