Significant Digits, Precision, and Accuracy

Reliability of Measurement

• Definition of Reliability:

  • Measurement reliability refers to how much a given measurement can be trusted.

  • Reliability depends primarily on the instrument used to make the measurement, including its precision, accuracy, and sensitivity.

  • Measurements must be reported responsibly and strictly within the operational limits of the instrument.

• Certainty and Reported Digits:

  • Scientific measurements must be reported to reflect appropriate certainty.

  • Generally, the greater the number of digits in a reported measurement, the greater the precision and certainty in that measurement.

  • Scientific measurements must include every certain digit plus only one estimated digit (which is inherently uncertain).

  • The combination of all certain digits and the single estimated digit are defined as the significant digits (or significant figures) of the measurement.

  • The number of digits reported depends directly on the resolution and calibration of the measuring device.

• Ruler Comparison Example (Illustration of Reliability):

  • Ruler A (Coarse Scale - Markings every $1\,\text{cm}$):

  • Measurement reading: $4.7\,\text{cm}$

  • Certain digits: $4$

  • Estimated digits: $7$

  • Implied uncertainty: $\pm 0.1\,\text{cm}$

  • Expected range of true length: $4.6\,\text{cm}$ to $4.8\,\text{cm}$

  • Ruler B (Fine Scale - Markings every $0.1\,\text{cm}$):

  • Measurement reading: $4.68\,\text{cm}$

  • Certain digits: $4.6$

  • Estimated digits: $8$

  • Implied uncertainty: $\pm 0.01\,\text{cm}$

  • Expected range of true length: $4.67\,\text{cm}$ to $4.69\,\text{cm}$

  • Comparison & Analysis:

  • Ruler B yields more significant digits than Ruler A, has greater precision, and carries lower uncertainty.

  • A measurement from Ruler A cannot be reported as $4.70\,\text{cm}$ or $4.700\,\text{cm}$ because Ruler A lacks the sensitivity to verify the hundredths or thousandths places; adding non-measurable trailing zeroes falsely claims precision that the device cannot deliver.

Precision vs. Accuracy of Repeated Measurements

• Fundamental Definitions:

  • Accuracy: How closely measured values agree with the true or actual value.

  • Precision: How closely a series of repeated measurements agree with one another. Precision is also directly connected to the number of significant digits in a measurement.

  • Repeated measurements are performed in laboratory settings to increase confidence in the experimental data.

• Experimental Case Study: Weighing a Lead Block Standard

  • Four students repeatedly weighed a standard lead block known to have a true mass of $10.00\,\text{g}$.

• Student Data Table:

  • Student A: Trial 1 = $10.72\,\text{g}$, Trial 2 = $9.89\,\text{g}$, Trial 3 = $8.94\,\text{g}$, Trial 4 = $10.41\,\text{g}$. Average = $9.99\,\text{g}$.

    • Precise? No (individual trials vary widely from $8.94\,\text{g}$ to $10.72\,\text{g}$).

    • Accurate? Yes (the average of $9.99\,\text{g}$ is extremely close to the actual mass of $10.00\,\text{g}$).

  • Student B: Trial 1 = $9.78\,\text{g}$, Trial 2 = $9.82\,\text{g}$, Trial 3 = $9.75\,\text{g}$, Trial 4 = $9.80\,\text{g}$. Average = $9.79\,\text{g}$.

    • Precise? Yes (all measurements are tightly clustered between $9.75\,\text{g}$ and $9.82\,\text{g}$).

    • Accurate? No (the average of $9.79\,\text{g}$ deviates systematically from the true mass of $10.00\,\text{g}$).

  • Student C: Trial 1 = $10.03\,\text{g}$, Trial 2 = $9.99\,\text{g}$, Trial 3 = $10.02\,\text{g}$, Trial 4 = $9.98\,\text{g}$. Average = $10.01\,\text{g}$.

    • Precise? Yes (measurements closely agree with each other).

    • Accurate? Yes (the average of $10.01\,\text{g}$ closely matches the true mass of $10.00\,\text{g}$).

  • Student D: Trial 1 = $8.07\,\text{g}$, Trial 2 = $12.02\,\text{g}$, Trial 3 = $5.03\,\text{g}$, Trial 4 = $1.23\,\text{g}$. Average = $6.59\,\text{g}$.

    • Precise? No (wildly scattered data points).

    • Accurate? No (the average of $6.59\,\text{g}$ is far from the true value).

Significant Figure Guidelines

• Rules for Identifying Significant Digits:

  1. All non-zero digits are significant:

  • $28.3\,^\circ\text{C}$ has $3$ significant figures.

  • $0.54\,\text{kg}$ has $2$ significant figures.

  1. Interior zeroes (zeroes between non-zero digits) are significant:

  • $20.03\,\text{K}$ has $4$ significant figures.

  • $2001\,\text{m}$ has $4$ significant figures.

  1. Leading zeroes (zeroes to the left of the first non-zero digit) are NOT significant:

  • They serve only as place-holders to locate the decimal point.

  • $0.0032\,\text{s}$ has $2$ significant figures.

  • $0.8\,\text{A}$ has $1$ significant figure.

  1. Ending zeroes after a decimal point are significant:

  • $45.000\,\text{g}$ has $5$ significant figures.

  • $3.600\,\text{km}$ has $4$ significant figures.

  1. Ending zeroes before an explicit decimal point are significant:

  • $2600.\,\text{mm}$ has $4$ significant figures.

  • $700,000.\,\text{ns}$ has $6$ significant figures.

  1. Ending zeroes before an implied decimal point are ambiguous:

  • Writing a value such as $4000\,\text{m}$ is an improper way to report a measurement because the estimated digit and exact uncertainty cannot be established.

  • Scientific notation must be used instead to remove ambiguity.

• Scientific Notation vs. Uncertainty Breakdown for $4000\,\text{m}$:

  • $4 \times 10^3\,\text{m}$:

  • Significant figures: $1$

  • Assumed uncertainty: $\pm 1000$

  • Range of true value: $3000$ to $5000\,\text{m}$

  • $4.0 \times 10^3\,\text{m}$:

  • Significant figures: $2$

  • Assumed uncertainty: $\pm 100$

  • Range of true value: $3900$ to $4100\,\text{m}$

  • $4.00 \times 10^3\,\text{m}$:

  • Significant figures: $3$

  • Assumed uncertainty: $\pm 10$

  • Range of true value: $3990$ to $4010\,\text{m}$

  • $4.000 \times 10^3\,\text{m}$:

  • Significant figures: $4$

  • Assumed uncertainty: $\pm 1$

  • Range of true value: $3999$ to $4001\,\text{m}$

Exact Numbers

• Definition:

  • Exact numbers carry no uncertainty and possess an unlimited (infinite) number of significant digits.

• Three Categories of Exact Numbers:

  1. Numbers from accurately counting discrete objects:

  • Examples: $25$ pennies, $230$ students.

  1. Numbers from defined quantities or unit conversions:

  • Examples: $1000\,\text{mm} = 1\,\text{m}$, $1\,\text{foot} = 12\,\text{in}$, $1\,\text{mile} = 5280\,\text{ft}$.

  1. Integral numbers that are built into mathematical equations:

  • Examples: $\text{Area of a triangle} = \frac{\text{base} \times \text{height}}{2}$, $\text{Diameter} = 2 \times \text{radius}$.

Significant Figures in Calculations

• General Principle:

  • The precision and uncertainty of a measured quantity cannot be improved simply by using it inside a mathematical calculation.

  • Example: Adding $3\,\text{g}$ ($\pm 1\,\text{g}$) to $0.00034\,\text{g}$ ($\pm 0.00001\,\text{g}$) equals $3.00034\,\text{g}$. Reporting $3.00034\,\text{g}$ incorrectly implies an uncertainty of $\pm 0.00001\,\text{g}$, which is invalid because the starting value of $3\,\text{g}$ is uncertain in the ones place.

  • The uncertainty of individual values must carry through all calculations; the final calculated result can never be more precise than the least precise input value.

  • Addition/subtraction follow different rules than multiplication/division.

Addition and Subtraction Rules

• Core Rule:

  • The precision of the final answer is dictated by the most uncertain value in the calculation.

  • This relates directly to the place value / number of decimal places of uncertainty.

• Rounding Rule:

  • Look at the first non-significant digit immediately past the rounding cutoff position:

  • If the digit is $0$ to $4$, round down.

  • If the digit is $5$ to $9$, round up.

• Worked Example 1:

  • Problem: $2.036 + (1.23 \times 10^2) - 0.502 = ?$

  • Expand scientific terms: $1.23 \times 10^2 = 123.$

  • Align values by decimal location:     \begin{array}{r@{}l} 2. & 036 \\ + 123. & \quad \text{(most uncertain term, uncertain in the ones place)} \\ - 0. & 502 \\ \hline 124. & 534 \end{array}

• Cutoff line: Ones decimal place (dictated by $123.$).

• Look at the first non-significant digit: $5$ in the tenths place.

• Round $124.534$ up to the ones place: $125$.

  • Worked Example 2:

• Problem: $357.6 - (8.5 \times 10^4) + 42.4 = ?$

• Scientific terms and individual uncertainties:

  • $357.6$ has uncertainty $\pm 0.1$ (tenths place).

  • $-8.5 \times 10^4 = -85000.$ has uncertainty $\pm 1000$ (thousands place; most uncertain term).

  • $+42.4$ has uncertainty $\pm 0.1$ (tenths place).

• Compute unrounded total: $-84600.0$

• Cutoff line: Thousands place (dictated by $-8.5 \times 10^4$).

• Convert back to scientific notation rounded to the thousands place: $-8.5 \times 10^4$

• Important Takeaway: Addition/subtraction precision is governed by place value uncertainty, not simply counting digits to the right of the decimal point.

Multiplication and Division Rules

• Core Rule:

  • The final result must contain the same number of significant figures as the measurement term with the fewest significant figures.

• Avoidance of Intermediate Rounding Errors:

  • To avoid cumulative rounding errors in multi-step calculations, round only the final answer—never round intermediate calculation steps.

• Worked Example:

  • Problem: $\frac{(3.6) \times (4.84 \times 10^3)}{0.5020} = ?$

• Significant figure counts per term:

  • $3.6$: $2$ significant figures (limiting term)

  • $4.84 \times 10^3$: $3$ significant figures

  • $0.5020$: $4$ significant figures

• Unrounded calculation: $34709.163... = 3.4709163 \times 10^4$

• Rounding to $2$ significant figures: $3.5 \times 10^4$

Significant Figures in Mixed Calculations

• Handling Combined Mathematical Operations:

  • When a calculation combines addition/subtraction with multiplication/division, track significant figures through each step following order of operations, but retain unrounded values until the end.

• Worked Example:

  • Problem: Evaluate $\frac{10.6 - 0.911}{1.11 \times 0.3240}$

  • Step 1 (Numerator Addition/Subtraction):

  • Compute $10.6 - 0.911$

  • $10.6$ has uncertainty $\pm 0.1$ (tenths place).

  • $0.911$ has uncertainty $\pm 0.001$ (thousandths place).

  • Result: $10.6 - 0.911 = 9.689$

- The subtraction result is limited to the tenths place ($\pm 0.1$), giving $9.689$ a total of **2 significant figures** (the digits $9.6$ are significant). Do not round yet; carry $9.689$ forward unrounded.

• Step 2 (Overall Multiplication/Division):

  • Perform division: $\frac{9.689}{1.11 \times 0.3240}$

  • Significant figure counts of factors:

    • Numerator $9.689$: $2$ significant figures (from Step 1)

    • Denominator term $1.11$: $3$ significant figures

    • Denominator term $0.3240$: $4$ significant figures

  • Unrounded quotient: $26.94083...$

  • Round to $2$ significant figures (dictated by the numerator):       $\text{Final Answer} = 27$