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.*

“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?

- 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?

- 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”?

- 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?

- Names
- Grades (ordered labels like beginner, intermediate, advanced)
- 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)

- Nominal
- Gradation of membership
- Ordinal
- Interval
- Log-interval
- Extensive ratio
- Cyclical ratio
- Derived ratio
- Counts
- Absolute

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

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

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

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

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: 10^{9}, 10^{12}

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

How many people lives in Turkey?

Be brave, take a guess, compare with something else

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

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 \[0.1, 0.3, 1, 3, 10, 30, 100,…\]

The speed of light is about 300000000 m/s

It is easy to miscount the number of “0”

Instead, we write 3×10^{8} m/s$

Better, in computers we write `3E8`

(This is called *exponential notation*)

Here `3`

is called *mantissa* and `8`

is
the *exponent*

Remember: we are looking for two numbers

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