Measurement Scales · Business Analysis Notes

Key ideas#

Experiments and measurements#

Tarasenko separates passive experiments (you observe without touching inputs) from active ones (you change some inputs on purpose). He also separates direct measurement (you observe the property itself) from indirect (you observe something linked to it).

The key rule for indirect data: use only the operations allowed by the scale of the property you care about, even if the observed value sits on a stronger scale. Time to finish a test is a ratio-scale number, but as a measure of ability it is only ordinal. Someone who finishes twice as fast is not “twice as able”.

The six basic scales#

Each scale allows everything the weaker ones allow, plus more.

Scale	What it captures	Valid operations and statistics	Everyday examples
Nominal	Same or different	Equality check; counts, mode	Customer IDs, diagnoses, product categories
Ordinal	Order	Comparison; median, percentiles, rank correlation (Spearman, Kendall)	Ratings, grades, priority levels
Interval	Equal differences, arbitrary zero	Subtraction; arithmetic on differences	Temperature in °C or °F, calendar years
Cyclic	Interval scale that wraps around	As interval, minding the wrap	Time of day, compass bearing
Ratio	Equal ratios, true zero	All arithmetic	Length, weight, money, duration
Absolute	Counts with a natural unit	Any operation	Number of items

What this means in practice#

Do not add or average ordinal values. Tarasenko describes a study that computed average school grades by summing them and had to be cancelled, because addition is not valid on an ordinal scale.
Do not divide interval values. “It warmed from 9°C to 18°C, twice as warm” is meaningless: in Fahrenheit the same change is 48.2°F to 64.4°F, nowhere near double.
Only ratio and absolute scales give “real” numbers you can use with any arithmetic.
Any order-preserving relabelling of an ordinal scale changes nothing. Add one star to every rating level and the information is the same.

Tarasenko also mentions discretized scales (continuous values rounded to an instrument’s precision), non-linear scales such as logarithmic ones, and membership scales for fuzzy categories, where something can belong partly to several classes at once.

How to apply it#

For every metric or criterion, write down its scale next to its definition.
For indirect measures, ask what scale the underlying property has, not just the number you record.
Choose summary statistics the scale allows: mode for nominal; median and distribution for ordinal; means for ratio data.
For cyclic data such as time of day, use circular methods or bucket by hour; never average naively.
When combining criteria from different scales in a decision, compare options per criterion or rank them, rather than adding raw scores.
Flag reports that break these rules before they reach decision makers.

Example#

The Workshop’s dashboard draft shows:

“Average satisfaction 4.2 stars”. Star ratings are ordinal. Report the median and the share of 4–5 star ratings instead.
“Average turnaround 3.6 days”. Duration is a ratio scale, so the mean is valid; add the 90th percentile to show the slow tail.
“Average drop-off time 12:00”. Drop-off time is cyclic. Late-evening drop-offs at the 24-hour lockers and early-morning ones would average to midday, which is wrong. Show a histogram by hour.
“Shop 7 is twice as good as shop 3”, because its rank is higher. Ranks are ordinal, so the ratio means nothing. Say “ranked higher”.

Pitfalls#

Turning ordinal labels into numbers and doing arithmetic → use medians and distributions.
Ratios on interval data → work with differences instead.
Trusting a precise number from an indirect measure → check the underlying property’s scale.
Throwing away information by downgrading a strong scale to a weak one without need → keep the raw ratio data.

Sources and further reading#

F. P. Tarasenko. Applied Systems Analysis (in Russian). KnoRus, 2010, ch. 5, section 5.8.