Accuracy Precision, percentage Error 2024 matches notes

Standards for Measurement

  • Accuracy & Precision

  • Significant Figures

  • Scientific Notation

Measurements

  • A measurement consists of two essential parts: a number and a unit.

  • Without a unit of measure, numbers have no meaning and cannot be contrasted or evaluated.

Accuracy & Precision

Precision

  • Refers to the closeness of individual measurements when compared with one another.

  • Indicates how fine or detailed a measurement is.

Accuracy

  • Indicates how closely individual measurements align with the accepted or true value.

Understanding Accuracy and Precision

Example: Precise but Not Accurate

  • Measurements: 95.0°C, 95.1°C, 95.3°C

  • True value: 100°C

  • Measurements are tight but far from the actual boiling point of water at standard pressure.

Example: Accurate but Not Precise

  • Measurements: 1.0°C, 1.2°C, -5.0°C

  • True value: 0.0°C

  • Average measurements are centered around the true boiling point but have significant scatter.

Example: Neither Accurate Nor Precise

  • Measurements: 10.01 atm, 0.25 atm, 234.5 atm

  • True value: 1.00 atm

  • Values vary significantly and do not approach the true value.

Example: Accurate and Precise

  • Measurements: 1.000 kg, 0.999 kg, 1.002 kg

  • True value: 1.000 kg

  • Measurements cluster closely around the true value, demonstrating both accuracy and precision.

Precision and Technique

  • Precision is defined by how closely measurements group together.

  • Conducting multiple measurements can reveal precision levels.

  • Poor precision often stems from operator error or inadequate measurement techniques.

Assessing Accuracy

  • Accuracy can be checked by comparing results obtained with different methods or instruments.

Issues Leading to Poor Accuracy and Precision

  • Errors could arise from human mistakes or equipment failure.

  • A common problem is equipment not zeroing correctly: for instance, scales should read zero when empty.

Experimental Example and Error Calculation

  • When measuring the boiling point of pure water, if a thermometer reads 99.1°C, but the accepted value is 100.0°C, the measurement is inaccurate.

  • Calculate error as: 99.1°C - 100.0°C = -0.9°C.

Accepted vs Experimental Value

  • Accepted value: The standard, correct measurement derived from trustworthy sources.

  • Experimental value: The measurement obtained through personal or laboratory means.

Percent Error Calculation

  • Percent error is expressed as:

    • Percent Error = |experimental value - accepted value| / accepted value x 100%

  • Example: For a boiling point measurement of 99.1°C:

    • Percent Error = |99.1 - 100| / 100 x 100% = 0.9%.

Significant Figures

  • Significant figures (sig figs) define reliable digits in a measurement, influencing how precise that measurement is.

  • More precise instruments yield readings with larger numbers of significant figures.

  • Every measurement inherently possesses some degree of uncertainty.

Examples of Uncertainty

  • When measuring the time a pencil takes to fall, the precision may differ between a stopwatch and wall clock.

  • Comparisons among instruments, like a graduated cylinder and a beaker, highlight the variance in precision and implications on significant figures.

Instruments: Examples and Precision

  • Graduated Cylinder:

    • Reading: 41.0 mL (3 sig figs) high precision.

  • Beaker:

    • Reading: 40. mL (2 sig figs) lower precision.

Scientific Notation

  • A method to express numbers in terms of powers of ten, aiding in the handling of very large or small numbers.

  • Typically adopted for numbers greater than 100 or less than 0.10.

  • It clearly indicates the number of significant figures in a measurement.

The Importance of Scientific Notation

  • Changes in powers of ten denote significant differences in size; for example, the diameter of Earth's orbit vs. an atom:

    • Earth’s orbit: 1.0 x 10^11 m

    • Atom diameter: 1.0 x 10^-10 m.

Writing in Scientific Notation

  1. Move the decimal to the right of the first non-zero digit.

  2. Multiply the resulting number by 10 raised to the power that indicates how many places the decimal shifted.

    • Left shift: Power is positive.

    • Right shift: Power is negative.

Example Conversions to Scientific Notation

  1. 789 g → 7.89 x 10² g (3 sig figs)

  2. 96,875 mL → 9.6875 x 10⁴ mL (5 sig figs)

  3. 0.0000133 J → 1.33 x 10⁻⁵ J (3 sig figs)

  4. 8.915 atm → 8.915 x 10⁰ atm (4 sig figs)

  5. 0.94°C → 9.4 x 10⁻¹ °C (2 sig figs)

Evaluating Accuracy or Precision

  • When determining if data is accurate or precise, consider only quantitative (numerical) data:

    1. Recipe requires 25 chocolate chips, actual numbers are 34, 35, 32.

    2. Percent NaCl recorded as 99%, 99%, 98%.

    3. Required mass of KF is 0.04 g; used amounts are 0.038, 0.039, 0.041, 0.040.

    4. Henry needs 500 points; he got 400, 480, 395 in three trials.

Rounding Rules

  • Round based on the digit following the specified rounding point:

    • If it's 5 or above, round up the prior digit.

    • If it's less than 5, truncate the number.

Rounding Examples

  • Round to the nearest tenth (2 sf):

    • 6.8212 → 6.8

    • 6.7777 → 6.8

    • 6.7499 → 6.7

    • 6.9521 → 7.0

  • Round to the nearest hundredth (3 sf):

    • 6.821 → 6.82

    • 6.777 → 6.78

    • 6.749 → 6.75

    • 6.952 → 6.95