Rounding and significant figures

Significant figures

The significant figures in a recorded value are the digits that carry information about the measurement. The rules:

• Nonzero digits are significant, and so are zeros between them: 1205 has four.
• Leading zeros are not: 0.004050 has four (4, 0, 5, 0).
• Trailing zeros after a decimal point are significant: 12.50 says the reading was resolved to 0.01, which 12.5 does not.
• Trailing zeros in a whole number are ambiguous: 1250 could have three or four. Write 1.250 × 10³ or 1.25 × 10³ to say which.

The least significant digit is the last one recorded. Its place value (0.01 for 12.50) is the resolution of the value as written.

Rounding rules

All the rules agree except on an exact tie, where the part dropped is exactly half of the last kept place.

• Round half up (away from zero): 2.345 becomes 2.35. It is what most people learn at school and what many spreadsheets do.
• Round half to even: a tie goes to the even neighbor, so 2.345 becomes 2.34 and 2.355 becomes 2.36. Over many values the ties go up and down equally often, so averages are not pushed upward. ASTM E29 and ISO 80000-1 use this rule for test data.
• Truncate: drop the digits. Truncation always moves toward zero, so it biases results. Use it only where a procedure calls for it.

Rounding to an increment such as 0.02 or 0.5 rounds to the nearest multiple of it, the way a reading is taken from a scale with those divisions.

The tool works on the decimal digits exactly as typed, not on a binary approximation, so 2.345 really is a tie.

Round once, at the end

Rounding twice can give a different answer from rounding once: 2.3449 rounded to 2.345 and then to three figures half up gives 2.35, but 2.3449 rounded straight to three figures is 2.34. Carry at least one extra digit through intermediate steps and round the final result.

Sums and products

• Adding and subtracting: the result is known only to the decimal place of the least precise term. 12.52 + 3.1 − 0.448 = 15.172, reported as 15.2.
• Multiplying and dividing: the result keeps as many significant figures as the factor with the fewest.
• Exact numbers, such as counts and defined constants (25.4 mm per inch), do not limit the result.

These rules are a rough way of not claiming more precision than the data has. A proper statement of uncertainty does the job better, and the reported value is then rounded to match the uncertainty.

On the exam

CCT IV.D.5 asks you to determine the resolution of calculations, including the number of digits and the least significant digit, and to round and truncate to a specified number of digits. II.C covers resolution and readability of measurement data. Expect to round a calculator result to a given number of significant figures, and to spot the answer choice that rounded or truncated the wrong way.

Common questions

How do you round to significant figures?

Count from the first nonzero digit, keep that many digits, and round on the next one. 26,814 to three significant figures is 26,800; 0.002946 to two is 0.0029; 45.51 to three is 45.5. Leading zeros never count.

Are trailing zeros significant?

After a decimal point, yes: 2.50 has three significant figures and says the value was read to the hundredth. In a whole number without a decimal point, such as 26,800, they are ambiguous. Write 2.68 × 10⁴ to show exactly three.

What is round half to even?

When the digits dropped are exactly 5, round to the neighbor that ends in an even digit: 2.45 becomes 2.4 and 2.55 becomes 2.6 at one decimal. Over many values it avoids the upward drift that always rounding 5 up creates, which is why many measurement methods and computers use it.

How many digits should a sum or a product keep?

A sum or difference keeps the fewest decimal places of its inputs. A product or quotient keeps the fewest significant figures. Carry extra digits through the working and round once, at the end.

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