Accuracy and Precision Notes

Accuracy & Precision, Random Errors & Systematic Errors

Key Concepts

  • Accuracy: How close a measurement is to the true or accepted value.
  • Precision: How close measurements of the same item are to each other.
  • Random Errors: Errors in measurement caused by factors that vary from one measurement to another.
  • Systematic Errors: Errors having a nonzero mean, so that their effect is not reduced when observations are averaged. These errors lead to predictable and consistent departures from the true value.

Accuracy and Precision

  • The distinction between accuracy and precision is critical in scientific endeavors.
  • It is possible to be precise but not accurate, and vice versa.
  • The best scientific observations are both accurate and precise.
Examples
  • Scenario A: Darts are neither close to the bulls-eye nor close to each other $\rightarrow$ neither accuracy nor precision.
  • Scenario B: Darts land close together but far from the bulls-eye $\rightarrow$ precision, but not accuracy.
  • Scenario C: Darts are equally distant from and spaced around the bulls-eye $\rightarrow$ mathematical accuracy (average of darts is in the bulls-eye).
  • Scenario D: Darts are scattered around the bulls-eye $\rightarrow$ data is accurate, but not precise.
  • Martin's Experiment:
    • First test: 5.2 grams
    • Second test: 1.3 grams
    • Third test: 8.5 grams
    • Average yield: 5.0 grams
    • Known yield: 5.1 grams
    • Analysis: The results are far from one another, but the average is close to the known value. Accuracy is present because 5.2 is close to the known value.

Random Errors & Systematic Errors

  • Random Error: Results from the experimenter's inability to take the same measurement in exactly the same way to get the exact same number.
  • Systematic Error: Errors fluctuate due to unpredictability or uncertainty in the measuring process, or variation in the quantity being measured.
Examples of Random Error
  1. Limitations of instruments: Estimating readings when they fall between marks on a scale.
  2. Environmental factors: Measuring the mass of a sample on an analytical balance where air currents or water entering/leaving the specimen affect the balance.
  3. Variations in procedure: Weighing yourself on a scale and positioning yourself slightly differently each time.
Examples of Systematic Error
  1. Observational error: Not reading the meniscus at eye level for a volume measurement, leading to consistently inaccurate readings (either low or high).
  2. Imperfect instrument calibration: Using an improperly calibrated thermometer that gives accurate readings within a certain temperature range but becomes inaccurate at higher or lower temperatures.
  3. Environmental interference: Measuring length with a metal ruler that gives different results at different temperatures due to thermal expansion.

Practice Problems

  1. Three people weigh a standard mass of 2.00 g on the same balance and each obtains a reading of 7.32 g. This implies a Systematic Error because the balance is not calibrated properly.
  2. Susan conducts an experiment five times and gets concentrations of 1.9M, 2.1M, 1.8M, 1.9M, and 2.2M. The known concentration is 2.0M. Susan's results are Accurate because the average of her results is close to the known value.
  3. If equipment is not calibrated properly, all measurements will be offset by the same fraction, indicating Systematic Error.
  4. Measuring something that falls between two markings on a scale and needing to round up or down indicates Random Error due to the limitation of the instrument and estimation.

Remember

Accuracy vs. Precision
  • Accuracy: How close a value is to its true value (e.g., an arrow close to the bull's-eye).
  • Precision: How repeatable a measurement is (e.g., how close a second arrow is to the first).
Random Error vs. Systematic Error
  • Main difference: Random errors cause fluctuations around the true value, while systematic errors cause predictable departures.
  • Random error: Causes one measurement to differ slightly from the next due to unpredictable changes.
  • Systematic error: Affects measurements by the same amount or proportion, provided the reading is taken the same way each time. It is predictable.
  • Random errors cannot be eliminated, but systematic errors can often be reduced.

Let’s Try (Multiple Choice Questions)

  1. Which of the following refers to the term accuracy?
    • C. the extent to which a value approaches its true value
  2. Which sentences below that describes the term precision?
    • D. The over-all quality of the data
  3. Which of the following statements describes when a measurement is repeatable?
    • A. high precision
  4. A chemist who regularly brings out a complex experiment is likely to have high in .
    • B. precision
  5. Which of the following is the reason why systematic errors occurred?
    • C. Both A and B
  6. Which of these is NOT true for random errors?
    • D. None of the above
  7. Systematic errors lead to a lack of ____.
    • A. accuracy in the measurement
  8. Random errors lead to a lack of__.
    • B. precision in the measurement
  9. Repeated measurements of a quantity can reduce the effects of ____.
    • A. random errors
  10. A group of measurements for which there is insignificant random error but significant systematic error is ____.
    • A. precise & biased