- 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

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

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

- resolution
- random errors
- systematic errors

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

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

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

**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

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

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)

High trueness, Low precision

High precision, Low trueness

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

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

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 ÷

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

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

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

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

The curve depends on the initial DNA 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

\[X(0) = X(C)⋅2^{-C}\]

DNA concentration crosses 50% at 13.73 cycles

DNA concentration crosses 5% at 10 cycles

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

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

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)

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

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

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

**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

Stephanie Bell. Centre for Basic, Thermal and Length Metrology National
Physical Laboratory. UK

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

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

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 \[\frac{s}{\sqrt{n}}\]