PHY C2: Errors & Uncertainties

Today we’ll be covering:

  • Accuracy & Precision
  • Uncertainty & Errors
  • Systematic & Random Errors
  • Absolute, Fractional, Percentage Uncertainties
  • Uncertainty in Derived Quantities

Let’s get down to business.

What is Accuracy?
The degree (or CLOSENESS) to which a MEAN MEASURED VALUE approaches the TRUE VALUE.
The Mean Measured Value is the Mean (average) calculated by dividing the sum of multiple measurements of the same value by the number of measurements taken.
The more accurate the instrument, the closer the measurements (& thus the Mean measurement) to the actual quantity measured.
Accuracy is affected by the presence of ERRORS in the reading/experiment.

What is Precision?
The degree (or CLOSENESS) to which the measured values are to each other.
In maths, this can be expressed as DEVIATION.
Precision describes the range of values taken from different measurements of the same quantity.
The smaller this range, the more Precise your measurement.
The larger this range, the higher the UNCERTAINTY of your measurement.

Let’s take an example to differentiate between Accuracy & Precision.
I have a pencil, with an actual length of 10 cm.
I take 3 measurements of its length using a metre rule.
The measurements are: 9.9cm, 10cm, 10.1cm.
The Mean Measured Value is 10cm.
The measurements are both ACCURATE & PRECISE, as they are close to each other AND the actual value.

Let’s say I took another metre rule, & measured the same pencil.
The measurements are: 9.5cm, 10cm, 10.5cm.
The measurements are now STILL ACCURATE but NOT AS PRECISE, as the measurements differ from each other by a larger value.

In general, using an instrument with smaller divisions on its scale INCREASES the PRECISION by reducing UNCERTAINTY.
At the same time, elimination of ZERO ERRORS increases ACCURACY.

What is Uncertainty?
The total range of values within which a measurement can lie.
An uncertainty of ±0.5 units means the measured value could be 0.5 units ABOVE the Mean Measured Value, or 0.5 units BELOW it.
Uncertainties are written after the Mean Measured Value when writing out the measurement, as such:

Uncertainty can be CAUSED by:

  1. Systematic Uncertainties
  2. Measurement Uncertainties

Uncertainty can be EXPRESSED as:

  1. Absolute Uncertainty
    – expressed as a NUMBER
    – the Range between the (Max Measured Value – Min Measured Value)/2
    – Example: ±0.5cm 
  2. Relative Uncertainty
    – expressed as a PROPORTION (decimal)
    – Absolute Uncertainty divided by the True Value (Mean Measured Value)
    – Example: ±0.05 
  3. Percentage Uncertainty
    – expressed as a Percentage
    – Relative Uncertainty x 100
    – Example: ±5%

These can be used to show the uncertainty of measurements of ONE quantity, OR the uncertainty of ALL measurements of an instrument across the instrument’s range.

If you have a SET OF MEASUREMENTS of the SAME quantity, here’s how you find the UNCERTAINTY in your MEASUREMENT:

  1. Find Mean Measured Value
  2. Identify Max Measured Value
  3. Identify Min Measured Value
  4. Find Abs. Uncertainty = (Max – Min) divided by 2

Answer: Mean Measured Value ± Abs. Uncertainty
To find Relative Uncertainty, divide the Abs Uncertainty by the Mean Measured Value
To find Percentage Uncertainty, multiply Relative Uncertainty by 100.

For example, let’s say I took 5 measurements of the temperature of a cup of water.
The measurements are: 49.8, 49,9, 50, 50.1, 50.2 degrees Celsius.
The Mean Measured Value is 50
The Max Measured Value is 50.2
The Min Measured Value is 49.8
The Abs Uncertainty = (50.2 – 49.8)/2 = 0.4/2 = ±0.2
The Measurement = 50±0.2 degrees Celsius

If you have a RANGE of MEASURABLE values that the instrument can read, & given the ABSOLUTE uncertainty of ANY measurement, you can identify the RELATIVE/PERCENTAGE Uncertainty like this:

  1. Find the Range of Measurable Values =
    Max Measurable Value – Min Measurable Value
  2. Identify the Abs. Uncertainty, & multiply it by 2
  3. Calculate Relative Uncertainty =
    2(Abs. Uncertainty)/Range

Answer: Given as a Proportion or Percentage

For example, let’s say I have a thermometer than can measure 40 to 100 degrees Celsius.
It can measure up to an uncertainty of ±0.5 degrees Celsius.
The Range is 60
The Range of Uncertainty: 1
The Relative Uncertainty is: 1/60
The Percentage Uncertainty is: 1/60 x 100

What is an Error?
The difference between a measured value & the TRUE value.

There are 2 types of Error:

  1. Systematic Error
    – caused by problems in the experiment/instrument
    – degree of error is CONSISTENT as long as the cause is present
    – cannot be eliminated by taking the Mean of the readings
    – ex: Zero Error, Reaction Time, Faulty Instrument, Wrong Assumption of Constants
  2. Random Error
    –  caused by fluctuations in the environment of an experiment
    – degree of error is INCONSISTENT & random
    – can be eliminated by taking the Mean of the readings
    – ex: Timing of a Pendulum, Fluctuating Pressure of Micrometer, Fluctuating Temperature

Combined Uncertainties
Let’s say we have a quantity C derived from base quantities A & B.
If there are uncertainties in the measurements of A & B, there will be an uncertainty in C.
Uncertainty can be denoted as Δ (delta).
Depending on the relation between A & B, the combined uncertainty can be found by different methods.


Addition & Subtraction
C = A + B
C = A – B
The combined ABSOLUTE uncertainty is simply the SUM of the ABSOLUTE uncertainties of the base units:
ΔC = ΔA + ΔB

Multiplication & Division
C = AB
C = A/B
The combined RELATIVE uncertainty is simply the SUM of the RELATIVE uncertainties of the base units:
(ΔC)/C = (ΔA)/A + (ΔB)/B

With this in mind, we can figure out the combined uncertainty of variables raised to powers.
Let’s say:
C = A²
C = A x A
ΔC/C = ΔA/A + ΔA/A
ΔC/C = 2(ΔA/A)

Multiplication by a Constant
C = kA
The combined ABSOLUTE uncertainty is the absolute uncertainty in the base quantity multiplied by the constant:
ΔC = ΔA x k

The combined RELATIVE uncertainty is the SAME as the base quantity’s relative uncertainty:

Since division is just multiplication by a fraction, the same rule applies to division.

A few worked examples: Sciencing: How to Calculate Uncertainty

Combining Uncertainties from Different Sources
Sometimes you’ll come across statements such as:
A manufacturer quotes the uncertainty of the instrument as ±1% ±2 digits.

This means that the instrument has a few different sources of uncertainty.
The ±1% means that the displayed value could vary by ±1%.
The ±2 digits means that the last DIGIT of the displayed figure has an uncertainty of ±2.
You also have to consider the uncertainty caused by fluctuations in the reading.

To find the total uncertainty of any measurement by the instrument, you have to add all these uncertainties together.

For example, if I have a set of measurements of a value ranging from 9.95 units to 10.05, my actual value is quoted as:
10.00 ±0.05 ±1% ±2 digits
10.00 ±0.05 ±(10.00 x 1/100) ±(0.02)
10.00 ±0.17 units

ANOTHER METHOD to find Combined Uncertainties: the MINIMAX METHOD
This method is handy in checking whether your calculations were correct.
You can also rely solely on this method, depending on your preference in calculations.
The method works by manually finding the Minimum & Maximum values of the Derived Quantity by substituting the highest & lowest possible values of the Base Quantities into the equation. Here’s a step-by-step:

  1. Identify the equation relating the base quantities with the derived quantity
  2. Identify the possible values of the base quantities that would produce the HIGHEST possible derived quantity, & substitute the values into the equation to find the MAX VALUE for the derived quantity
  3. Identify the possible values of the base quantities that would produce the LOWEST possible derived quantity, & substitute the values into the equation to find the MIN VALUE for the derived quantity
  4. Find the difference between the MAX & the MIN value of the derived quantity
  5. Divide this difference by 2

Uncertainties of Gradients
When plotting a graph of measured values, you can take into account the Uncertainties by drawing a BAR instead of a POINT, like so:
Image result for uncertainty gradient

The bar shows the Maximum & Minimum possible values, & the cross (‘x’) shows the Mean Measured Value.

  1. Finding Minimum Possible Gradient:
    Draw the line with the SMALLEST gradient (least slope) that fits within the error bars of the data
  2. Finding Maximum Possible Gradient:
    Draw the line with the LARGEST gradient (steepest slope) that fits within the error bars of the data
  3. Finding Line of Best Fit:
    Draw the line passing through as many MEAN points, & is mostly equidistant from the Max & Min Lines
  4. Calculate Uncertainty in Gradient:
    • Find the difference between MIN & MAX Gradients
    • Divide this difference by 2
  5. Write the Value of the Gradient as:
    Best Fit Gradient ± Uncertainty in Gradient

Other Resources:

⇐ Previous in Physics: Measurements
⇒ Next in Physics: Kinematics

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