Errors and Uncertainties

General physics

Learning Objectives: Master Your Understanding

Cambridge AS A Level Physics Curriculum (1.3):

1.3.1 Understand and explain the effects of systematic errors (including zero errors) and random errors in measurements
1.3.2 Understand the distinction between precision and accuracy

Language Objectives: Communicate Like a Scientist

Use scientific vocabulary related to measurements and uncertainties accurately
Explain the difference between accuracy and precision using appropriate terminology
Describe sources of error in experimental procedures using scientific language
Analyze and interpret data while discussing uncertainties and limitations

Key Terms: Build Your Scientific Vocabulary

English Term	Russian Translation	Kazakh Translation
Systematic Error	Систематическая ошибка	Жүйелі қате
Random Error	Случайная ошибка	Кездейсоқ қате
Precision	Точность	Дәлдік
Accuracy	Правильность	Нақтылық
Zero Error	Нулевая ошибка	Нөлдік қате
Uncertainty	Неопределенность	Белгісіздік
Measurement	Измерение	Өлшеу
Calibration	Калибровка	Калибрлеу

Concept Cards: Visual Learning Tools

Systematic Error

Errors that affect measurements in the same way each time. They cause all measurements to be shifted by the same amount from the true value.

Example: A ruler that starts at 0.2 cm instead of 0.0 cm

Random Error

Unpredictable variations in measurements that occur due to uncontrolled factors. They cause measurements to scatter around the true value.

Example: Human reaction time variations in timing experiments

Precision vs Accuracy

Precision: How close repeated measurements are to each other

Accuracy: How close measurements are to the true value

You can be precise but not accurate, and vice versa!

Glossary: Essential Definitions

Systematic Error: A consistent, repeatable error that affects all measurements in the same way, causing them to differ from the true value by the same amount each time.

Translation

Kazakh: Жүйелі қате — барлық өлшеулерге бірдей әсер ететін, оларды нақты мәннен бірдей мөлшерде ауытқытатын тұрақты, қайталанатын қате.

Random Error: Unpredictable variations in measurements caused by factors that vary in an unknown way from one measurement to the next.

Translation

Kazakh: Кездейсоқ қате — бір өлшеуден екіншісіне белгісіз жолмен өзгеретін факторларға байланысты өлшеулердегі болжамсыз ауытқулар.

Precision: A measure of how close repeated measurements are to each other, regardless of whether they are close to the true value.

Translation

Kazakh: Дәлдік — қайталанған өлшеулердің бір-біріне қаншалықты жақын екенінің өлшемі, олардың нақты мәнге жақын болуына қарамастан.

Accuracy: A measure of how close a measurement is to the true or accepted value.

Translation

Kazakh: Нақтылық — өлшеудің нақты немесе қабылданған мәнге қаншалықты жақын екенінің өлшемі.

Zero Error: A type of systematic error where an instrument gives a non-zero reading when it should read zero.

Translation

Kazakh: Нөлдік қате — аспап нөлді көрсету керек кезде нөлдік емес көрсеткішті беретін жүйелі қате түрі.

Theory: Understanding Errors and Uncertainties

Types of Errors in Measurements

When we make measurements in physics, we inevitably encounter errors. Understanding these errors is crucial for reliable scientific work.

1. Systematic Errors

Systematic errors are consistent errors that affect all measurements in the same way. They cause all readings to be displaced from the true value by the same amount.

Examples of Systematic Errors:

Zero Error: When an instrument doesn’t read zero when it should
Calibration Error: When an instrument is not properly calibrated
Environmental Factors: Constant temperature or pressure effects

Kazakh Translation

Жүйелі қателер — барлық өлшеулерге бірдей әсер ететін тұрақты қателер. Олар барлық көрсеткіштердің нақты мәннен бірдей мөлшерде ауытқуына себеп болады.

2. Random Errors

Random errors are unpredictable variations that cause measurements to scatter around the true value. They cannot be corrected but can be reduced by taking multiple measurements.

Sources of Random Errors:

Human reaction time variations
Electrical noise in instruments
Small environmental fluctuations

Kazakh Translation

Кездейсоқ қателер — өлшеулердің нақты мән айналасында шашыраңқы болуына себеп болатын болжамсыз ауытқулар. Оларды түзету мүмкін емес, бірақ көп өлшеу арқылы азайтуға болады.

Precision vs Accuracy

It’s important to distinguish between precision and accuracy:

Precision: How reproducible your measurements are
Accuracy: How close your measurements are to the true value

Kazakh Translation

Дәлдік пен нақтылықты ажырату маңызды:
• Дәлдік: Өлшеулеріңіздің қаншалықты қайталанатындығы
• Нақтылық: Өлшеулеріңіздің нақты мәнге қаншалықты жақындығы

Reducing Errors

To improve the quality of measurements:

Systematic Errors: Calibrate instruments, check for zero errors
Random Errors: Take multiple measurements and calculate the average

Memory Exercise: Practice Key Terms

Exercise 1: Match the definition to the term:

1. An error that affects all measurements in the same way → ___________

2. How close repeated measurements are to each other → ___________

3. Unpredictable variations in measurements → ___________

4. How close measurements are to the true value → ___________

Answer

1. Systematic Error
2. Precision
3. Random Error
4. Accuracy

Exercise 2: Complete the sentences with appropriate terms:

1. A _______ error occurs when a measuring instrument doesn’t read zero when it should.

2. To reduce _______ errors, we can take multiple measurements and find the average.

3. _______ can be improved by proper calibration of instruments.

Answer

1. Zero
2. Random
3. Accuracy

Video Learning: Watch and Learn

Additional Resources:

Worked Examples: Problem Solving Practice

Example 1: Identifying Error Types

Problem: A student uses a ruler to measure the length of different objects. The ruler has a manufacturing defect where the 0 cm mark is actually at 0.2 cm. What type of error is this, and how does it affect measurements?

Solution

Solution:
This is a systematic error, specifically a zero error.
Effect: All measurements will be 0.2 cm larger than the true value.
Correction: Subtract 0.2 cm from all measurements.

Pronunciation Guide

systematic [ˌsɪstəˈmætɪk] - жүйелі
zero error [ˈzɪəroʊ ˈerər] - нөлдік қате
measurements [ˈmeʒərmənt] - өлшеулер

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Example 2: Precision vs Accuracy Analysis

Problem: A student measures the mass of an object 5 times and gets: 48.2g, 48.1g, 48.3g, 48.2g, 48.1g. The true mass is 50.0g. Comment on the precision and accuracy of these measurements.

Solution

Solution:
Average measurement: (48.2 + 48.1 + 48.3 + 48.2 + 48.1) ÷ 5 = 48.18g
Precision: High — measurements are very close to each other (range: 0.2g)
Accuracy: Low — average is 1.82g away from true value
Conclusion: The measurements are precise but not accurate, suggesting a systematic error.

Example 3: Error Calculation

Problem: Calculate the percentage error when measuring a length of 25.0 cm as 24.6 cm.

Percentage Error = |measured value — true value| / true value × 100%

Solution

Solution:
Given: True value = 25.0 cm, Measured value = 24.6 cm
Percentage Error = |24.6 — 25.0| / 25.0 × 100%
= 0.4 / 25.0 × 100%
= 1.6%

Interactive Investigation: Explore with Simulations

Investigation Questions:

Add data points and observe how the line of best fit changes. What happens when you add an outlier?
Try adjusting the error bars. How do larger uncertainties affect your interpretation?
Compare different curve fitting options. Which gives the best representation of your data?

Investigation Answers

1. Outliers can significantly affect the line of best fit, pulling it away from the general trend of the data.
2. Larger error bars indicate greater uncertainty, making it harder to determine precise relationships between variables.
3. The best curve fit depends on the data pattern — linear for proportional relationships, polynomial for more complex relationships.

Collaborative Learning: Work Together

Group Activity Instructions:

Work in pairs to complete the interactive matching exercise above
Discuss your answers with your partner before submitting
Take turns explaining different types of errors to each other
Create your own examples of systematic and random errors from daily life

Individual Assessment: Test Your Understanding

Question 1: A digital balance always shows 0.05 g when nothing is placed on it. What type of error is this?

A) Random error
B) Systematic error
C) Precision error
D) Accuracy error

Answer

B) Systematic error
This is a zero error, which is a type of systematic error because it consistently affects all measurements in the same way.

Question 2: A student measures the acceleration due to gravity five times and obtains the values: 9.78, 9.82, 9.79, 9.81, 9.80 m/s². The accepted value is 9.81 m/s². Which statement best describes these measurements?

A) High precision, high accuracy
B) High precision, low accuracy
C) Low precision, high accuracy
D) Low precision, low accuracy

Answer

A) High precision, high accuracy
The measurements are clustered closely together (high precision) and the average (9.80 m/s²) is very close to the accepted value (high accuracy).

Question 3: Which of the following is the best way to reduce random errors in an experiment?

A) Calibrate the instruments
B) Take multiple readings and calculate the mean
C) Use more expensive equipment
D) Perform the experiment in a controlled environment

Answer

B) Take multiple readings and calculate the mean
Random errors can be reduced by taking multiple measurements and calculating the average, as the random variations tend to cancel out.

Question 4: A stopwatch consistently reads 0.2 seconds when it should read zero. If this stopwatch is used to measure a time interval of 10.0 seconds, what will be the measured value?

A) 9.8 seconds
B) 10.0 seconds
C) 10.2 seconds
D) 10.4 seconds

Answer

C) 10.2 seconds
This is a zero error. The stopwatch adds 0.2 seconds to all measurements, so the measured value will be 10.0 + 0.2 = 10.2 seconds.

Question 5: Calculate the percentage uncertainty in a measurement of 25.0 ± 0.5 cm.

A) 2.0%
B) 2.5%
C) 5.0%
D) 0.5%

Answer

A) 2.0%
Percentage uncertainty = (uncertainty / measured value) × 100%
= (0.5 / 25.0) × 100% = 2.0%

Additional Resources: Extend Your Learning

Recommended Websites:

Practice Papers and Resources:

Reflection: Think About Your Learning

Take a moment to reflect on what you’ve learned:

What was the most challenging concept in this lesson?

Consider: Was it understanding the difference between systematic and random errors, or distinguishing between precision and accuracy?
How can you apply this knowledge in laboratory work?

Think about: What steps will you take to identify and minimize errors in your experiments?
What questions do you still have about errors and uncertainties?

Consider: Are there any aspects of error analysis that you’d like to explore further?
Rate your confidence level (1-10) in:
- Identifying systematic errors: ___/10
- Identifying random errors: ___/10
- Understanding precision vs accuracy: ___/10
- Calculating percentage errors: ___/10
What will you do next to improve your understanding?

Consider: Will you practice more problems, watch additional videos, or discuss with classmates?

Remember: Understanding errors and uncertainties is fundamental to all scientific work. The skills you’ve learned today will be essential throughout your physics studies and beyond!