Types of Data and Levels of Measurement

In this lesson you’ll learn to classify data as categorical or quantitative, tell discrete from continuous, and identify the four levels of measurement. This matters because the data type decides which summaries and charts are even legal.

Before you compute anything, you have to know what kind of data you have. Averaging zip codes produces a number, and that number is nonsense. The classification below is what stops you from doing that.

Categorical (or qualitative) data records which group something belongs to. Eye color, country, brand preference, yes/no answers.

You can count how many fall in each category and find the most common one, but you cannot average them.

Watch out for categories coded as numbers. If a survey stores “Male = 1, Female = 2, Nonbinary = 3,” the data is still categorical. The mean of 1.7 means nothing at all. This is one of the most common real-world data mistakes.

Quantitative data records an amount, and arithmetic on it is meaningful. Height, income, test scores, number of children.

Quantitative data splits again:

• Discrete - countable, with gaps. Number of siblings, cars in a lot, defects in a batch. You can have 3 siblings, never 3.4.
• Continuous - measurable on an unbroken scale. Height, weight, time, temperature. Between 70 and 71 kg there are infinitely many possible values.

The test is the same one from discrete mathematics: between any two values, is there always another?

A practical note: continuous data always arrives rounded, because instruments have limits. Height recorded as 175 cm is really “somewhere near 175.” It’s still continuous data, just recorded coarsely.

This is a finer classification, and it tells you exactly which operations make sense.

1. Nominal - names only, no order. Blood type, city, favorite color.

Allowed: counts, mode, “same or different.”

2. Ordinal - ordered, but the gaps between values aren’t equal or aren’t known. Survey scales (poor / fair / good / excellent), letter grades, race finishing position.

Allowed: everything nominal allows, plus median and percentiles. The catch: you can say “good” beats “fair,” but not that the gap from fair to good equals the gap from good to excellent. Averaging ordinal data is common in practice and technically shaky.

3. Interval - ordered with equal gaps, but no true zero. Temperature in Celsius or Fahrenheit, calendar years, IQ scores.

Allowed: addition and subtraction. Differences are meaningful: 30°C is 10 degrees warmer than 20°C.

Not allowed: ratios. 30°C is not “twice as hot” as 15°C, because 0°C doesn’t mean “no heat,” it’s just where water freezes.

4. Ratio - ordered, equal gaps, and a true zero meaning “none of it.” Height, weight, income, time elapsed, count of anything.

Allowed: everything, including ratios. 60 kg really is twice 30 kg, because 0 kg means no mass.

The escalating summary:

Level	Ordered?	Equal gaps?	True zero?	Center you can use
Nominal	no	no	no	mode
Ordinal	yes	no	no	mode, median
Interval	yes	yes	no	mode, median, mean
Ratio	yes	yes	yes	all, plus ratios

Each level allows everything the level above it allows, plus one more thing.

The data type determines which display is honest:

• Categorical - bar chart, pie chart. Bars have gaps because the categories aren’t a continuum.
• Quantitative - histogram, boxplot, dotplot. Histogram bars touch because the values are a continuum.

Using a histogram for categories or a bar chart with touching bars for measurements both imply a continuity that isn’t there.

Example 1: Classify each variable.

1. Jersey number on a football shirt
2. Number of goals scored in a match
3. Time to run 5 km
4. Satisfaction rating from 1 to 5
5. Temperature in Fahrenheit
6. Annual salary

Solution.

1. Categorical, nominal. It’s a label. The average jersey number is meaningless.
2. Quantitative, discrete, ratio. Countable, and 0 goals genuinely means none.
3. Quantitative, continuous, ratio. 0 seconds means no elapsed time, and 20 minutes is twice 10.
4. Categorical-ish, ordinal. Ordered, but the gap from 1 to 2 need not equal 4 to 5.
5. Quantitative, continuous, interval. No true zero, so ratios fail.
6. Quantitative, continuous, ratio. 0 income means none, and 80,000 is twice 40,000.

Example 2: The zip code trap.

A dataset stores zip codes as integers. A colleague computes the mean zip code as 48,213. What went wrong?

Solution. Zip codes are nominal data that happen to look numeric. The digits encode geography, not quantity. There’s no sense in which one zip code is “more” than another, and the mean lands on an unrelated arbitrary location.

The general rule: just because a value is stored as a number doesn’t make it quantitative. Ask whether arithmetic on it means anything.

Example 3: Interval versus ratio.

Yesterday was 10°C, today is 20°C. Is today twice as warm?

Solution. No. Celsius is an interval scale, so only differences are meaningful: today is 10 degrees warmer.

To see the problem, convert to Fahrenheit: 50°F and 68°F. If “twice as warm” were real, the ratio would survive the conversion, but $68/50 = 1.36$, not 2. The ratio changes with the scale, which proves it wasn’t measuring anything.

The Kelvin scale is ratio, because 0 K is true absence of thermal energy. 300 K really is twice 150 K.

Example 4: Pick the right summary.

A survey records: favorite streaming service, hours watched per week, and a 1-to-5 recommendation score. What’s the best measure of center for each?

Solution.

• Favorite service - nominal, so mode only. The most-chosen service.
• Hours watched - ratio, so mean works, though the median may be better if a few heavy viewers skew it.
• Recommendation score - ordinal, so median is the defensible choice. Reporting a mean of 3.8 is extremely common and rests on assuming the gaps are equal.

Example 5: Discrete or continuous?

1. Number of emails received today
2. Weight of a package
3. Shoe size
4. Number of people in a household

Solution.

1. Discrete. Whole emails only.
2. Continuous. Any value in a range, limited only by the scale’s precision.
3. Discrete, and arguably ordinal. Sizes come in fixed steps like 8, 8.5, 9, and the steps aren’t a true continuum.
4. Discrete. Whole people.

Choosing the wrong data type is one of the most common ways a dashboard lies. Averaging a satisfaction scale, averaging categories coded as numbers, or computing a mean of ordinal survey responses all produce plausible-looking figures that don’t support the conclusions drawn from them.

Database design depends on it. Storing a category as an integer for efficiency is fine, but the schema should make clear it’s a label, or someone downstream will average it. Statistical software has explicit types (R calls categorical variables “factors”) precisely to prevent this.

Machine learning requires the distinction constantly. Categorical features need encoding (one-hot, for instance) before a model can use them, because feeding raw category codes tells the model that category 3 is somehow between 2 and 4.

Medical scales are a live controversy. Pain rated 1 to 10 is ordinal, yet clinical studies routinely average it. It’s usually treated as acceptable, but it’s a known compromise rather than a solved problem.

Survey design lives here too. Choosing between a yes/no question, a five-point scale, and a numeric entry determines what analysis becomes possible later, so the decision has to be made before any data exists.