# Why do we add an extra decimal place when we calculate the range in statistics?

From Mario Triola's textbook:

I understand that we add a decimal place when we calculate the median (it is possible to have (a+b)/2 ). I absolutely do NOT understand why we do that when we round off the range. It's just a subtraction, a simple algebraic operation.

When we calculate the mode, an exception was made for it. I expected to see the same exception for the range. • Maybe I misunderstand, but this just seems wrong. If you have some values with some precision, you can only work within that (or a lower) precision. You can't say much about what happens outside that precision, which is exactly what you're doing when adding decimal places. If 55.6 and 14.1 are rounded, their actual values can be 55.64 and 14.05, and subtracting those would give you 41.59, which is very different from 41.50. 41.5 would also be incorrect in that case, so it's something to be careful with regardless, but 41.50 is off by far more in its least significant digit (9 versus 1). – Bernhard Barker May 9 at 18:06
• I have never heard of such rules. – John Coleman May 9 at 22:22

## 1 Answer

Round-off rules are mostly arbitrary. Those you posted are reasonable, but not universally accepted.

Various statistical software packages show different numbers of decimal places, sometimes more than you would want to show in a publication or report. (Sometimes statistical software rounds P-values larger than $$0.20 = 20\%$$ because full accuracy is not used in making decisions whether to reject.)

Whenever you may need to use a value for subsequent computation (like taking the square root of a variance to get SD) then it is a good idea to keep an extra decimal place or two. Ranges and interquartile ranges are sometimes used in subsequent computations (for example, to make boxplots or for tests). In a chi-squared test, do not round 'expected counts' to integers.

It is not a good idea to round original data, except possibly to save space in a publication, and then only after all computations are finished--and keep a copy of original data. Never round data before doing a non-parametric test: some of them are very sensitive to ties, and rounding can induce ties.

Ordinarily, boundaries of CIs should not have more than one extra place beyond the original data (to avoid a false pretension of accuracy).

The formula $$S^2 = \frac{1}{n-1}[\sum_{i=1}^n X_i^2\, - n \bar X^2]$$ can be extremely sensitive to rounding; so round nothing until you get the value of $$S^2,$$ then round as usual. Or use $$S^2 = \frac{1}{n-1}\sum_{i=1}^n (X_i -\bar X)^2$$ instead.