1. Accuracy
- Definition: Accuracy refers to the closeness of a measured value to the true or accepted value. It determines how correct a measurement is.
- Example: If the true length of a metal rod is 10 cm and the measured values are 9.8 cm, 9.9 cm, and 10.1 cm, these values are close to the true value. Hence, they are considered accurate.
- Real-Life Example:
Consider a thermometer that measures body temperature. If a properly calibrated thermometer reads 98.6°F for a healthy person, it is said to be accurate since 98.6°F is considered the normal body temperature.
2. Precision
- Definition: Precision refers to how closely repeated measurements agree with each other. It indicates the reproducibility or consistency of the measurements.
- Example: Suppose you measure the length of a rod five times and get 10.1 cm each time. These measurements are highly consistent, showing high precision, even if they are not accurate compared to the true length (10 cm).
- Real-Life Example:
In target shooting, if a person hits the same spot on a target multiple times (even if it is not the bullseye), they are said to have good precision. All shots are close to each other but may not be at the exact center (accurate).
Key Difference Between Accuracy and Precision:
- Accuracy is about being correct (close to the true value).
- Precision is about being consistent (getting similar results).
3. Errors in Measurement
Errors refer to the deviation of the measured value from the true or accepted value. These errors can arise due to various reasons.
a. Systematic Errors
- Definition: Systematic errors are consistent and repeatable errors that occur due to flaws in the measurement system. These errors affect the accuracy of the measurement.
- Sources:
- Instrumental Errors: Faulty instruments, improper calibration, or wear and tear over time.
- Environmental Factors: Changes in temperature, humidity, or pressure that affect measurement.
- Personal Errors: Errors made by the observer, such as parallax errors (misreading due to the wrong viewing angle).
- Example:
Suppose a weighing scale always reads 0.5 kg more than the actual weight. This error will be systematic because it affects every measurement by the same amount. - Correction: Systematic errors can be minimized by proper calibration of instruments, accounting for environmental conditions, and careful observation.
b. Random Errors
- Definition: Random errors are unpredictable fluctuations in measurements that occur due to unknown or uncontrollable factors. These errors affect the precision of the measurement.
- Sources:
- Variations in environmental conditions (e.g., slight changes in wind, temperature, etc.).
- Limitations of the observer (e.g., difficulty in making perfect readings).
- Example:
If you repeatedly measure the time taken for a pendulum to swing back and forth, you may get slightly different times, such as 2.01 s, 2.03 s, and 2.00 s. These differences are due to random errors. - Correction: Random errors can be reduced by taking multiple measurements and averaging the results.
c. Gross Errors
- Definition: Gross errors are large mistakes made by humans, such as recording the wrong data or using the wrong measuring instrument.
- Sources:
- Misreading instruments (e.g., writing 15 cm instead of 12 cm).
- Faulty experimental procedure (e.g., using a ruler instead of a tape measure for large objects).
- Example:
A person mistakenly writes down the length of a rod as 5.3 cm when it is actually 3.5 cm. This is a gross error due to human negligence. - Correction: Gross errors can be eliminated by double-checking measurements and following proper procedures.
4. Types of Errors Based on Causes
Errors can also be classified into two major categories based on their cause:
a. Absolute Error
- Definition: Absolute error is the difference between the measured value and the true value.
- Formula: Absolute error = Measured value – True value
- Example:
If the true weight of an object is 5 kg and the measured weight is 4.8 kg, the absolute error is 0.2 kg.
b. Relative Error
- Definition: Relative error is the absolute error expressed as a fraction of the true value. It provides a sense of how significant the error is relative to the correct value.
- Formula: Relative error = Absolute error / True value
- Example:
If the true weight of an object is 5 kg and the absolute error is 0.2 kg, the relative error is 0.2/5 = 0.04 (or 4%).
5. Combination of Errors
When different quantities are combined (added, subtracted, multiplied, or divided), their errors also combine. Understanding how errors propagate through different operations helps in calculating the overall error.
a. Addition and Subtraction of Errors
- Rule: When quantities are added or subtracted, their absolute errors are added.
- Example:
If you are adding two lengths, L1 = 10 cm (with an error of 0.1 cm) and L2 = 5 cm (with an error of 0.05 cm), the total length is L = L1 + L2 = 10 + 5 = 15 cm. The total error will be 0.1 cm + 0.05 cm = 0.15 cm.
b. Multiplication and Division of Errors
- Rule: When quantities are multiplied or divided, the relative errors are added.
- Example:
If the mass of an object is 5 kg with an error of 2% and the acceleration due to gravity is 9.8 m/s² with an error of 1%, the weight (W = m × g) will have a total error of 2% + 1% = 3%.
c. Power and Exponential Errors
- Rule: When a quantity is raised to a power, the relative error is multiplied by that power.
- Example:
If the length of a cube’s side is measured with a 2% error, the volume (which is the side length cubed) will have an error of 3 × 2% = 6%.
6. Minimizing Errors
To ensure better accuracy and precision in experiments, the following steps can be taken:
- Calibration: Regularly calibrate instruments to maintain accuracy.
- Repeat Measurements: Take multiple measurements to reduce random errors.
- Use Proper Techniques: Avoid gross errors by following proper measurement techniques and checking for parallax.
- Environmental Control: Minimize the impact of external conditions like temperature and pressure.
