Rules for Rounding
The standard rules for rounding a value are well known. But first, some definitions:

• The most significant digit is the leftmost digit (not including leading
zeros).

• If you are rounding to n significant digits, then the least significant
digit is the n ^{th} digit from the most significant digit. Note that the least
significant digit can be a zero.

• The first non-significant digit is the n+1 ^{st} digit.

The rules for rounding are:

1) If the first non-significant digit is less than 5, then the least significant digit
remains unchanged.

2) If the first non-significant digit is greater than 5, the least significant digit
is incremented by 1.

3) if the first non-significant digit is 5, the least significant digit can either be
incremented or left unchanged (see discussion on the right).

4) All non-significant digits are removed.

Some Jive for the Non-significant Five
When the first non-significant digit is five, you might be told (by your instructor, your manager, or by procedure)
to increment the least significant digit by one if it is odd and leave
it unchanged if it is even (alternate to rule #3).

As a matter of fact, it doesn't actually matter if you increment on odd, even, or never if you are
only rounding a single number. However when you are rounding a series of numbers that will
be used in a calculation, treating all numbers the same will result in over or understating
the values, thus accumulating a rounding error.

Since odd and even numbers will tend to be
distributed evenly, this accumulation error will tend to stay smaller by using the odd/even rule.

On average, you would do just as well by flipping a coin!

How Far to Take It?
Observed values should be rounded to the number of digits that most accurately conveys the
uncertainty of a measurement.

I)
Usually this means rounding to the number of significant digits in the quantity (i.e., the
number of digits that are known exactly, plus one).

II)
When this cannot be applied

(

e.g., when the certainty range crosses a power of 10
Suppose an object is found that has a weight of 2.98±0.05 g.
Its true weight would therefore be in the range of 2.93 to 3.03 g.

When judging how to round this value, you determine the number of digits that are known
exactly (i.e., are certain). In this example you find there are none, since every digit is uncertain
(even the most significant digit of the min and max range value is different).

Since the initial "2" is the leftmost digit whose value is uncertain, this implies
that the result should be rounded to one significant digit (rounding rule I, 0 + 1 = 1 ), yielding 3 g

In the alternative, one could "bend the rules" and round to two significant digits,
resulting in 3.0 g for the value.

In this case, an examination of the implied uncertainties of the two candidate values
(3 and 3.0) and comparing them with the uncertainty of the original measurement (2.98)
is a useful exercise.

Rounded Value Implied Min Implied Max Absolute Uncertainty Relative Uncertainty
2.98 2.975 2.985 ±0.005 (0.01) 1 in 298 (0.36%)
3 2.5 3.5 ±0.5 (1) 1 in 3 (33%)
3.0 2.95 3.05 ±0.05 (0.1) 1 in 30 (3%)

Clearly in this example, the only reasonable course is to round off to two digits.

),

then we round in such a way that the relative implied uncertainty in the
result is as close as possible to that of the measured value.

See also: Uncertainty , Significance