BMS 1110 · Common Core (Tronc commun)Section 4.2 · Organising and describing data

Section 18 of 30

Levels of measurement

Classify a clinical variable as nominal, ordinal, interval or ratio, state what arithmetic its level allows, and pick a statistic and a graph that the level makes legal.

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Why this matters

The level of measurement is the first decision in every question about data, and it is invisible on the chart. It decides whether you may quote a mean or only a median, whether a bar chart or a histogram is honest, and whether a sentence such as “twice as hot” means anything at all. A nurse who reports a mean pain score, or a ratio of two temperatures, has made a mathematical error the reader cannot see; a nurse who names the level first cannot make it.

What you need to know first

You need only the idea, met in section 4.1, that a variable is something you record for each patient. No formula is required here: every rule in this section is a question you ask about the recorded values, and the four questions are written out in full in the first tutorial. The section does name statistics before they are computed elsewhere — the median and the percentiles (summaries that use position, computed in 4.5 and 5.1), the mean and the standard deviation (summaries that use the size of the gaps, computed in 4.5 and 5.3), and the coefficient of variation (the standard deviation divided by the mean, computed in 5.4). Here you only need to know which of them the level makes legal.

Words to know (6)

Level of measurement: The set of mathematical statements a recorded variable genuinely supports: naming only, ranking, adding differences, or multiplying ratios. Blood group supports naming only, while body weight supports multiplication as well.
Nominal scale: A scale whose values are categories with no natural order, so the only legal arithmetic is counting how many fall in each category. Blood group recorded as A, B, AB or O is nominal: A is not more than O.
Ordinal scale: A scale whose categories can be ranked, but where the gap between neighbouring categories is not known to be equal to the next gap. Pain recorded as mild, moderate or severe is ordinal: the order is real, the size of each step is not.
Interval scale: A numerical scale with equal gaps between values but no true zero, so differences are meaningful while ratios are not. Temperature in °C is interval: 40 − 30 = 10 °C has a meaning, but 40 ÷ 20 = 2 does not.
Ratio scale: A numerical scale with equal gaps and a true zero, so differences, means and ratios are all meaningful. Weight in kg is ratio, which is why 80 kg really is twice 40 kg.
True zero: The value of a scale that means none of the quantity is present; only then may one value be called a multiple of another. 0 kg is no mass, but 0 °C is not no heat; the first scale has a true zero and the second does not.

One idea at a timeIdea 1 / 6

In one sentence Nominal means categories; ordinal adds order; interval adds equal differences; ratio adds a zero that means none of the measured quantity.

Each level permits one more statement: name, order, add, multiply

A level of measurement is not a property of the patient; it is a property of the scale you record with. The same clinical fact can sit at different levels depending on how you write it: fever can be recorded as present or absent (nominal), as mild, moderate or severe (ordinal), or as 38.6 °C (interval). Each step up the ladder adds exactly one permitted statement. Nominal permits “this value is different from that one”. Ordinal adds “this value is greater than that one”. Interval adds “the difference here equals the difference there”, which is what gives addition and subtraction a meaning. Ratio adds “this value is a multiple of that one”, which is what gives multiplication and division a meaning. Because the permitted statements accumulate, the ladder is nested: any statement that is legal at a lower level stays legal at every higher one.

For example

A temperature of 38.6 °C and a chart marked “fever present” carry the same clinical news, but the first permits differences (38.6 − 37.0 = 1.6 °C) while the second permits only counting.

Pause and check yourself

A ward records each patient’s pain as none, mild, moderate or severe, then counts how many patients fall in each category. Which permitted statements is the ward using, and which is it not using?

Show the answer

It is naming the four categories and ordering them (none < mild < moderate < severe), and it is counting. It is not using differences or ratios, because the step from mild to moderate is not known to equal the step from moderate to severe.

Read detailed explanations

Why unequal gaps break the mean: Take five patients whose pain is recorded as mild, moderate, severe, severe and moderate on a three-point verbal scale. If you code mild = 1, moderate = 2 and severe = 3, the arithmetic mean of the codes is (1 + 2 + 3 + 3 + 2) ÷ 5 = 2.2. Now re-code the same three categories as mild = 1, moderate = 5 and severe = 9. The order is unchanged — mild is still below moderate, which is still below severe — and not one patient has moved. The mean becomes (1 + 5 + 9 + 9 + 5) ÷ 5 = 5.8. The mean changed because the coding changed, which means it was never a property of the patients. This is the coding artefact at the heart of the ordinal problem: any strictly increasing relabelling of the categories, called a monotone transformation, leaves both the median and the mode exactly where they were, while the mean and the standard deviation move freely. The median survives because it uses only position in the ordered list; the mean does not, because it uses the distance between the numbers you chose. The same argument applies to summed questionnaire scores, where several ordinal items are added to make a total. Summation silently assumes that each step of every item is worth the same amount, an assumption that is defensible only when the scale has been built and tested for it. In ward reporting, the safe habit is to give the median with the interquartile range, to state the wording of the scale and its direction, and to keep the counts per category. Rank-based, or nonparametric, tests were designed for exactly this situation: they compare the positions of the observations in the ordered list rather than the size of the gaps.
The true-zero test, and two temperatures that fail it: To separate an interval scale from a ratio scale, run one test: say out loud what zero means. If zero means “none of the quantity is present”, the scale is ratio. If zero is a reference point that someone chose, the scale is interval. Mass passes the test, because 0 kg is no mass and a negative mass is impossible for a body. Elapsed duration passes it too, because 0 hours is no time gone by. Celsius fails: 0 °C is the temperature at which water freezes under ordinary pressure, and heat is plainly still present, which is why 20 °C is not half of 40 °C in any physical sense. Convert both readings to kelvin — the thermodynamic scale whose zero is absolute, the point at which molecular motion reaches its minimum — and the numbers become 293.15 K and 313.15 K, so their quotient is about 1.07. The ratio of two pure numbers, by contrast, is built to survive conversion: 80 kg × 2.2046 ≈ 176 lb and 40 kg × 2.2046 ≈ 88 lb give 176 ÷ 88 = 2, exactly as 80 ÷ 40 = 2. Fahrenheit compounds the difficulty for beginners because its degree is smaller: a difference of 1 °C equals a difference of 1.8 °F, so even differences must be converted before they are compared. Two further examples of a conventional zero are worth keeping: a calendar year, where year zero is a convention rather than the beginning of time, and a pH value, where 0 does not mean no acidity at all. The pH scale is logarithmic, however: equal steps in pH are equal ratios of hydrogen-ion concentration rather than equal amounts of acidity, so it is cited here only for its conventional zero. Whenever a scale can take negative values in ordinary use, treat its zero as a reference point rather than an absence, and restrict your statements to differences.
From level to statistic: the decision rule a ward audit follows: An audit team decides four things in a fixed order, and the level of measurement settles the first three of them. First the summary: nominal data are summarised by frequencies and the mode; ordinal data by the median and the interquartile range; interval and ratio data by the mean, and both of those levels also permit the standard deviation. Second the display: nominal data use a bar chart or a pie chart with the categories in any order; ordinal data use a bar chart with the categories in their natural order; interval or ratio data use a histogram, a line graph or a scatter plot. Third the test: nominal counts are compared with a chi-square test of association; ordinal groups with rank-based methods such as the Mann–Whitney test for two independent groups; interval or ratio groups with t-tests or analysis of variance, which compare means. Choosing among these tests is the subject of section 6.4. Fourth, and this one no level can supply, the clinician asks whether the difference the statistic shows is large enough to change care. A single number on a ward dashboard is rarely of one pure level. Clinical early-warning scores add points awarded for several ordinal parameters — respiratory rate, oxygen saturation, systolic blood pressure, pulse, temperature and level of consciousness — and compare the total with an escalation threshold, commonly 5 or 7 points. Adding those points assumes equal gaps between them, a pragmatic decision defended by the fact that the total is used only to trigger a review, never to claim that one patient is twice as sick as another. Aggregated scales of this kind are called summated rating scales, and the rule for using them is to keep the claim as modest as the assumption: a threshold, a rank or a direction, rather than a ratio. Reporting the level beside the number, in a footnote if necessary, lets any reader re-check which statistics and which displays were licensed by the data.

Explore the diagram

Look at the whole drawing first. Then select one numbered part and connect it to the idea you just read.

The four levels of measurement

A ladder of four scales, each permitting one more statement than the one below. Nominal values may be named only, ordinal values may be ranked, interval values may be added as differences, and ratio values may be multiplied as ratios. Each band gives a clinical example, the legal statistics and the illegal move. The bottom band applies the true-zero test to the standard pair: 40 °C is not twice 20 °C, but 80 kg is twice 40 kg.

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01 / 5① Nominal: name only

If the drawing is wider than this panel, scroll it sideways. Labels stay readable; the numbered explanations below show the same ideas.

① Nominal: name only

Nominal values are labels. Blood group A, B, AB and O tells you whether two patients are the same or different, and nothing more, because there is no meaningful ranking of those four categories. The only legal arithmetic is counting, the only legal average is the mode, and the honest display is a bar chart or a pie chart. A clinic that stores A as 1, B as 2, AB as 3 and O as 4 has made digits, not measurements; a reported “mean blood group” is a statement about the coding, not about the patients.

Schematic ladder drawn to a fixed grid; the bands are equal in height for legibility, not because the four levels are equally spaced. The bottom band is a decision test, not a measurement scale.

Key points to remember

The four levels are nominal, ordinal, interval and ratio, and each adds one permitted statement: name, order, add a difference, multiply a ratio.
Nominal values are categories with no order; only counts, the mode, a bar chart or a pie chart are legal.
Ordinal values can be ranked but their gaps are not known to be equal; the median and the percentiles are legal, the mean is not.
An interval scale has equal gaps and no true zero, so differences and means are legal and ratios are not; temperature in °C is the standard example.
A ratio scale has equal gaps and a true zero, so every statistic is legal, including ratios; 0 kg is no mass, and 80 kg is twice 40 kg.

Go deeper

Worked example

Classifying three variables from one ward round. During a ward round a nurse records three things for every patient: blood group, a pain score from 0 to 10, and oral temperature in °C. Before the values go into the audit, decide the level of each variable and the one summary number that may be quoted for it.

Start with the naming question: can the values only be told apart? Blood group values are A, B, AB and O, and there is no meaningful order among them, so blood group is nominal.
Move to the ordering question: can the values be put in a meaningful order? A pain score from 0 to 10 can (0 < 2 < 6), so the variable is at least ordinal.
Now ask whether the gaps are equal. On a pain scale the step from 2 to 4 need not equal the step from 4 to 6, so pain acts only as ordinal and the mean of the category codes needs additional assumptions.
Temperature in °C has equal gaps on the Celsius scale: 30 − 20 = 10 °C and 40 − 30 = 10 °C. But 0 °C is the freezing point of water, not the absence of heat, so it is interval, not ratio.
Choose the legal summaries: counts and mode for blood group, median with interquartile range for pain, and mean with standard deviation for temperature.
Finish with the ratio check on temperature: 40 ÷ 20 = 2 looks like arithmetic but is not, because the Celsius zero is a convention. The same division on two weights, 80 kg ÷ 40 kg = 2, would be legal.

One ward round, three levels and three different summaries. The level is decided from the scale you recorded with, never from whether the value happens to be written as a number.

Tutorial 1 — assigning a level to five clinical variables

A student nurse must classify five variables recorded on the same day: blood group; pain severity recorded as mild, moderate or severe; oral temperature in °C; body weight in kg; and ward of admission. Each level must be justified in one line.

Rule used: Ask the four questions in order and stop at the first one whose answer is no. 1. Are the values only names? Then nominal. 2. Can they be ranked, but the gaps between neighbouring values are not equal? Then ordinal. 3. Are the gaps equal, but zero does not mean “none of the quantity”? Then interval. 4. Are the gaps equal and does zero mean “none”? Then ratio.

Write the four questions on your page before classifying anything. The order matters, because each level includes everything the level below it allows.
Reveal this step
Questions in order: names only gives nominal; ranked with unequal gaps gives ordinal; equal gaps but no true zero gives interval; equal gaps with a true zero gives ratio.
Classify blood group using the first question only.
Reveal this step
The values A, B, AB and O are labels with no order, so the answer to question 1 is yes: the variable is nominal. The only legal arithmetic is counting, and the mode is the only average available.
Classify pain severity recorded as mild, moderate or severe. Which question decides, and what is the level?
Reveal the answer
The values can be ranked, mild < moderate < severe, so question 1 fails. But the step from mild to moderate is not known to equal the step from moderate to severe, so the deciding question is 2 and the level is ordinal: the median is legal, the mean is not.
Classify oral temperature in °C and say which question fails.
Reveal this step
Differences are equal — 30 − 20 = 10 °C and 40 − 30 = 10 °C — so questions 2 and 3 pass. Question 4 fails, because 0 °C is the freezing point of water and not the absence of heat. Temperature in °C is interval.
Classify body weight in kg, then say what extra statement it allows that temperature does not.
Reveal the answer
Weight has equal gaps and a true zero, because 0 kg is no mass at all, so it is a ratio scale. It alone permits a ratio statement: 80 kg is twice 40 kg, because 80 ÷ 40 = 2, and that statement stays true in pounds.
Classify ward of admission, then finish with the legal average for each of the five variables.
Reveal this step
Ward is nominal, names with no order, so counts and mode. Blood group nominal: mode. Pain severity ordinal: median and interquartile range. Temperature interval: mean and standard deviation. Weight ratio: mean, standard deviation and ratios.

Answer: Blood group and ward of admission are nominal, pain severity is ordinal, temperature in °C is interval, and weight in kg is ratio. Only weight permits a “twice as” statement.

Keep this habit: Always run the four questions in order: names, order, equal gaps, true zero. The first question that fails gives the level. Naming the level before you calculate is what prevents an illegal average from reaching the report.

Tutorial 2 — “twice as hot” and “twice as heavy”

A colleague writes two lines in a ward report. Line one: “yesterday’s 40 °C was twice the 20 °C recorded in the cool room.” Line two: “the 80 kg patient is twice the 40 kg patient.” Decide which line the mathematics supports, and reword the other one honestly.

Rule used: A ratio statement, “A is n times B”, is legal only on a scale with a true zero, where 0 means none of the quantity. Then n is found by dividing the larger measurement by the smaller, in the same unit: n = A ÷ B. On a scale without a true zero, only differences may be compared: A − B, in the same unit.

Carry out both divisions exactly as written, 40 ÷ 20 and 80 ÷ 40, and compare the two answers.
Reveal this step
40 ÷ 20 = 2 and 80 ÷ 40 = 2. The arithmetic is identical in both lines, so arithmetic alone cannot decide which line is honest; the scale has to decide.
What does zero mean on each scale? Say it in words for °C, for K and for kg.
Reveal the answer
0 °C is the freezing point of water, a chosen reference point with heat still present. 0 K is absolute zero, no thermal energy at all. 0 kg is no mass. Only kelvin and kilogram have a true zero.
Convert the two Celsius temperatures to kelvin and divide again. What happens to the ratio of 2?
Reveal the answer
20 °C = 20 + 273.15 = 293.15 K and 40 °C = 40 + 273.15 = 313.15 K. Then 313.15 ÷ 293.15 ≈ 1.07, not 2. The value 2 was an accident of where the Celsius zero was placed.
Re-express the two weights in pounds and divide, to show the contrast with temperature.
Reveal this step
80 kg × 2.2046 ≈ 176 lb and 40 kg × 2.2046 ≈ 88 lb. Then 176 ÷ 88 = 2, unchanged. A ratio survives a change of unit exactly when the zero means none of the quantity.
Write the honest version of the temperature line, keeping the same two readings.
Reveal this step
“Yesterday’s 40 °C was 20 °C warmer than the 20 °C recorded in the cool room.” The difference 40 − 20 = 20 °C is legal on an interval scale; the ratio is not, because the zero is a convention.
The weight line stays as written. Which property of the kilogram scale makes it legal, and which statistic is legal on weight but not on temperature?
Reveal the answer
The kilogram has a true zero, so 80 ÷ 40 = 2 is a real statement about mass. A mean is legal on both scales because it uses differences, but the coefficient of variation, the standard deviation divided by the mean, is legal only where the mean is measured from a true zero, as with weight.

Answer: Line two is legal: 80 kg ÷ 40 kg = 2, and the ratio is unchanged when the weights are expressed in pounds. Line one must be reworded as a difference: 40 °C − 20 °C = 20 °C.

Keep this habit: Before writing “twice as”, ask what zero means. If zero is a chosen reference point, the most you may state is a difference. If zero means none of the quantity, division gives a ratio that survives every change of unit.

Now you try

Attempt each one on paper first. Write the formula, substitute the numbers, then check the unit of your answer.

Classify each of these four variables and name one statistic that is legal for it: blood group; pain recorded as mild, moderate or severe; weight in kg; temperature in °C.
Hint
Run the four questions in order for each variable, and stop at the first answer that is no. A number written on the page is not evidence of a level.
Show the worked answer
Blood group is nominal: the values are labels with no order, so counts and the mode are legal. Pain is ordinal: it can be ranked but the gaps are unequal, so the median and interquartile range are legal. Weight in kg is ratio: equal gaps and a true zero, so the mean, standard deviation and ratios are legal. Temperature in °C is interval: equal gaps but a conventional zero, so the mean and standard deviation are legal and ratios are not.
An audit reports that the mean pain score on a 0–10 numerical scale was 4.6 on ward A and 2.3 on ward B, and concludes that pain on ward A is twice that on ward B. Identify both errors and state what should be reported instead.
Hint
There are two separate mistakes: one about the mean, one about the ratio. The order of the scores is the only safe information.
Show the worked answer
First, the mean of an ordinal score needs additional assumptions, because the step from 2 to 4 need not equal the step from 4 to 6; renumbering the scale can move the mean without moving a single patient. Second, even an interval scale would not license a ratio without a true zero, and 0 on a pain scale is a point on the scale, not the absence of all pain sensation. Report the median with the interquartile range on each ward, for example median 5 with interquartile range 3 to 7 against median 2 with interquartile range 1 to 4, and state which scale was used.
A researcher records tumour stage as I, II, III or IV, codes the stages 1, 2, 3 and 4, and calculates a mean stage of 2.4. Explain the problem and give a legal summary of the same data.
Hint
Stage is a rank, and the codes 1 to 4 were chosen by the researcher. Position, not gap size, is the reliable information.
Show the worked answer
Stage is ordinal: the order is real, but the gap between stage I and stage II is not known to equal the gap between stage III and stage IV, and the codes 1 to 4 are a choice. Any increasing renumbering, for example 1, 3, 6 and 10, leaves the order intact but moves the mean, so 2.4 is an artefact of the coding rather than a property of the patients. Summarise with the median stage and its interquartile range, or with the number and percentage of patients in each stage, and always give the total with the counts.

Common misconception

Beginners believe that a value written as a number can always be averaged, because arithmetic works on numbers. The scale, not the notation, decides. Blood groups coded 1 to 4 are numbers on the screen, but a mean of 2.4 is a label that merely looks like arithmetic. Ask what the number means before asking what may be done with it.

Check your reasoning

Explain why changing the zero of a temperature scale changes the ratio 40 °C ÷ 20 °C, while changing the unit of weight from kilograms to pounds leaves 80 kg ÷ 40 kg unchanged.

Reveal an answer

A ratio compares each value with zero. In Celsius the zero is a chosen reference point, the freezing point of water, so moving to kelvin shifts both readings by 273.15 and changes their quotient: 313.15 ÷ 293.15 ≈ 1.07. In kilograms zero means no mass at all, and a change of unit multiplies both values by the same factor: 80 kg becomes 176 lb and 40 kg becomes 88 lb, so 176 ÷ 88 is still 2. Comparing with zero survives a change of unit only when zero is the true absence of the quantity.

Where this comes from

BMS 1110 Applied Mathematics and Medical Biostatistics lecture notes (supplied) · Section “1.5 Levels of Measurement”, numbered lines 591–629: nominal scale with blood group (593–603), ordinal scale with pain and disease severity (604–613), interval scale with Celsius (614–619), ratio scale with weight and the statement that 80 kg is twice 40 kg (620–629).
Note on the course code · The supplied cover page reads “Course Number: BMS 110”. The curriculum catalogue in this project records “BMS 1110”; the catalogue is treated as authoritative and the notes’ cover value is retained here for provenance.
Beyond the supplied notes: the true-zero test, monotone re-labelling and the statistic-selection rule · The notes define the four scales but give no rule for choosing a statistic or a graph, no account of how renumbering changes the mean of ordinal data, and no mention of the kelvin scale, rank-based tests, summated rating scales or early-warning score aggregation.

Lesson progress is saved on this device for the current learner. Practice answers stay separate and are not stored.

About this course and its sources

Bilingual lessons organised from the supplied BMS 1110 Applied Mathematics and Medical Biostatistics lecture notes (Mr. Ebangan Prosper Achuanoh). Diagrams are original educational schematics; sections that go beyond the supplied notes say so in their sources. Mathematics, clinical and French-language review are pending.