6.3 Types of Data and Measurement Scales
Key Takeaways
- Qualitative data describes categories; quantitative data measures amounts, and splits into discrete counts and continuous measurements.
- The four measurement scales are nominal, ordinal, interval, and ratio, in increasing order of information content.
- Only ratio scales have a true zero, so only ratio data supports meaningful ratios such as 'twice as long'.
- Continuous data requires far smaller samples than attribute data to detect the same change, so convert to attribute form only when the process genuinely is pass or fail.
- Ordinal data such as a satisfaction rating has ordered categories but unequal intervals, so computing a mean of ordinal ratings is technically improper even though it is common practice.
The primary split
| Class | Definition | Examples | Analysis implications |
|---|---|---|---|
| Qualitative (categorical) | Describes a category or attribute | Defect type, supplier, shift, pass/fail | Counts and proportions; chi-square tests; p, np, c, u charts |
| Quantitative discrete | Counts of whole events; cannot be subdivided meaningfully | Defects per unit, calls per hour, scratches per panel | Poisson or binomial models; c and u charts |
| Quantitative continuous (variable) | Measurements on a continuum, limited only by gauge resolution | Length, mass, time, temperature, pressure | Normal-theory methods; t, F, ANOVA; X-bar/R, X-bar/s, ImR charts |
The continuous/discrete boundary is occasionally subtle. Money is formally discrete (there is a smallest unit) but is treated as continuous because the increments are tiny relative to the range. Conversely, a count of defects with a range of 0 to 3 must be treated as discrete no matter how the data is stored.
The four measurement scales
| Scale | Properties | Permissible statistics | Examples |
|---|---|---|---|
| Nominal | Categories with no order | Mode, frequency, chi-square | Defect type, machine ID, colour, region |
| Ordinal | Ordered categories, unequal or unknown intervals | Median, percentiles, rank tests | Likert ratings, severity 1-5, hardness scales, first/second/third |
| Interval | Ordered, equal intervals, arbitrary zero | Mean, standard deviation, differences | Celsius and Fahrenheit temperature, calendar dates, IQ scores |
| Ratio | Ordered, equal intervals, true zero | All of the above plus ratios and coefficient of variation | Length, mass, time, count, kelvin temperature, cost |
Each scale contains all the properties of those above it plus one more, so information increases down the table.
The true-zero test
The distinguishing feature of a ratio scale is a zero that means "none of the quantity". This is what licenses ratio statements.
- 40 C is not twice as hot as 20 C, because 0 C is the freezing point of water rather than the absence of thermal energy. Celsius is interval.
- 400 K is twice 200 K, because 0 K is the absence of thermal energy. Kelvin is ratio.
- A 40-minute cycle time is twice a 20-minute cycle time, because zero minutes means no time elapsed. Time is ratio.
The practical consequence is that coefficient of variation, $CV = \sigma / \mu$, and percentage change are meaningful only on ratio scales. Reporting that a temperature "increased 25%" is meaningless if the units are Celsius.
Ordinal data and the mean
Ordinal categories are ordered but the gaps between them are not known to be equal: the distance from "very dissatisfied" to "dissatisfied" is not necessarily the same as from "satisfied" to "very satisfied". Strictly, the mean of ordinal ratings is not defined, and the median or the distribution across categories should be reported instead.
In practice, Likert data is frequently averaged and the exam may accept that convention, but the defensible analysis reports the median and the full distribution, and uses non-parametric tests -- Mann-Whitney, Kruskal-Wallis, Wilcoxon -- rather than t-tests and ANOVA when comparing ordinal groups.
Why the classification drives everything
| Decision | Continuous data | Attribute data |
|---|---|---|
| Control chart | X-bar/R, X-bar/s, individuals and moving range | p, np, c, u |
| Capability index | Cp, Cpk, Pp, Ppk | Process sigma from DPMO or proportion defective |
| Two-group comparison | Two-sample t-test | Two-proportion test or chi-square |
| Multi-group comparison | ANOVA | Chi-square test of independence |
| Relationship between two variables | Correlation, regression | Contingency table, logistic regression |
| Typical sample size to detect a moderate shift | 25 to 50 | 300 to 1,000+ |
That last row is the reason continuous data is preferred wherever a choice exists. Detecting a change in a proportion around 5% with reasonable power requires several hundred observations per group; detecting an equivalent shift in a continuous mean typically requires a few dozen. Attribute data records only whether the unit crossed a line; continuous data records how far it was from the line, and that distance is information you have already paid to collect.
Converting between types
Continuous to attribute is a one-way loss. Recording "shaft diameter = 20.08 mm" and then storing only "in specification: yes" discards the information that the process is running near the upper limit. Wherever the underlying characteristic is measurable, capture the measurement and apply the specification afterwards during analysis.
Attribute to continuous is generally impossible, but sometimes a genuinely continuous surrogate exists behind an apparent attribute:
| Apparent attribute | Continuous surrogate |
|---|---|
| Leak: pass/fail | Leak rate in cc per minute |
| Late: yes/no | Hours early or late relative to promise |
| Colour match: acceptable/not | Delta-E colour difference |
| Weld: good/bad | Peel strength in newtons |
| Form complete: yes/no | Number of fields incorrect, or time to correct |
Finding the surrogate is one of the highest-value moves available in the Measure phase, because it can cut the required sample size by an order of magnitude.
Classification worksheet
For any characteristic, answer in order:
- Is it a category or a quantity? Category means qualitative.
- If a quantity, can it take any value between two points, limited only by the gauge? Yes means continuous; no means discrete.
- Which scale? Ordered? Equal intervals? True zero?
- Given the answers, which chart, which index, which test, and what sample size?
A team records ambient temperature in degrees Celsius and reports that the process area is 'twice as warm' at 40 C as at 20 C. What is wrong with this statement?
A team currently records weld quality as pass or fail and must detect a small shift in performance. A destructive test can measure peel strength in newtons. What is the principal advantage of switching to the continuous measure?
Which combination correctly pairs the data type with an appropriate control chart?