General Chemistry I (CHEM-UA), Lab 2 | Source: Data Analysis Lab Handout (Geggier, NYU)
Tags: significant figures, sig figs, SFs, rounding, scientific notation, decimal places, precision in calculations
Difficulty: Introductory | Prerequisites: Accuracy, Precision, and Experimental Errors study notes.
When you record a measurement, the number of digits you write down tells the reader how precise your instrument was. Significant figures are the language scientists use to communicate that precision. This topic follows directly from accuracy and precision: once you understand that every measurement has uncertainty, you need a consistent set of rules for carrying that uncertainty through calculations. Get these rules wrong and your final answer claims a precision your data never had, which costs marks on every worksheet and lab report.
Significant figures (SFs) are the digits in a measurement that carry meaning, including all certain digits plus one uncertain digit. When you calculate with measured values, the answer cannot be more precise than the least precise input. Multiplications and divisions follow the least number of SFs; additions and subtractions follow the least number of decimal places.
Significant figures (SFs)
The number of digits in a measurement that are known with certainty, plus one additional digit that expresses uncertainty. SFs reflect the precision of the instrument used.
Think of it as: every digit you are allowed to trust, plus one you are guessing at.
Leading zeros
Zeros that appear to the left of the first non-zero digit (e.g. the zeros in 0.036). These are placeholders only and are never significant.
Trailing zeros
Zeros at the end of a number. To the right of a decimal point, they are significant (0.5000 has 4 SFs). To the left of a decimal point with no decimal shown, they are ambiguous (100 could be 1, 2, or 3 SFs).
Scientific notation
A way of writing numbers as a coefficient multiplied by a power of ten (e.g. 2.0 x 10⁻⁵). All digits in the coefficient are significant. This format removes ambiguity about trailing zeros.
All non-zero digits are significant. Example: 22.15 has 4 SFs.
Leading zeros are not significant. Example: 0.036 has 2 SFs.
Zeros between non-zero digits (captive zeros) are significant. Example: 305 has 3 SFs.
Trailing zeros to the right of the decimal point are significant. Example: 0.5000 has 4 SFs.
In scientific notation, all digits of the coefficient are significant. Example: 2.0 x 10⁻⁵ has 2 SFs.
Definitions and counted numbers have an infinite number of SFs. Example: 100 cm = 1 m (exact), and "4 oxygen atoms per formula unit of KMnO₄" (counted).
Numbers with trailing zeros only to the left of the decimal point are ambiguous. Write 100 mL in scientific notation to clarify: 1 x 10² mL (1 SF), 1.0 x 10² mL (2 SFs), or 1.00 x 10² mL (3 SFs).
Multiplication and division
The result has the same number of SFs as the least precise piece of data in the calculation.
Example: 2.00 x 5 = 10 = 1 x 10¹. The number 5 has 1 SF, so the answer is reported with 1 SF.
Addition and subtraction
The result has the same number of decimal places as the least precise piece of data.
Example: 0.2738 + 0.340 + 0.12 = 0.7338 → rounded to 0.73. The number 0.12 has only 2 decimal places, so the answer is rounded to 2 decimal places.
Mixed calculations (both types in one expression)
Evaluate the addition/subtraction steps first and assess their SF count. Then treat that intermediate result as a new piece of data for the multiplication/division step.
Example: (0.45 + 0.60) / 0.5000
Addition step: 0.45 + 0.60 = 1.05 (3 SFs, because both inputs have 2 decimal places, giving the sum 2 decimal places)
Division step: 1.05 / 0.5000 = 2.10 (3 SFs, because 1.05 has 3 SFs and 0.5000 has 4 SFs; take the smaller)
Final answer: 2.10
This shows that inputs with 2, 2, and 4 SFs can yield an answer with 3 SFs.
If the first nonsignificant digit is 5, 6, 7, 8, or 9, round up. Example: 0.5256 x 2.0 = 1.0512 → 1.1.
If the first nonsignificant digit is 0, 1, 2, 3, or 4, round down. Example: 0.5244 x 2.0 = 1.0488 → 1.0.
Do not round intermediate answers. Keep at least 2 extra digits throughout your calculations and round only the final answer.
The number of digits you write down must match the precision of the instrument you used. If you measure 25 mL with a volumetric pipette (precision to ±0.02 mL), record it as 25.00 mL, not 25 mL. Writing 25 mL implies you used a low-precision instrument like a beaker.
"Trailing zeros never count." They do when they are to the right of a decimal point. 0.5000 has four significant figures, not one. The zeros communicate that you measured to the ten-thousandths place.
"My calculator gives the right number of SFs." It does not. Calculators do not track significant figures. You must apply the rules yourself and round the final answer manually.
"I should round each step as I go." Rounding intermediate results introduces cumulative "round-off" error. Keep extra digits throughout and round only at the end.
"Addition and multiplication follow the same SF rule." They do not. Multiplication/division uses the fewest SFs; addition/subtraction uses the fewest decimal places. Mixing them up is one of the most common errors on worksheets.
⚠️ Worksheet I Question 2 tests SF counting directly (0.00070, 0.200, 1.043 x 10⁻⁶, and the ambiguous "100").
⚠️ Worksheet I Question 3 tests SF rules in calculations, including a mixed-operation problem. Work through each step separately.
⚠️ You will lose marks for writing too many or too few significant figures in any calculated result across all worksheets.
⚠️ Remember that "100" without a decimal point is ambiguous. Write "ambiguous" if asked for its number of SFs, unless context gives you more information.
How many SFs does 0.00070 have? (2)
True or False: 0.200 has the same number of SFs as 0.2. (False: 0.200 has 3 SFs, 0.2 has 1 SF)
Fill in the blank: For multiplication and division, match the fewest __________. (significant figures)
Fill in the blank: For addition and subtraction, match the fewest __________. (decimal places)
True or False: You should round intermediate results to the correct number of SFs before continuing. (False: keep extra digits and round only the final answer)
Q: How many significant figures does 1.043 x 10⁻⁶ have?
A: 4 SFs. In scientific notation, all digits of the coefficient are significant.
Q: Calculate 0.020 + 9.216 + 8.0340 with the correct number of significant figures.
A: 0.020 has 3 decimal places, 9.216 has 3, and 8.0340 has 4. The limiting term is 3 decimal places. Sum = 17.2700 → 17.270. The answer is 17.270 (note: 0.020 limits to the thousandths place, so 3 decimal places).
Q: Calculate (2.846 x 0.0049) − 6.6 x 10⁻³ + (0.035 x 0.0289). Give the answer with the correct number of SFs.
A: First evaluate each multiplication: 2.846 x 0.0049 = 0.013945 (2 SFs → 0.014); 0.035 x 0.0289 = 0.0010115 (2 SFs → 0.0010). Now add/subtract: 0.014 − 0.0066 + 0.0010 = 0.0084. The least precise term in the addition has its last significant digit in the thousandths place, so round to 0.008.
Q: A group of students measures the specific heat capacity of aluminium as 0.465 J/(g·K). The accepted value is 0.450 J/(g·K). Calculate the percent error.
A: % error = |0.450 − 0.465| / 0.450 x 100% = 0.015 / 0.450 x 100% = 3.33...%. With 2 SFs in the numerator (0.015) and 3 SFs in the denominator (0.450), the answer is reported as 3.3%.
Q: Why must "100" be written in scientific notation to communicate its precision?
A: Because the trailing zeros are ambiguous. 100 could mean 1 SF (1 x 10²), 2 SFs (1.0 x 10²), or 3 SFs (1.00 x 10²). Without the decimal point or scientific notation, the reader cannot tell how many digits are meaningful.
Significant figures are the bridge between raw measurement and every calculated quantity in this course. When you compute a mean or standard deviation (covered in the next set of notes), the SF rules determine how many digits to report. When you calculate percent error, the answer must also be rounded correctly. Graphing, too, depends on SFs: trendline parameters like slope and y-intercept should be reported to the same number of SFs as the data used to create the graph.
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