A-Level IAL Unit 1 03 Errors and Uncertainties

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Unit 1 03 Errors and Uncertainties

1. Learning Objectives

  • Subject Content:
    • Distinguish between Precision and Accuracy.
    • Identify Random and Systematic errors.
    • Calculate absolute, fractional, and percentage Uncertainties.
    • Combine uncertainties in calculations (addition, multiplication, powers).
  • Language Goals:
    • Describe data sets using “precise but not accurate” etc.
    • Explain the “source of error” in experiments.

2. Key Terminology

Term Definition
Accuracy How close a reading is to its true value.
Precision How close the readings are to each other.
Random Error Unpredictable variation between measurements.
Systematic Error Constant bias in one direction (e.g., zero error).
Absolute Uncertainty The range () of a measurement.
Percentage Uncertainty

3. Accuracy vs. Precision

  • Accuracy: How close the measured value is to the “true value”. Affected by systematic errors.
  • Precision: The “concentration” (spread) between multiple measured values. Affected by random errors.

Exam Tip: A measurement can be very precise (consistent) but inaccurate (wrong) if there is a zero error in the instrument.

4. Types of Errors

Type Cause How to Reduce
Random Error Fluctuations in conditions, human reaction time, parallax error (variable). Repeat and Average the readings.
Systematic Error Poorly calibrated instruments, zero error, background interference. Recalibrate or check against a known standard.
Caution

Repeating measurements does NOT reduce systematic errors.

This is a very important principle in experimental physics. Simply put: Repeated measurements can only “dilute” random fluctuations, but cannot eliminate a fixed bias.

1. The Core Reason: Directionality of Error

Systematic errors are consistent and unidirectional. This means that no matter how many times you measure, the result will always deviate in the same direction (e.g., always larger than the true value, or always smaller).

When we take the average of repeated measurements, we are trying to cancel out random errors that fluctuate up and down. However, since a systematic error is “wrong in the same way” every time, averaging them preserves the error rather than eliminating it.

2. An Intuitive Example: The Uncalibrated Scale

Imagine you have an electronic scale that is not zeroed; it reads 0.1 kg even when nothing is on it (this is a systematic error).

  • True Weight: An apple weighs 0.5 kg.
  • 1st Measurement: Reads 0.6 kg (0.5 + 0.1).
  • 2nd Measurement: Reads 0.6 kg.
  • 100th Measurement: Still reads 0.6 kg.

If you add up these 100 results and calculate the average, the result is still 0.6 kg. Repeating the measurement did not bring you closer to the true 0.5 kg. You have simply measured a wrong (inaccurate) value very precisely.

3. Mathematical Representation

Suppose the true value is , and the systematic error is (a constant).

Each measured result is:

If you measure times and take the average ():

Conclusion:

The average still contains the full systematic error .

4. Only Random Errors Can Be Reduced by Repetition

Unlike systematic errors, Random Errors are unpredictable; sometimes they are high, sometimes low.

  • The first time might measure 0.52 kg (+0.02).
  • The second time might measure 0.48 kg (-0.02). After averaging many measurements, the +0.02 and -0.02 tend to cancel each other out, bringing the result closer to the true value.

Summary

  • Systematic Error: Repeated measurements do not help because the error is constant. Solution: Calibrate instruments or improve experimental design.
  • Random Error: Repeated measurements help because the errors cancel out.

5. Combining Uncertainties

In A-level exams, you need to calculate the uncertainty of final results based on formulas.

Rule 1: Addition and Subtraction

  • or
  • Rule: Add the Absolute Uncertainties.

Rule 2: Multiplication and Division

  • or
  • Rule: Add the Percentage Uncertainties.

Rule 3: Powers


  • Rule: Multiply percentage uncertainty by the power index.

6. Rules for Significant Figures (s.f.)

In A-level Physics, the use of significant figures is not a suggestion but a grading standard.

A. Uncertainty and s.f.

  1. Uncertainty is usually 1 s.f.

    • Example: Write , instead of .

  2. The value must match the precision of the uncertainty. (The decimal places of the measured value must align with the decimal places of the uncertainty).

    • Correct: (Two decimal places aligned)
    • Incorrect: (Not aligned)

B. Calculation Rules

  1. Multiplication/Division: The significant figures of the result should match the data with the least significant figures in the question (usually 2 or 3 digits).
  2. Intermediate Steps: During intermediate calculations, keep one extra significant figure (e.g., keep 4 s.f.) to avoid rounding errors, and only round off in the final answer.
Value Number of s.f. Reason
0.0052 2 Leading zeros don’t count
5.200 4 Trailing zeros after decimal count
502 3 Zeros between digits count

7. Practice Exercises

Q1. A voltmeter has a zero error of . This error is:

A. Random and affects precision.

B. Random and affects accuracy.

C. Systematic and affects precision.

D. Systematic and affects accuracy.

Q2. The length of a table is measured as and its width is .

Calculate the percentage uncertainty in the area of the table.

Q3. The density of a metal cube is calculated using the mass and the length of its side :

The percentage uncertainty in is , and the percentage uncertainty in is .

What is the percentage uncertainty in the calculated density?

Q4. A student measures the time for 20 oscillations of a pendulum to be with a stopwatch. The uncertainty in the timing is .

What is the absolute uncertainty in the period (the time for one oscillation)?

Practice Exercises Analysis

Q1. Voltmeter Zero Error

  • Analysis: A Zero error is caused by the instrument itself not being calibrated, meaning every measurement will be larger or smaller by a fixed amount. This “constant bias” is a Systematic error. Since it causes the measured value to deviate from the true value, it affects Accuracy.
  • Correct Answer: D

Q2. Percentage Uncertainty in Area

  • Formula:
  • Step 1: Find % uncertainty of L.
  • Step 2: Find % uncertainty of W.
  • Step 3: Combine using Multiplication Rule (Add % uncertainties).
  • Final Answer: (Rounded to 2 s.f.)

Q3. Percentage Uncertainty in Density ()

  • Step 1: Identify power rule for . The % uncertainty in is .
  • Step 2: Combine with mass using Division Rule (Add % uncertainties). .
  • Final Answer:

Q4. Absolute Uncertainty in Period

  • Data: for oscillations.
  • Step 1: Calculate Period . .
  • Step 2: Calculate Absolute Uncertainty in (). When you divide the total time by , the absolute uncertainty is also divided by . .
  • Final Answer: .