Negative Numbers in Statistics
Negative numbers cause most of the mistakes in descriptive statistics, and almost all of them come from one thing: forgetting that a negative number is smaller than any positive number when the data is sorted.
Sorting
Negative values sit to the left of zero on the number line, so ascending order places them first: −15, −5, 0, 5, 10. This decides the median, the quartiles and the five-number summary, so sort before you do anything else.
Adding
Keep the sign attached to each value while you add: −5 + 10 − 15 + 20 = 10. Writing the sum out with brackets around the negatives, as in (−5) + 10 + (−15) + 20, makes the arithmetic much harder to get wrong.
Worked example: mean and median with mixed signs
For −5, 10, −15, 20:
- Sum = 10, count = 4, so the mean is 10 ÷ 4 = 2.5.
- Sorted: −15, −5, 10, 20. The two middle values are −5 and 10, so the median is (−5 + 10) ÷ 2 = 2.5.
- The mean and median agree here, which happens when the data is roughly symmetrical.
Worked example: the range with negatives
For −5, −10, 3, 7 the maximum is 7 and the minimum is −10, so the range is 7 − (−10) = 17. Subtracting a negative number adds, and a range of 17 is wider than the 12 you would get by mistakenly using 7 − 5.
Quartiles and standard deviation with negatives
Quartiles follow the same sorting rule. For −10, −5, 0, 5, 10 the quartiles are −5, 0 and 5, so the interquartile range is 5 − (−5) = 10. Standard deviation works on squared distances from the mean, and squaring always gives a positive number, so the standard deviation is never negative no matter how negative the data is.
Common mistakes
- Sorting −10 after 3 because the digits 1 and 0 look small.
- Writing 7 − 10 instead of 7 − (−10) when finding the range.
- Dropping minus signs when pasting data out of a spreadsheet.
- Assuming a mean of zero means the dataset is empty. A mean of zero simply means the negatives and the positives balance out.