Class 6: Statistical uncertainty

Methodology of Scientific Research

Andrés Aravena, PhD

March 24, 2022


What is a measurement?

  • A measurement tells us about a property of something
    • It gives a number to that property
  • Measurements are always made using an instrument of some kind
    • Rulers, stopwatches, weighing scales, thermometers, etc.
  • The result of a measurement has two parts: a number and a unit of measurement

Measurement Good Practice Guide No. 11 (Issue 2). A Beginner’s Guide to Uncertainty of Measurement. Stephanie Bell. Centre for Basic, Thermal and Length Metrology National Physical Laboratory. UK

What is not a measurement?

There are some processes that might seem to be measurements, but are not. For example

  • Counting is not normally viewed as a measurement
  • Tests that lead to a ‘yes/no’ answer or a ‘pass/fail’ result
  • Comparing two pieces of string to see which one is longer

However, measurements may be part of the process of a test

Measurements have uncertainty

Uncertainty of measurement is the doubt about the result of a measurement, due to

  • resolution
  • random errors
  • systematic errors

Observational error

Every time we repeat a measurement with a sensitive instrument, we obtain slightly different results

  • Systematic error which always occurs, with the same value, when we use the instrument in the same way and in the same case

  • Random error which may vary from observation to another

Error versus uncertainty

Do not to confuse error and uncertainty

Error is the difference between the measured and the “true” value

Uncertainty is a quantification of the doubt about the result

Whenever possible we try to correct for any known errors

But any error whose value we do not know is a source of uncertainty

Where do errors and uncertainties come from?

Flaws in the measurement can come from:

  • The measuring instrument – instruments can suffer from errors including wear, drift, poor readability, noise, etc.

  • The item being measured – which may not be stable (measure the size of an ice cube in a warm room)

  • The measurement process – the measurement itself may be difficult to make. Measuring the weight of small animals presents particular difficulties

  • ‘Imported’ uncertainties – calibration of your instrument has an uncertainty

Where do errors and uncertainties come from?

  • Operator skill – One person may be better than another at reading fine detail by eye. The use of an a stopwatch depends on the reaction time of the operator

  • Sampling issues – the measurements you make must be representative. If you are choosing samples from a production line, don’t always take the first ten made on a Monday morning

  • The environment – temperature, air pressure, humidity and many other conditions can affect the measuring instrument or the item being measured

Reading v/s measurement

A reading is one observation of the instrument

A measurement may require several reads

For example, to measure a length, we make two reads, and we calculate the difference

The measurement will accumulate the uncertainty

Precise ≠ Accurate

Precision, accuracy, and trueness

In this context, people has defined the following ideas

  • accuracy: closeness of measurements to the true value
  • precision: closeness of the measurements to each other
  • trueness: closeness of the mean of a set of measurement to the true value

BS ISO 5725-1: “Accuracy (trueness and precision) of measurement methods and results - Part 1: General principles and definitions.”, p.1 (1994)

Precision v/s trueness

High trueness, Low precision

High trueness, Low precision

High precision, Low trueness

High precision, Low trueness

In a graphic

Historical note

In some old material, people say “accuracy” in place of trueness

Other people say bias

These words are still common in science and technology

Be aware of this discrepancy


Knowing the uncertainty

Uncertainty of a single read

For a single read, the uncertainty depends at least on the instrument resolution

For example, my water heater shows temperature with 5°C resolution: 50, 55, 60,…

If it shows 55°C, the real temperature is somewhere between 53°C and 57°C

We write 55°C ± 2.5°C

For a single read, \(Δx\) is half of the resolution

The easy rule

Last class we saw the easy rule for error propagation

\[ \begin{aligned} (x ± Δx) + (y ± Δy) & = (x+y) ± (Δx+Δy)\\ (x ± Δx) - (y ± Δy) & = (x-y) ± (Δx+Δy)\\ (x ± Δx\%) \times (y ± Δy\%)& =xy ± (Δx\% + Δy\%)\\ (x ± Δx\%) ÷ (y ± Δy\%)& =x/y ± (Δx\% + Δy\%) \end{aligned} \]

Here \(Δx\%\) represents the relative uncertainty, that is \(Δx/x\)

We use absolute uncertainty for + and -, and relative uncertainty for ⨉ and ÷

Be careful whit absolute and relative

It is easy to get confused with relative errors

Instead of \((x ± Δx\%)\) it is better to write \[x(1± Δx/x)\]

Mathematical notation was invented to make things clear, not confusing


Let’s verify the formulas of the previous slide

Remember that we assume that \(Δx/x\) is small

General formula for a function

Assuming that the errors are small compared to the main value, we can find the error for any “reasonable” function

For any smooth function \(f,\) we have \[f(x±Δx) = f(x) ± \frac{df}{dx}(x)\cdot Δx + \frac{d^2f}{dx^2}(x+\varepsilon)\cdot \frac{Δx^2}{2}\] When \(Δx\) is small, we can ignore the last part, so

Mean value theorem

If \(f\) is smooth, there is a value \(c\) between \(a\) and \(b\) such that \[\frac{f(b)-f(a)}{b-a}=\frac{df}{dx}(c)\]


\[(x±Δx)^2\] \[\ln(x±Δx)\] \[\log_{10}(x)\] \[\exp(x±Δx)\]


qPCR depends on initial concentration

The curve depends on the initial DNA concentration

Finding the initial concentration

We care only about the exponential phase

The signal increases 2 times on every cycle

\[X(C) = X(0)⋅2^C\]

So we can find the initial concentration

\[X(0) = X(C)⋅2^{-C}\]

CT: cycle where Signal crosses threshold

DNA concentration crosses 50% at 13.73 cycles

Different threshold

DNA concentration crosses 5% at 10 cycles

Standard curve: CT changes with concentration

Start with a large concentration of template, and dilute it several times. Measure the CT of each dilution

Errors may compensate

Probabilistic uncertainty propagation

These rules are “pessimistic”. They give the worst case

In general the “errors” can be positive or negative, and they tend to compensate

(This is valid only if the errors are independent)

In this case we can analyze uncertainty using the rules of probabilities

Variance quantifies uncertainty

In this case, the value \(Δx\) will represent the standard deviation of the measurement

The standard deviation is the square root of the variance

Then, we combine variances using the rule

“The variance of a sum is the sum of the variances”

(Again, this is valid only if the errors are independent)

The probabilistic rule

\[ \begin{align} (x ± Δx) + (y ± Δy) & = (x+y) ± \sqrt{Δx^2+Δy^2}\\ (x ± Δx) - (y ± Δy) & = (x-y) ± \sqrt{Δx^2+Δy^2}\\ (x ± Δx\%) \times (y ± Δy\%)& =x y ± \sqrt{Δx\%^2+Δy\%^2}\\ \frac{x ± Δx\%}{y ± Δy\%} & =\frac{x}{y} ± \sqrt{Δx\%^2+Δy\%^2} \end{align} \]

Confidence interval for the measurement

When using probabilistic rules we need to multiply the standard deviation by a constant k, associated with the confidence level

In most cases (but not all), the uncertainty follows a Normal distribution. In that case

  • \(k=1.96\) corresponds to 95% confidence
  • \(k=2.00\) corresponds to 98.9% confidence
  • \(k=2.57\) corresponds to 99% confidence
  • \(k=3.00\) corresponds to 99.9% confidence

Not all uncertainties are alike

Previously we considered one kind of uncertainty: the instrument resolution

This is a “one time” error

  • We notice it immediately
  • It does not change if we measure again

Two ways to estimate uncertainties

  • Type A - uncertainty estimates using statistics
    • (usually from repeated readings)
  • Type B - uncertainty estimates from any other information.
    • past experience of the measurements, calibration certificates, manufacturer’s specifications, calculations, published information, common sense

In most measurement situations, uncertainty evaluations of both types are needed

Measurement Good Practice Guide No. 11 (Issue 2). A Beginner’s Guide to Uncertainty of Measurement.
Stephanie Bell. Centre for Basic, Thermal and Length Metrology National Physical Laboratory. UK

Standard deviation of discretization

Standard deviation of rectangular distribution is \[u=\frac{a}{\sqrt{3}}\] when the width of the rectangle is \(2a\)

Standard deviation of thermal noise

Standard deviation of noise can be estimated from the data: \[s=\sqrt{\frac{1}{n-1}\sum_i (x_i - \bar x)^2}\]

Standard deviation of average

If the measures are random, their average is also random

It has the same mean but less variance

Standard error of the average of samples is \[\frac{s}{\sqrt{n}}\]

More samples give higher precision, but not better accuracy