Methodology of Scientific Research

Levels of Measurement

Psychologist Stanley Smith Stevens proposed a taxonomy of Levels of Measurement

• Nominal scale
• Ordinal scale
• Interval scale
• Ratio scale

Stevens, S.S. (1946). On the Theory of Scales of Measurement. Science, 103 2684, 677-80.

Nominal scale

“Giving names to things”

• Qualitative
• Assigning names to things
• Group similar things together
• Taxonomies
• Ontologies
• Mathematical operations: test for equality, test for set membership
• Central tendency: mode

Can you name examples?

Ordinal scale

• Semi qualitative, Rank
• Example: ‘completely agree’, ‘mostly agree’, ‘indifferent’, ‘mostly disagree’, ‘completely disagree’
• There is order, but intervals can be different
• Real differences between adjacent ranks may not be equal
• Mathematical operations: less-than, sort
• Central tendency: median
• And all the properties of Nominal

Can you give more examples?

Interval scale

• Difference between items, but not the ratio between them
• Temperature in the Celsius scale
• date measured from an arbitrary epoch (such as AD)
• location in Cartesian coordinates
• direction measured in degrees from north
• Mathematical operations: Plus, Minus, ratio of differences
• Central tendency: arithmetic mean, standard deviation

What is the meaning of “Twice the temperature”?

Ratio scale

• Has a meaningful (unique and non-arbitrary) zero value
• Most measurements in physical sciences and engineering
• mass, length, duration, plane angle, energy, and electric charge
• values have units of measurement
• Mathematical operations: multiplication, division
• Central tendency: geometric mean, harmonic mean, coefficient of variation

Examples?

Alternative: Mosteller and Tukey’s typology (1977)

• Names
• Ranks (orders with 1 being the smallest or largest, 2 the next smallest or largest, and so on)
• Counted fractions (bound by 0 and 1)
• Counts (non-negative integers)
• Amounts (non-negative real numbers)
• Balances (any real number)

Chrisman’s typology (1998)

• Nominal
• Ordinal
• Interval
• Log-interval
• Extensive ratio
• Cyclical ratio
• Derived ratio
• Counts
• Absolute

Summary of “Levels of Measurement”

Incremental progress Measure property Mathematical operators Advanced operations Central tendency
Nominal Classification, membership =, ≠ Grouping Mode
Ordinal Comparison, level >, < Sorting Median
Interval Difference, affinity +, − Yardstick Mean, Deviation
Ratio Magnitude, amount ×, / Ratio Geometric mean, Coefficient of variation

Wikipedia: Levels of Measurement

Estimating quantities

Training or intuition

Our brain has two “operation modes”

• Most of the time we use the intuitive system
• Rational thinking needs extra effort

Instead of thinking, we “guess” and take a shortcut

Today we will start training our intuition

Upper and lower bounds

When estimating a value, we usually can guess that the real value is somewhere between two values

In other words, we guess lower and upper bounds $$L$$ and $$U$$

Choose the smallest value that seems right,
then the largest one

Powers of 10

At first, we approximate measurements by powers of 10
(after choosing the appropriate units)

We even have names for some of them them:

• deci, centi, milli, micro, nano, pico

• deca, hecto, kilo, mega, giga, tera, peta, exa

The “billion dollars problem”

In USA, a billion is a thousand millions

In the rest of the world, a billion is a million millions
(often called “a milliard”)

There are two conventions: short scale and long scale

To avoid confusion, better use giga or tera

or use scientific notation: 109, 1012

Order of magnitude

The order of magnitude of a value is its power of 10

More precisely, is the integer part of the logarithm base 10

We say that two quantities are in the same order of magnitude if their ratio is between 0.1 and 10

i.e. if each one is less than ten times the other

Exercise

How many people lives in Turkey?

Be brave, take a guess, compare with something else

(no Google, no books, no Internet, only guess)

Middle points

Instead of going $10^{-1}, 10^{0}, 10^{1}, 10^{2}, 10^{3}$ we can increment the exponent by 0.5 $10^{-1}, 10^{-0.5}, 10^{0}, 10^{0.5}, 10^{1}$

Since $$10^{0.5} = \sqrt{10}≈ 3.16≈3$$ we can say $0.1, 0.3, 1, 3, 10, 30, 100,…$

Writing large numbers

The speed of light is about 300000000 m/s

It is easy to miscount the number of “0”

Better, in computers we write 3E8

(This is called exponential notation)

Here 3 is called mantissa and 8 is the exponent

How many people lives in Istanbul?

Remember: we are looking for two numbers

Combining upper and lower bounds

We guessed lower and upper bounds $$L$$ and $$U$$

The center of this interval is thr geometric mean $\sqrt{L⋅U}$

This can be approximated taking the average of the mantisas and the average of the exponents