%20scale(-1%201)'/%3e%3c!--%20power%20symbol%20--%3e%3cpath%20fill='none'%20stroke='%23fff'%20stroke-width='28'%20stroke-linecap='round'%20d='M212.3,219.9%20A68,68%200%201%200%20299.7,219.9'/%3e%3cpath%20fill='none'%20stroke='%23FBBF24'%20stroke-width='24'%20stroke-linecap='round'%20d='M256,200%20V284'/%3e%3c/svg%3e){kind=small}
Physics · Chapter 2: Physical Quantities and Measurement
Errors and Uncertainties
Systematic and random errors, uncertainty, accuracy and precision, and percentage uncertainty for Class 11 and A Level.
Key termRememberExampleWatch out
01Concept
Every measurement has some doubt in it. The difference between the measured value and the true value is the error. Errors are not mistakes; they come from the instrument, the method and the person measuring. We cannot remove them completely, but we can reduce them.
Types of error
| Systematic error | Random error | |
|---|---|---|
| What it does | Makes every reading wrong by the same amount in the same direction | Makes readings scatter above and below the true value |
| Causes | Zero error, a wrongly marked scale, a slow clock, always reading at an angle | Reaction time, parallax from different positions, small changes in conditions |
| How to reduce it | Check and correct zero error, calibrate the instrument, use a better method | Repeat the measurement and take the mean |
| Does repeating help? | No: every repeat has the same error | Yes |
Uncertainty
The uncertainty of a measurement shows the range within which the true value is likely to lie. It is usually taken as the least count of the instrument and written with ±.
- Metre rule: (25.4 ± 0.1) cm
- Vernier callipers: (2.46 ± 0.01) cm
- Screw gauge: (0.52 ± 0.01) mm
Accuracy and precision
- Accuracy: how close a measurement is to the true value. Reduced by systematic errors.
- Precision: how close repeated measurements are to each other, and how fine the instrument's least count is. Reduced by random errors.
Percentage uncertainty (Class 11 / A Level)
Percentage uncertainty = (absolute uncertainty ÷ measured value) × 100%. (25.4 ± 0.1) cm → 0.1 ÷ 25.4 × 100 = 0.39%
- Adding or subtracting quantities: add the absolute uncertainties.
- Multiplying or dividing: add the percentage uncertainties.
- Raising to a power: multiply the percentage uncertainty by the power. If r has 2% uncertainty, the area πr² has 4%.
02Virtual Lab
Try this topic for yourself in the virtual lab. It opens on its own page with steps, tasks and questions.
03Key Formulas
Percentage uncertainty (Class 11 / A Level)
- = absolute uncertainty
- = measured value
Uncertainty in a power (Class 11 / A Level)
04Real World Examples
Cricket umpire's DRS
Ball-tracking shows a small margin of uncertainty; when the ball only clips the stumps, the decision stays with the umpire ("umpire's call") because the measurement is not certain enough.
Petrol pumps
The Weights and Measures department checks petrol pumps with standard measures. A pump that always gives a little less is a systematic error, and it is fined.
Medical tests
Labs repeat important blood tests, because a single reading may have a random error.
05Practice Questions
Try to answer in your own words first, then open the model answer.
Short questions
Q1.Differentiate between systematic and random errors.Show answer
Answer
| Systematic error | Random error |
|---|---|
| Same size and direction each time | Readings scatter both ways |
| Caused by zero error, faulty instrument | Caused by reaction time, parallax |
| Reduced by correcting the instrument | Reduced by repeating and taking the mean |
Q2.What is uncertainty? Write the length 18.7 cm measured with a metre rule with its uncertainty.Show answer
Answer
Uncertainty is the range within which the true value is likely to lie. Length = (18.7 ± 0.1) cm.
Q3.Differentiate between accuracy and precision.Show answer
Answer
- Accuracy: closeness of a measurement to the true value.
- Precision: closeness of repeated measurements to each other (and fineness of the scale).
Q4.A student's stopwatch always starts 0.2 s late. What kind of error is this? How can it be corrected?Show answer
Answer
It is a systematic error. It is corrected by adding 0.2 s to every reading or by using a stopwatch that works properly. Repeating the reading will not remove it.
Q5.(Class 11 / A Level) A length is (50.0 ± 0.5) cm. Find the percentage uncertainty.Show answer
Answer
- % uncertainty = 0.5 ÷ 50.0 × 100
- = 1%
Long questions
Q6.What are errors in measurement? Describe systematic and random errors, their causes and how to reduce them. Explain accuracy and precision with a diagram.Show answer
Answer
An error is the difference between the measured value and the true value.
- Systematic errors shift every reading by the same amount in the same direction. Causes: zero error, incorrect calibration, a fault in the method. Reduce them by checking for zero error, correcting the readings and calibrating instruments.
- Random errors make readings scatter. Causes: reaction time, parallax, changing conditions. Reduce them by repeating the measurement many times and taking the mean.
Accuracy is closeness to the true value; precision is closeness of the readings to each other. Draw four targets: accurate and precise, precise but not accurate, accurate but not precise, and neither.