Class 4: Significant Figures

Methodology of Scientific Research

Andrés Aravena, PhD

February 29, 2024

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 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

The German way

Measure twice, cut once

The Chilean way

Measure once, cut twice

False precision

Dinosaur Bones

Some tourists in the Museum of Natural History are marveling at some dinosaur bones. One of them asks the guard, “Can you tell me how old the dinosaur bones are?”

The guard replies, “They are 3 million, four years, and six months old.”

“That’s an awfully exact number,” says the tourist. “How do you know their age so precisely?”

The guard answers, “Well, the dinosaur bones were three million years old when I started working here, and that was four and a half years ago.”

Being honest

Lets be honest about what we know and what we do not know

We write the values that have real meaning

3 million years means 3±0.5 ⨉ 106

Adding 4.5 years is meaningless

Significant Figures

Which figures are significant?

  • All non-zero digits are significant
    • In 1234 all digits are significant
    • Same in 12.34 and 1.234
  • Zeros surrounded by non-zero are significant
    • Same in 1204. Four significant digits

Which figures are significant ?

  • Zeros to the left are not significant
    • Four significant digits in 0.0001234
  • Zeros to the right are not significant unless there is a decimal point
    • 12340 has four significant digits
    • 123.40 has five significant digits
    • 1234.0 has five significant digits
    • 12340. has five significant digits

Be aware of notation

Notice that 1234.0 is not the same as 1234

Also, 12340. is not the same as 12340

This is a convention but not everybody uses it

It is much safer to use scientific notation

Using scientific notation

It is safer to represent numbers in scientific notation \[ \begin{aligned} 1234.0 & = 1.2340\cdot 10^3\\ 1234 & = 1.234 \cdot 10^3\\ 12340. & = 1.2340 \cdot 10^4\\ 12340 & = 1.234 \cdot 10^4 \end{aligned} \]

All digits in the mantissa are significant

(mantissa is the number being multiplied by 10 to the power of the exponent)

Writing our result

  • Last class we measured the volume of the stone ball
  • Round the uncertainty to a single figure
    • Instead of \(2013.765 ± 78.345\) write \(2013.765 ± 80\)
  • Then round the main value to the last well known place
    • Instead of \(2013.765\) ± 80 write \(2010 ± 80\)
  • The digits “3.765” were a white lie. Let’s not fool ourselves

Mechanical calculators

Calculating by mind and hand

Slide rule

This is the tool used to build nuclear reactors and going to the moon

(and all bridges and buildings)


A simple slide rule for multiplication and division