Significant Figures, Precision, and Accuracy — Study Notes

Accuracy vs. Precision

Precision and accuracy are common everyday terms, but in science they have distinct meanings.
Precision: how closely repeated measurements agree with each other (the spread/consistency among measurements).
Accuracy: how closely those measurements agree with the true value (the truth of what’s being measured).
Key distinction: precision relates to agreement among measurements; accuracy relates to agreement with the true value.
Everyday intuition can confuse the two, but they are not the same.

The Bullseye Metaphor (True Value)

Imagine a dartboard with a bullseye representing the true value (the truth you’re trying to measure).
A measurement is accurate if it lands near the bullseye; it is precise if multiple measurements cluster tightly together, regardless of how close they are to the bullseye.
Diagram concepts described in words:
- Case A: measurements cluster near the bullseye (high accuracy and high precision).
- Case B: measurements cluster tightly but far from the bullseye (high precision, low accuracy).
- Case C: measurements are spread out and far from the bullseye (low accuracy and low precision).

Scenarios: Understanding the Three Cases

Case 1: High precision and high accuracy
- Repeated measurements are close to each other and near the true value.
Case 2: High precision but low accuracy
- Measurements are very repeatable (close to each other) but far from the true value (e.g., a miscalibrated instrument).
Case 3: Low precision and low accuracy
- Measurements are all over the place and far from the true value; shows a fundamental measurement issue.

Exact Numbers vs Measured Numbers

Exact numbers: defined and have no uncertainty; do not limit significant figures in calculations.
Examples of exact numbers:
- The mass of a dozen eggs: 12\ \text{eggs} = \text{exact}
- Conversion factors: 1\ \text{kg} = 1000\ \text{g} = \text{exact}
- Conversion: 1\ \text{inch} = 2.54\ \text{cm} = \text{exact}
Measured numbers: have uncertainty and require significant figures in calculations.
Non-exact example discussed: thermometer readings (e.g., 30°C and 25°C on a thermometer scale) are not exact; the reading like 27°C is an estimate with limited precision.
Leap-year caveat for time-based quantities (e.g., microseconds in a week) can make some quantities non-exact due to real-world calendar variations.
Pages in a book are considered exact within the context given (counting printed pages).

Significance Figures: Core Rules (Overview)

Significance figures tell us how many digits in a number are meaningful based on measurement precision.
Goal: Only keep as many digits as the measurement justifies; avoid implying more precision than the data provides.
In calculations, rounding is guided by the number of significant figures in the original numbers.
Scientific notation can remove ambiguity for how many digits are significant.

Significance Figure Rules (Practical Rules)

Rule 1: Non-zero digits are always significant.
- Example: 1,342.6 has five significant figures: 1, 3, 4, 2, 6.
Rule 2: Zeros between non-zero digits are significant (zeros between numbers are significant).
- Examples: 1,005 has four significant figures; 7.03 has three significant figures.
Rule 3: Leading zeros are not significant (ignore leading zeros).
- Examples: 0.02 has one significant figure; 0.0026 has two significant figures.
Rule 4: Trailing zeros are significant if there is a decimal point; otherwise, trailing zeros are ambiguous.
- Examples: 0.0200 has three significant figures (2, 0, 0) because there is a decimal point and trailing zeros are significant; 0.04100 has four significant figures (4, 1, 0, 0).
- Ambiguity example: 10,300 can be interpreted as anywhere from three to five significant figures unless written in scientific notation, which clarifies the count.
- Scientific notation clarifies significant figures independently of the power of 10:
- 1.03 \times 10^{4} has 3 significant figures.
- 1.030 \times 10^{4} has 4 significant figures.
- 1.0300 \times 10^{4} has 5 significant figures.
Note on ambiguity: In many textbooks, trailing zeros without a decimal point may not be significant; to avoid ambiguity, use scientific notation.

Examples: Identifying Zeros as Significant

Problem A: 0.041
- Leading zeros are not significant; none of the zeros are significant. Result: 0 significant figures.
Problem B: 0.00410
- Ignore leading zeros; trailing zero after decimal is significant. Result: 3 significant figures.
Problem C: 0.04100
- Ignore leading zeros; trailing zeros with decimal are significant. Result: 4 significant figures.
Problem D: 4.010 × 10^4
- Look at the 4.010 part: 4, 0, 1, 0 are all significant (0 between 4 and 1 is significant; trailing 0 with decimal is significant). Result: 4 significant figures.

Significance Figure Counts: Worked Examples

A: 6.070 \times 10^{-15} → 4 significant figures (the 0s between nonzero digits are significant; trailing zeros with decimal are significant).
B: 0.00038400 → 5 significant figures (ignore leading zeros; trailing zeros are significant due to decimal point).
C: 17.000 → 5 significant figures (decimal point makes trailing zeros significant).
D: 8.0 \times 10^{8} → 2 significant figures (the 0 after decimal is significant).
E: 463.805 → 6 significant figures (all digits, including the zero between 3 and 8, are significant).
F: 3,000 (assumed measured) → 1 significant figure (trailing zeros are not significant unless a decimal point is present).
G: 301 → 3 significant figures (the zero is between non-zero digits).
H: 30. (a decimal point after the number) → 2 significant figures (the decimal makes the trailing zero significant).

Rounding to Five Significant Figures (Practice)

Given numbers, round to five significant figures:
- (A) 156.852 → five sig figs are the first five digits (1, 5, 6, ., 8, 5). The next digit is 2 (<5), so it rounds down: 156.85
- (B) 156.842 → five sig figs: the fifth digit is 2, next is 4 (<5) so 156.84
- (C) 156.849 → five sig figs: the fifth digit is 9, next is 0 or not shown? Actually next is 9; since it's 9, round the fifth digit up: 156.85
- (D) 156.899 → five sig figs: the fifth digit is 9 and the next digit is 9, so the fifth becomes 9 and the number rounds to 156.90 (note the trailing zero is significant because of the decimal).

Rounding to Three Significant Figures and Writing in Scientific Notation

Task: Round the following values to three significant figures, then rewrite in scientific notation.
A: 102.53070 → three significant figures: 1 0 2 (the third digit is 2; the next digit is 5, so it rounds up). Result: 103.0? Wait: with three sig figs, the rounded value is 1.03\times 10^{2}.
- Scientific notation: 1.03 \times 10^{2}
B: 656.9 → three sig figs: 6, 5, 7 (since the next digit is 6, rounds the 5 up? The transcript indicates 657). So: 6.57 \times 10^{2}
C: 0.008543210 → three sig figs: 8, 5, 4 (leading zeros ignored). Rounding based on next digit (3) yields 0.00854. Scientific notation: 8.54 \times 10^{-3}
D: 0.00002578 → three sig figs: digits are 2, 5, 7; rounding based on next digit (8) makes 7 become 8: 0.000258. Scientific notation: 2.58 \times 10^{-4}
E: -0.035720202 → three sig figs: digits 3, 5, 7; next digit is 2 (<5), so remains 7. Value: -0.0357. Scientific notation: -3.57 \times 10^{-2}

Practical Takeaways and Next Steps

This lesson covered:
- Accuracy vs. precision definitions and the bullseye metaphor.
- The concept of exact numbers vs. measured (not exact) numbers.
- The rules for counting significant figures, including handling of leading/trailing zeros and the role of decimal points.
- How to count significant figures in a variety of example numbers.
- Rounding to a specified number of significant figures and expressing results in scientific notation to avoid ambiguity.
The next step: learning how to perform calculations with significant figures and determine the correct rounding of final answers based on the significant figures in the starting data.