Uncertainty of Measurements

Uncertainty of Measurements (Section 1.6)

This section focuses on how to treat and report numbers in scientific measurements, particularly addressing their uncertainty.

Exact Numbers

Exact numbers are values known with infinite precision and no uncertainty. They are derived in two ways:

• Counting: Any value that can be counted is exact.

  • Example: Counting fingers on a hand results in $5$ fingers. There is no possibility of $4.5$ or $5.5$ fingers; the number is precisely $5$.

• Definitions: Defined values are also exact.

  • Example 1: The metric prefix 'milli' is defined as $1 \times 10^{-3}$. Therefore, $1$ millimeter ($mm$) is exactly equal to $1 \times 10^{-3}$ meters ($m$). Equivalently, $1000$ millimeters equals exactly $1$ meter. These are precise definitions without any inexactness.

Inexact Numbers (Measured Values)

In contrast to exact numbers, measured values are inherently inexact and contain uncertainty. This uncertainty manifests in two primary ways:

Reading Analog Scales

• Description: Analog scales (e.g., graduated cylinders, rulers, thermometers) show continuous gradations. When measuring, the liquid meniscus or object edge may fall between the smallest marked intervals.

• Rule for Reporting: When reading an analog scale, you can estimate one digit beyond what you are certain of.

  • Example (Graduated Cylinder): Imagine a graduated cylinder where lines are marked for $25$ mL and $26$ mL, with nine smaller lines in between, making each small line represent $0.1$ mL.

    • A mark indicates $25.4$ mL, and the next mark indicates $25.5$ mL.

    • If the meniscus of the liquid falls between $25.4$ mL and $25.5$ mL, you are certain it's greater than $25.4$ and less than $25.5$.

    • You then estimate the final digit (to the hundredths place) by judging its position between the $0.1$ mL gradations.

    • If the meniscus appears to be about a third of the way up from $25.4$, you might report the volume as $25.43$ mL.

    • The last digit ($3$ in $25.43$ mL) is an estimate and thus inherently uncertain, but it's an accepted practice to provide this extra digit of precision.

• Characteristics: Analog scales represent an infinite range of possible values. As an instrument is filled, every possible value is theoretically achieved at some instant. This leads to the need for estimation and reporting with one digit of uncertainty.

Reading Digital Scales

• Description: Digital scales (e.g., electronic balances, digital thermometers) provide a direct numerical readout.

• Rule for Reporting: With digital scales, all digits displayed should be recorded and are considered meaningful. It is assumed that there is some uncertainty in the last reported digit.

  • Example (Digital Balance): If a digital balance reads $4.3101$ grams ($g$), all these digits are recorded.

  • Inherent Fluctuation: The last digit on a precision digital balance often fluctuates due to minor environmental factors (air currents, vibrations) or manufacturing imprecision. This fluctuation signifies the inherent uncertainty in that last digit.

  • Practical Use: In practice, one might have to 'eyeball' and choose a stable reading if display jumps, or refer to manufacturer specifications for actual precision (e.g., $\pm 0.05$ degrees).

• Focus of Uncertainty Discussion: This discussion about uncertainty in measurements primarily concerns how well our instruments measure values when used correctly, not errors arising from improper instrument use (e.g., misreading a scale, misplacing a ruler).

Precision vs. Accuracy

When evaluating the quality of a measured value, it's crucial to distinguish between accuracy and precision.

• Accuracy:

  • Definition: Accuracy refers to how close a measurement is to the true or correct value.

  • Impact of Errors: Inaccurate measurements occur when the instrument is not used properly or there's a systematic error.

    • Example 1: Measuring a distance with a ruler that is not aligned to the zero mark will result in an inaccurate reading.

    • Example 2: Leaving a thumb on a balance while measuring mass will yield an inaccurate mass.

  • Dartboard Analogy: Hitting the 'bullseye' on a dartboard represents an accurate measurement.

• Precision:

  • Definition: Precision refers to how closely multiple repeat measurements agree with one another.

  • Reliability of Digits: It also speaks to the number of reliable, consistent digits a measurement can be known to.

  • Dartboard Analogy: All throws landing close together, regardless of their distance from the bullseye, represent a precise set of measurements.

• Combinations:

  • Precise but Inaccurate: Measurements are consistent with each other (tight cluster on the dartboard) but consistently far from the true target (e.g., ruler misaligned but consistent readings).

  • Accurate but Imprecise: Measurements are, on average, close to the true value, but individual measurements are scattered (e.g., darts all over the board, but centered around the bullseye).

  • Ideal Scenario: Both precise and accurate – repeatable measurements that are also close to the true value (all darts clustered in the bullseye).

• Relationship to Uncertainty: The discussion on the uncertainty of measurements primarily focuses on the precision of our measurements – determining how many reliable digits can be reported.

Significant Figures (Sig Figs)

Significant figures are the digits in a measured value that reliably convey its precision. The fundamental assumption is that the last reported digit in any measured value has some degree of uncertainty.

• Example from Analog Scale: If a thermometer reads $27^<br>\circ C$, and it's known to be between $25^<br>\circ C$ and $30^<br>\circ C$, then $27$ is reported. This value has two significant figures, and the '$7$' is understood to have some uncertainty.

Rules for Identifying and Counting Significant Figures

1. All non-zero digits are significant.

   • Example: $25$ has $2$ significant figures.

2. Zeros between non-zero digits are significant.

   • Example: $205$ has $3$ significant figures.

3. Zeros at the beginning of a number (leading zeros) are never significant. These are placeholder zeros, indicating magnitude but not precision.

   • Example: $0.0001$ has $1$ significant figure (the '$1$'). The leading zeros are placeholders.

4. Zeros at the end of a number (trailing zeros) are significant if the number contains a decimal point.

   • Without a Decimal Point: Trailing zeros without a decimal point are considered placeholder zeros and are not significant, as their purpose is to define the magnitude.

     • Example: $100$ has $1$ significant figure (the '$1$'). It's ambiguous whether the zeros indicate precision or just the magnitude of the hundreds place.

   • With a Decimal Point: If a decimal point is explicitly present, all trailing zeros are significant, indicating that they were measured or known to that level of precision.

     • Example: $100.$ (with a decimal point) has $3$ significant figures, indicating precision to the ones place.

     • This also applies if the zero is explicitly written to indicate precision after a decimal, even if it's the only non-leading zero.

Examples for Counting Significant Figures

Here are examples combining these rules, as presented in the video:

• $\mathbf{25.017}$:

  • All non-zero digits ($2, 5, 1, 7$) are significant.

  • The zero between $5$ and $1$ is significant.

  • Total: $5$ significant figures.

• $\mathbf{0.002017}$:

  • The leading zeros ($0.00$) are not significant (placeholders).

  • Non-zero digits ($2, 1, 7$) are significant.

  • The zero between $2$ and $1$ is significant.

  • Total: $4$ significant figures.

• $\mathbf{30.0}$:

  • The non-zero digit ($3$) is significant.

  • The zero between $3$ and the final zero after the decimal is significant.

  • The final zero ($.0$) is significant because it's a trailing zero after a decimal point.

  • Total: $3$ significant figures.

• $\mathbf{0.00010}$:

  • The leading zeros ($0.000$) are not significant (placeholders).

  • The non-zero digit ($1$) is significant.

  • The final zero ($0$) is significant because it's a trailing zero after a decimal point.

  • Total: $2$ significant figures.

• $\mathbf{1.020}$:

  • The non-zero digits ($1, 2$) are significant.

  • The zero between $1$ and $2$ is significant.

  • The final zero ($.0$) is significant because it's a trailing zero after a decimal point.

  • Total: $4$ significant figures.

This understanding of significant figures is crucial for performing calculations and correctly propagating uncertainty, which will be covered in subsequent discussions.