Mathematics
Discrete and Continuous Data
Eight everyday measurements sorted into two columns, the shoe-size case that catches people out, and the test that works better than 'is it a whole number'.
Discrete and continuous data
Discrete data can only take separate fixed values that you count, while continuous data comes from measuring and can take any value within a range.
- Also called
- Countable and measurable data
- Where students meet it
- Grade 6 statistics, the first time a class has to decide between a bar chart and a histogram, and again whenever grouped data appears.
The short answer
Discrete data is counted and can only take separate fixed values, such as the number of goals in a match. Continuous data is measured and can take any value in a range, such as the time of a race. Shoe size is discrete even with half sizes.
An example
Goals: 0, 1, 2, 3 | Race time: 12.4 s, 12.41 s, 12.407 s
A match can end 2–1 but never 2.4–1. Between 2 goals and 3 goals there is nothing at all, so goals are discrete. A race time of 12.4 seconds is only a rounded version of something more exact; between 12.4 and 12.41 there are 12.403 and 12.4072 and everything else. The values do not run out, so time is continuous.
Eight measurements, sorted
Working through real cases is faster than memorising a rule. Four of these are discrete, four are continuous, and one of them argues with itself.
Shoe size is the one that catches people. Half sizes make it look continuous, but sizes 7 and 7.5 exist and nothing between them does — you cannot buy a size 7.2 shoe. Discrete does not mean whole number; it means the possible values are separate.
- Number of siblings — discrete. You cannot have 2.4 brothers, though the mean can be 2.4.
- Height — continuous. Recorded as 1.62 m, but 1.6247 m is a real height.
- Shoe size — discrete, half sizes included.
- Time to run 100 m — continuous, whatever the stopwatch shows.
- Cars passing a gate in an hour — discrete.
- Mass of a mango — continuous.
- Marks out of 20 on a spelling test — discrete.
- Temperature at noon — continuous.
The test that actually works
Do not ask "is it a whole number?". Ask: take two neighbouring values that really occur, and is there a value between them that could also occur? For shoe sizes 7 and 7.5 the answer is no. For heights 1.62 m and 1.63 m the answer is yes, and it stays yes however finely you slice.
The other reliable clue is how the number arrived. Counting gives discrete data, measuring gives continuous. This is also why continuous data looks discrete once it is written down: every measurement is rounded, so a list of times to one decimal place contains repeats. The rounding is a limit of the instrument, not a property of time.
Money is the honest exception. Prices in rupees or pounds stop at the smallest coin, so strictly they are discrete, but with so many possible values so close together they are treated as continuous in grouped tables and on histograms. Both answers are defensible; say which you are using.
Why the label changes the diagram
The classification is not a vocabulary exercise. It decides how the data may be drawn and how a class interval may be written. Discrete data takes a bar chart or a vertical line chart, with gaps between the bars, because there is genuinely nothing in between them. Continuous data takes a histogram, with the bars touching, because the classes join edge to edge.
It also decides the notation. Discrete classes can be written 0–4, 5–9, 10–14 with no ambiguity. Continuous classes cannot leave a hole, so they are written 0 ≤ t < 5, 5 ≤ t < 10, and a value of exactly 5 has one home rather than two.
Common questions
Is shoe size discrete or continuous?
Discrete. Half sizes make it look otherwise, but the sizes form a separate list — 7, 7.5, 8 — with nothing available between them. Foot length is continuous; shoe size is a set of labels attached to ranges of foot length, and only those labels exist.
Is age discrete or continuous?
Continuous. Age increases smoothly and you are 15 years, 4 months and 11 days old at this moment. It is almost always recorded as a whole number of years, which is why a class labelled "15" covers everyone from their fifteenth birthday until the day before their sixteenth.
Is money discrete or continuous?
Technically discrete, since amounts step in units of the smallest coin, but treated as continuous in practice because the steps are tiny compared with the range. Grouped tables of prices are written with inequalities, like 0 ≤ p < 500, exactly as continuous data would be.
Which chart should I use for each?
Discrete data goes on a bar chart or vertical line chart with gaps between the bars. Grouped continuous data goes on a histogram with the bars touching, and if the classes have different widths the vertical axis must be frequency density rather than frequency.
Can discrete data have a decimal average?
Yes, and this trips people up. The mean number of children per family can be 2.4 even though no family has 2.4 children. The mean is a calculated summary of the data, not one of the values in it, so it is not restricted to the values the data can take.
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