Before you can summarise or display a set of data, you need to know what kind of data it actually is — the classifications here decide which charts, averages and tests are even valid to use later in the course.
8 min read4 worked examples
Quantitative and qualitative data
Quantitative data takes numerical values — anything that can genuinely be counted or measured, such as height, shoe size, or number of siblings. Qualitative data (sometimes called categorical data) describes a non-numerical characteristic, such as eye colour, favourite subject, or type of vehicle. A useful test: if the "average" of the data wouldn't mean anything (there's no sensible average hair colour), the data is qualitative.
Discrete and continuous data
Quantitative data splits further into two types. Discrete data can only take specific, separate values within a range — usually whole numbers, like the number of goals scored in a match or the number of pets someone owns. Continuous data can take any value within a range, limited only by the precision of the instrument used to measure it — height, time, and mass are all continuous, since a stopwatch or ruler could always in principle measure more precisely.
The quickest test
Ask: "could this variable take a value like 2.5, 2.05, or 2.0001?" If a fractional value between two possible readings genuinely makes sense, the data is continuous. If the variable jumps straight from one whole value to the next with nothing sensible in between (you can't score 2.5 goals), it's discrete.
Grouped frequency tables
When a continuous variable has been measured for a large sample, individual values are usually grouped into classes (intervals) rather than listed one by one. Each class is described by three related quantities:
Class boundaries: the true minimum and maximum values that belong in that class, once you account for the rounding used when the data was recorded.
Midpoint: the average of the two class boundaries — used as a representative value for every item in that class when estimating statistics like the mean.
Class width: the difference between the upper and lower class boundaries.
Watch out — boundaries are not the same as the labels
A class labelled "40–49" for data rounded to the nearest whole number doesn't have boundaries $40$ and $49$ — a value of $49.6$ would round to $50$ and so belongs in the next class, while a value of $39.5$ rounds to $40$ and belongs in this class. The true boundaries here are $39.5$ and $49.5$. Always work out how the data was rounded before stating class boundaries.
Worked example 1
Classifying variables
For each variable, state whether it is (i) quantitative or qualitative, and (ii) if quantitative, discrete or continuous: (a) the number of cars in a household; (b) the make of car a household owns; (c) the time taken for a runner to finish a race.
MethodApply the two tests in turn to each variable
1
(a) Quantitative and discrete — the number of cars is a count, and can only take whole-number values (0, 1, 2, ...).
A count of separate items.
2
(b) Qualitative — a car's make (Ford, Toyota, ...) is a category, not a number.
No numerical value or meaningful average.
3
(c) Quantitative and continuous — a finishing time could, in principle, be measured to any precision (minutes, seconds, hundredths of a second).
Any value within a range is possible.
▪ classified
Examiner's eye
A number attached to something isn't automatically quantitative — a football shirt's squad number (7, 10, 23, ...) is qualitative, since it's just a label with no meaningful average or ordering by size.
Worked example 2
Class boundaries, midpoint and width
The masses, $m$ grams, of 60 apples from an orchard are recorded to the nearest gram and grouped as shown.
Mass (g)
Frequency
80–99
9
100–119
22
120–139
19
140–159
10
Find the class boundaries, midpoint and class width for the class "100–119".
MethodAccount for rounding to the nearest gram, then apply the definitions
1
Since masses are rounded to the nearest gram, a mass of $99.5$g rounds to $100$g and a mass of $119.5$g rounds to $119$ or $120$g — so the true class boundaries are $99.5$g and $119.5$g.
Halfway between consecutive whole-number labels.
2
Midpoint $= \dfrac{1}{2}(99.5+119.5) = 109.5$g.
Average of the two boundaries.
3
Class width $= 119.5-99.5 = 20$g.
Upper boundary minus lower boundary.
▪ solved
Worked example 3
A rounding trap
The lengths, $l$ cm, of 40 fish caught in a survey are measured to the nearest 0.5 cm and grouped, with one class labelled "15.5–16.5". Explain why the boundaries of this class are not simply 15.5 and 16.5, and state the correct boundaries.
MethodWork out what values round to 15.5 and to 16.5
1
Data is rounded to the nearest $0.5$cm, so the rounding "gap" either side of a recorded value is $0.25$cm, not $0.5$cm.
Half of the rounding interval (0.5 cm).
2
A true length of $15.25$cm rounds to $15.5$cm, so $15.25$cm is the lower boundary. A true length of $16.75$cm rounds to $16.5$cm (just), while $16.76$cm would round to $17$cm — so $16.75$cm is the upper boundary.
Boundaries sit halfway to the next recorded value in either direction.
3
The correct class boundaries are $15.25$cm and $16.75$cm, giving a class width of $1.5$cm, not $1$cm.
The labels 15.5 and 16.5 are recorded values, not boundaries.
▪ explained
The key insight
The rule "add or subtract half the rounding unit" always applies, whatever that unit is — half of $1$ when rounding to the nearest whole number, half of $0.5$ when rounding to the nearest $0.5$, and so on. Reading the rounding precision from the data is the first step, every time.
Worked example 4
Working backwards from a midpoint · Challenge
A class of continuous data, rounded to the nearest whole number, has midpoint $54.5$ and class width $10$. Find the class boundaries and the range of whole-number values that would be recorded in this class.
MethodUse the midpoint and width to reconstruct both boundaries
1
Midpoint is the average of the boundaries, and width is their difference, so: lower boundary $= 54.5 - \dfrac{10}{2} = 49.5$; upper boundary $= 54.5+\dfrac{10}{2}=59.5$.
Midpoint ± half the width.
2
Check: $\dfrac{1}{2}(49.5+59.5)=54.5$ ✓ and $59.5-49.5=10$ ✓.
Both original values recovered.
3
Since data is rounded to the nearest whole number, the recorded values in this class run from $50$ to $59$ inclusive.
Whole numbers strictly between the boundaries.
▪ solved
Look deeper
Class boundaries, midpoint and width are three views of the same two numbers — given any two of the three pieces of information (boundaries, midpoint, width), the third can always be recovered algebraically, without needing to see the original recorded labels at all.
Common mistake
Don't assume class width is always a "nice" number just because the class labels look tidy — always calculate it from the true boundaries, after accounting for rounding, rather than reading it directly off the stated class labels.
Check yourself
A class of data, rounded to the nearest whole number, is labelled "60–74". Find the class boundaries, midpoint and class width. (Hint: rounding to the nearest whole number means the boundary sits 0.5 either side of the recorded value.)