1.3 Accuracy, Precision & Errors
Why Measurement Quality Matters
In science, we need to know how good our measurements are. Two key concepts help us describe this: accuracy and precision.
In science, we need to know how good our measurements are. Two key concepts help us describe this: accuracy and precision.
ā” Key Concept:
ā¢ Accuracy: How close a measurement is to the true value
ā¢ Precision: How close repeated measurements are to each other
These are NOT the same thing, as you can be precise but inaccurate.
ā¢ Precision: How close repeated measurements are to each other
These are NOT the same thing, as you can be precise but inaccurate.
Accuracy
Close to True Value
Your average result matches what it should be
Precision
Consistent Results
Your repeated measurements cluster together
š” Real Example:
The true value of gravitational acceleration is g = 9.81 m/sĀ²
ā¢ If your experiment gives 9.80 m/sĀ² ā Accurate
ā¢ If you get 9.80, 9.81, 9.79, 9.80 repeatedly ā Precise
ā¢ If you get 8.50, 8.51, 8.49, 8.50 ā Precise but NOT Accurate
ā¢ If your experiment gives 9.80 m/sĀ² ā Accurate
ā¢ If you get 9.80, 9.81, 9.79, 9.80 repeatedly ā Precise
ā¢ If you get 8.50, 8.51, 8.49, 8.50 ā Precise but NOT Accurate
The Target Analogy
The easiest way to understand accuracy and precision is with a target (dartboard) analogy.
The easiest way to understand accuracy and precision is with a target (dartboard) analogy.
ā” Target Analogy:
ā¢ Bullseye = The true value
ā¢ Accurate = Shots hit near the bullseye
ā¢ Precise = Shots are clustered together
ā¢ Accurate = Shots hit near the bullseye
ā¢ Precise = Shots are clustered together
Low Accuracy, Low Precision
Scattered everywhere
Low Accuracy, High Precision
Clustered but off-center
High Accuracy, Low Precision
Average is correct, but scattered
High Accuracy, High Precision
ā The ideal result
Example: Measuring Gravity
True value: g = 9.81 m/sĀ²
Student A's results: 9.80, 9.82, 9.79, 9.81
ā Average = 9.805 m/sĀ² ā Accurate AND Precise ā
Student B's results: 8.50, 8.52, 8.48, 8.50
ā Average = 8.50 m/sĀ² ā Precise but NOT Accurate ā
Student C's results: 8.00, 11.00, 9.50, 10.50
ā Average = 9.75 m/sĀ² ā Accurate average but NOT Precise
Student A's results: 9.80, 9.82, 9.79, 9.81
ā Average = 9.805 m/sĀ² ā Accurate AND Precise ā
Student B's results: 8.50, 8.52, 8.48, 8.50
ā Average = 8.50 m/sĀ² ā Precise but NOT Accurate ā
Student C's results: 8.00, 11.00, 9.50, 10.50
ā Average = 9.75 m/sĀ² ā Accurate average but NOT Precise
šÆ Accuracy or Precision Practice
Errors That Affect Accuracy
A systematic error is a repeating error that affects all your measurements in the same way.
A systematic error is a repeating error that affects all your measurements in the same way.
Systematic Errors
š Causes:
ā¢ Zero error on equipment (scale reads 0.1 kg when empty)
ā¢ Faulty equipment (stretched spring, slow clock)
ā¢ Flawed experimental method
ā¢ Environmental factors (temperature affecting measurements)
ā¢ Zero error on equipment (scale reads 0.1 kg when empty)
ā¢ Faulty equipment (stretched spring, slow clock)
ā¢ Flawed experimental method
ā¢ Environmental factors (temperature affecting measurements)
ā ļø Effect:
Reduces ACCURACY ā all readings shift in one direction (all too high OR all too low)
Reduces ACCURACY ā all readings shift in one direction (all too high OR all too low)
ā
How to Fix:
ā¢ Calibrate equipment before use
ā¢ Subtract zero errors from readings
ā¢ Fix or replace faulty equipment
ā¢ Cannot be fixed by repetition
ā¢ Calibrate equipment before use
ā¢ Subtract zero errors from readings
ā¢ Fix or replace faulty equipment
ā¢ Cannot be fixed by repetition
Example 1: Zero Error on a Scale
A scale shows 0.05 kg when nothing is on it (zero error).
Measurements taken: 2.55, 2.56, 2.54, 2.55 kg
Actual mass: Each reading is 0.05 kg too high.
Fix: Subtract 0.05 kg from each reading
Corrected values: 2.50, 2.51, 2.49, 2.50 kg
Measurements taken: 2.55, 2.56, 2.54, 2.55 kg
Actual mass: Each reading is 0.05 kg too high.
Fix: Subtract 0.05 kg from each reading
Corrected values: 2.50, 2.51, 2.49, 2.50 kg
Example 2: Reaction Time with Stopwatch
A student always starts the stopwatch 0.2 s late.
All time measurements will be 0.2 s shorter than they should be.
This is systematic because it's the same error every time.
Fix: Use light gates instead of manual timing
All time measurements will be 0.2 s shorter than they should be.
This is systematic because it's the same error every time.
Fix: Use light gates instead of manual timing
Graph showing Systematic Error
All points are shifted above the true line
šÆ Identify Systematic Errors
Errors That Affect Precision
A random error is an unpredictable error that causes measurements to scatter around the true value.
ā¢ Affects ACCURACY
ā¢ All readings shift one way
ā¢ Fix equipment/method
ā¢ Cannot fix by repeating
ā¢ Affects PRECISION
ā¢ Readings scatter randomly
ā¢ Take more readings
ā¢ Calculate the mean
A random error is an unpredictable error that causes measurements to scatter around the true value.
Random Errors
š Causes:
ā¢ Human reaction time variations
ā¢ Reading between scale divisions
ā¢ Fluctuations in conditions (slight air currents)
ā¢ Small, unpredictable variations in technique
ā¢ Human reaction time variations
ā¢ Reading between scale divisions
ā¢ Fluctuations in conditions (slight air currents)
ā¢ Small, unpredictable variations in technique
ā ļø Effect:
Reduces PRECISION ā readings scatter randomly (some too high, some too low)
Reduces PRECISION ā readings scatter randomly (some too high, some too low)
ā
How to Fix:
ā¢ Take multiple readings
ā¢ Calculate the mean (average)
ā¢ Use more sensitive equipment
ā¢ CAN be reduced by repetition.
ā¢ Take multiple readings
ā¢ Calculate the mean (average)
ā¢ Use more sensitive equipment
ā¢ CAN be reduced by repetition.
Example: Timing a Pendulum
A student times 10 swings of a pendulum 5 times:
The readings scatter due to reaction time (random error).
Fix: Calculate the mean:
Mean = $(15.2 + 14.8 + 15.1 + 14.9 + 15.0) Ć· 5 = 15.0$ s
The mean is more reliable than any single reading.
15.2 s
14.8 s
15.1 s
14.9 s
15.0 s
The readings scatter due to reaction time (random error).
Fix: Calculate the mean:
Mean = $(15.2 + 14.8 + 15.1 + 14.9 + 15.0) Ć· 5 = 15.0$ s
The mean is more reliable than any single reading.
š§® Mean Calculator:
Enter your measurements to find the mean:
Enter values and click Calculate
Systematic Errors
ā¢ Same error every timeā¢ Affects ACCURACY
ā¢ All readings shift one way
ā¢ Fix equipment/method
ā¢ Cannot fix by repeating
Random Errors
ā¢ Different each timeā¢ Affects PRECISION
ā¢ Readings scatter randomly
ā¢ Take more readings
ā¢ Calculate the mean
šÆ Error Type Practice
Practical Tips for Better Measurements
ā” Summary of Fixes:
For Systematic Errors (Accuracy):
ā¢ Calibrate instruments before use
ā¢ Check for and correct zero errors
ā¢ Use correct technique
For Random Errors (Precision):
ā¢ Take multiple readings (at least 3-5)
ā¢ Calculate and use the mean
ā¢ Use more precise instruments
ā¢ Calibrate instruments before use
ā¢ Check for and correct zero errors
ā¢ Use correct technique
For Random Errors (Precision):
ā¢ Take multiple readings (at least 3-5)
ā¢ Calculate and use the mean
ā¢ Use more precise instruments
š Scenario: Measuring the Length of a Wire
Systematic error: Ruler doesn't start at zero ā all measurements 2mm too long
Fix: Measure from 10cm mark and subtract 10cm, or use a different ruler
Random error: Wire isn't perfectly straight, hard to judge exact endpoint
Fix: Take 5 measurements and calculate the mean
š Scenario: Timing a Ball Rolling Down a Ramp
Systematic error: Always starting the timer late
Fix: Use light gates for automatic timing
Random error: Reaction time varies each trial
Fix: Repeat 5+ times and take the mean
| Feature | Systematic Error | Random Error |
|---|---|---|
| Pattern | Same direction always | Scattered randomly |
| Affects | Accuracy | Precision |
| Example Cause | Zero error on scale | Reaction time |
| Fixed by Repeating? | No ā | Yes ā |
| Solution | Fix equipment/method | Take mean of readings |
Real Life Uses:
ā¢ Medicine: Blood pressure monitors must be calibrated (systematic) and multiple readings taken (random)
ā¢ Manufacturing: Quality control checks for both accuracy and precision
ā¢ Sports: Electronic timing eliminates human reaction time errors
ā¢ Science: Experiments report both average values and uncertainty ranges
ā¢ Cooking: Kitchen scales need zeroing before use
ā¢ Medicine: Blood pressure monitors must be calibrated (systematic) and multiple readings taken (random)
ā¢ Manufacturing: Quality control checks for both accuracy and precision
ā¢ Sports: Electronic timing eliminates human reaction time errors
ā¢ Science: Experiments report both average values and uncertainty ranges
ā¢ Cooking: Kitchen scales need zeroing before use
šÆ Identify and Fix