Track 1 · Statistical Thinking · Lesson 4 of 10
Measurement scales
Classify variables as nominal, ordinal, interval or ratio, and choose only the arithmetic and summaries each scale supports.
An HR dashboard proudly reports that average agreement with “My workload is manageable” rose from 2.8 to 3.0 on a 1–5 scale. Is that a meaningful improvement, or arithmetic on labels? The answer depends on the variable's measurement scale.
The measurement scale of a variable tells you which comparisons and calculations mean something. There are four classic levels, each adding a property to the one before.
| Scale | What the values allow | Examples | Meaningful summaries |
|---|---|---|---|
| Nominal | Equal / different only | Department, blood type, payment method | Counts, percentages, mode |
| Ordinal | Plus order (more / less) | Job level, 1–5 agreement rating, race finishing position | Plus median and percentiles |
| Interval | Plus equal differences; zero is arbitrary | Temperature in °C, calendar year | Plus differences, mean, standard deviation |
| Ratio | Plus a true zero, so ratios make sense | Salary, weight, waiting time, distance, number of defects | Plus ratios (“twice as much”), percentage change |
Interval vs ratio is the subtle pair. On an interval scale, differences are meaningful but zero is a convention. 0 °C is the freezing point of water, not “no temperature”, so 30 °C is not twice as hot as 15 °C. On a ratio scale zero means none of the quantity: 0 SAR is no money, so 20,000 SAR really is twice 10,000 SAR.
To compare temperatures as ratios, convert to kelvin, which has a true zero: .
Worked example
Is it twice as hot?
A warehouse was at 15 °C at dawn and 30 °C in the afternoon. A report says “the temperature doubled”. By what factor did the absolute temperature actually increase?
Celsius is an interval scale, so the ratio 30/15 = 2 has no physical meaning.
Convert to kelvin: 15 °C = 288.15 K and 30 °C = 303.15 K.
Ratio: . The absolute temperature rose by about 5%, not 100%. What is meaningful on the Celsius scale is the difference: it became 15 degrees warmer.
About 1.05 times — a 15-degree rise, not a doubling.
Likert items — “strongly disagree (1) … strongly agree (5)” — are ordinal. We know 4 means more agreement than 3, but not that the step from 3 to 4 equals the step from 1 to 2. That is why the safest summaries are the distribution of answers, the median, and the percentage who agree (4 or 5). Means of Likert items are common in practice and can be useful for tracking, but they assume equal spacing and should be reported alongside the distribution.
Show the data as a table
| Q1 | Count |
|---|---|
| 1 | 47 |
| 2 | 150 |
| 3 | 181 |
| 4 | 223 |
| 5 | 163 |
Of the 764 employees who answered this item, 386 chose 4 or 5, so 50.5% agree; the median answer is 4. The mean code is 3.40, a number that corresponds to no answer anyone gave. Note the denominator: 778 people responded to the survey, but 14 skipped this item, and percentages should be based on those who answered it (and the skips reported).
Dataset
Employee engagement survey
778 rows · one row = one survey response (one employee) · synthetic data
| Department | Tenure | Work arrangement | Q1 | Q5 | Q6 |
|---|---|---|---|---|---|
| Customer Service | 1-3 years | Onsite | 3 | 1 | 2 |
| Customer Service | 1-3 years | Onsite | 3 | 3 | 2 |
| Technology | 3-5 years | Remote | 5 | 3 | 4 |
| Sales | 1-3 years | Onsite | 4 | 3 | 2 |
| Customer Service | 1-3 years | Onsite | 3 | 1 | 3 |
| Sales | 5-10 years | Hybrid | 4 | 4 | 4 |
Quick check
Runners' finishing positions in a race (1st, 2nd, 3rd, …) are on which scale?
Exercises
The lesson is complete when every exercise is answered correctly. Hints are free; viewing a worked solution before solving means the exercise won't count towards mastery.
0 of 7 exercises correct (not saved — sign in to keep your progress)
Exercise 1 · Practice, difficulty 1 of 3
Name the scale
Classify each variable by its measurement scale.
Exercise 2 · Practice, difficulty 2 of 3
Which statements make sense?
Select all the statements that are meaningful given the scale of the variable.
Exercise 3 · Practice, difficulty 1 of 3
Levels of measurement
Order the scales from the least information (fewest meaningful operations) to the most.
Exercise 4 · Guided, difficulty 2 of 3
The real temperature ratio
A logistics manager reports that a loading bay went from 15 °C in the morning to 45 °C at noon and writes that it became “3.0 times hotter”. Convert to kelvin () and compute the true ratio of absolute temperatures (noon ÷ morning), to 3 decimal places.
Exercise 5 · Application, difficulty 2 of 3
Summarising a Likert item
In the engagement_survey data, summarise q5_workload (“My workload is manageable”, 1 = strongly disagree … 5 = strongly agree) the ordinal way. Skipped answers are blank (missing) — leave them out.
(a) What is the median answer? (b) What percentage of those who answered chose 1 or 2 (disagree)? Give 1 decimal place.
Data: engagement_survey · Open in the lab to explore and calculate.
Exercise 6 · Practice, difficulty 2 of 3
Pick a summary
Match each variable to the most informative summary that its scale allows.
Exercise 7 · Interpretation, difficulty 3 of 3
A rise in mean satisfaction
A bank's branch network reports that mean customer satisfaction on a 1–5 scale (1 = very dissatisfied, 5 = very satisfied) rose from 3.4 to 3.6 after a service programme, and calls this “a 6% improvement in satisfaction”. In two or three sentences, explain what is questionable about this report and what you would show instead.
What you can now do
- Classify a variable as nominal, ordinal, interval or ratio
- Say which comparisons — order, differences, ratios — a scale supports
- Summarise Likert items with distributions, percentages agreeing and medians
- Spot invalid claims such as “twice as hot” in °C or percentage changes in rating codes