Psychologist Stanley Smith Stevens proposed a taxonomy of Levels of Measurement
Stevens, S.S. (1946). On the Theory of Scales of Measurement. Science, 103 2684, 677-80.
“Giving names to things”
Can you name examples?
Can you give more examples?
What is the meaning of “Twice the temperature”?
Examples?
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”
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: 109, 1012
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 3×108 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