Types of data

Three states a data set passes through. Raw data is the list exactly as recorded — no order imposed, for example the marks of 20 students in the order their scripts were checked. It is complete but unreadable at a glance. Arrayed data is the same values rearranged in ascending or descending order of magnitude; sorting is almost always the first physical step towards finding a range or a median. Grouped data goes one step further, bucketing values into class intervals such as $10$–$20$ or $20$–$30$ and recording how many observations fall in each — used once there are too many distinct raw values to list one at a time.

The variable decides everything downstream. A variable is the quantity being recorded, and every variable is one of exactly two kinds. A discrete variable takes only specific, separate values — almost always whole numbers — obtained by counting: number of siblings, goals scored, books on a shelf. It can never sit between two consecutive counted values (there is no such thing as $2.5$ siblings). A continuous variable can take any value in a range, obtained by measuring: height, mass, time, temperature. Between any two continuous readings there is always, in principle, another possible reading, limited only by the precision of the instrument used.

Why the label matters in practice. The discrete/continuous label controls how a frequency table is built later in the chapter. Discrete data is tabulated value by value, because each value can genuinely repeat (three students can each have exactly $1$ sibling). Continuous data is tabulated by class interval, because two measured heights are almost never identical to several decimal places — grouping is not optional convenience here, it is the only way the data becomes readable at all. Getting the label wrong at this stage produces a frequency table later that is either uselessly long (grouping should have been used) or artificially coarse (individual values should have been kept).

Tallying converts a list into a table. Tabulation using tally marks is the standard hand method: each observation adds one stroke ($\vert$) against its value or class, and the fifth stroke is drawn diagonally across the previous four ($\cancel{\vert\vert\vert\vert}$) so that counting proceeds in fast, visually checkable groups of five. Once every observation has been tallied, the total tally count against a value (or class) is its frequency, usually written $f$. A completed frequency table should always be checked against the count of the original raw list:

where: $N$ is the total number of observations in the raw data set, and $\sum_i f_i$ is the sum of the frequency column across every row of the finished table. If the two numbers disagree, a tally stroke has been mis-drawn or mis-counted somewhere, and the table must be re-checked before it is used for anything further — this single running check is the cheapest error-catcher in the whole chapter.

This topic has no formula to derive because it is a classification skill, not a calculation — but the classification is not arbitrary, and the reasoning behind it is worth making explicit. The discrete/continuous split traces back to how a number was produced. Counting is a discrete operation: it moves in steps of one whole unit and can never land between two consecutive whole numbers, so anything obtained by counting inherits that property and is discrete. Measuring, by contrast, reads a continuous scale (a ruler, a thermometer, a stopwatch), and any two readings on a continuous scale can, in principle, be refined further — a height of $41.5$ cm could always be re-measured to $41.53$ cm with a finer instrument — so anything obtained by measuring is continuous. This is exactly why the size of a data set, or the number of distinct values it contains, is irrelevant to the classification. A variable with only four possible values ($0, 1, 2, 3$ siblings) is discrete because it was counted, not because it has few values; a variable with a thousand distinct measured heights is still continuous, because it was measured, not because it has many values. The test is always the origin of the number — counting or measuring — never how many different numbers show up. Once that origin is identified correctly, the rest of the chapter (which tabulation method to use, whether class intervals are needed) follows automatically.

The number of siblings of 15 students in a class, recorded in the order the teacher asked each student, is $2, 0, 1, 3, 2, 1, 0, 2, 1, 1, 2, 3, 0, 1, 2$.

Step 1. Identify the data type and the variable. The numbers are listed exactly as collected, with no ordering imposed, so this is raw data. The variable "number of siblings" can only take whole-number values obtained by counting, so it is a discrete variable — it can never be $2.5$.

Step 2. Arrange the raw data as an array. Sorting the 15 values in ascending order gives $0, 0, 0, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 3, 3$. Sorting alone already makes the repeats visible, before any table is drawn.

Step 3. Tally into a frequency table. Each value from $0$ to $3$ gets one tally stroke per occurrence in the array, with every fifth stroke crossed through the previous four.

Step 4. Check the total against $N$. Using $N = \sum_i f_i$: $3 + 5 + 5 + 2 = 15$, which matches the original count of 15 students, confirming no tally stroke was lost or duplicated. The finished frequency table shows $1$ sibling and $2$ siblings are the most common responses (each with $f = 5$), information invisible in the raw list.

Competitive stretch

A gardener separately measures the heights (in cm) of 10 saplings: $42.3, 40.1, 45.6, 41.0, 43.8, 40.1, 44.2, 42.3, 41.5, 43.0$. Height is obtained by measuring on a scale, so it is a continuous variable, unlike siblings above. Arrayed, the values are $40.1, 40.1, 41.0, 41.5, 42.3, 42.3, 43.0, 43.8, 44.2, 45.6$ — almost every value is distinct, so a value-by-value tally table would give mostly $f = 1$ rows and reveal nothing. This is precisely the signal that the data should instead be grouped into class intervals such as $40$–$42$, $42$–$44$, $44$–$46$, the technique developed next in the chapter under grouped frequency distributions.

• Confusing discrete with "few distinct values" — the correct test is how the number was produced (counted versus measured), never how many different values appear in the set.
• Treating "arrayed" as a synonym for "grouped" — an array is raw data sorted in order; it has not yet been bucketed into class intervals.
• Forgetting the cross-out-the-fifth convention for tally marks, which slows counting and invites miscounts once a frequency exceeds about ten.
• Skipping the $N = \sum_i f_i$ check — a table that "looks right" can still have a lost or doubled tally stroke that only the total catches.

Ask the tutor: unsure whether a quantity like "shoe size" or "number of runs scored" counts as discrete or continuous — ask the AI tutor to walk through the counted-versus-measured test with you.