Maths · 10 Statistics

10.3 Discrete and continuous data

Before you start

What a variable is: the thing being recorded, such as height or number of pets.
Rounding a value to the nearest whole number, to 1 decimal place, or to the nearest 10.
Reading the inequality signs $<$ and $\leq$ , and writing a range such as $5 \leq x < 10$ .

Syllabus objectives covered (1)

Distinguish between discrete and continuous data.

Part 1 of 5

Which data the question is about

Some data are words or categories: eye colour, favourite sport, type of car. These are categorical (or qualitative) data. You cannot do arithmetic with them.
Other data are numbers: number of brothers, height, time. These are numerical (or quantitative) data.
The question 'discrete or continuous?' is a question about numerical data. Before you answer it, check that the values really are numbers that measure or count something.
Numerical data come in two kinds, and the difference is about what values are possible, not about what values happen to be in the table. Hold on to that idea: it decides every awkward case later in this lesson.

Part 2 of 5

Discrete data: separate values, nothing in between

Discrete data can take only particular, separate values. Between two neighbouring values, no other value is possible.
Counting produces discrete data. The number of brothers and sisters someone has can be $0, 1, 2, 3, \dots$ but never $2.4$ . Between $2$ and $3$ there is nothing a count can land on.
Picture the possible values as dots on a number line, with empty space between the dots.
graph diagram — not drawn yet
Two horizontal number lines, one above the other, each marked 0 to 5. The top line, labelled 'discrete: number of siblings', has a solid dot at each of 0, 1, 2, 3, 4, 5 and nothing between them. The bottom line, labelled 'continuous: height in metres', has the whole stretch from 0 to 5 shaded as a thick solid bar.
Discrete values are separate points with gaps between them; continuous values fill the whole stretch, so every point on it is possible.
The trap: discrete does not mean 'whole numbers only'. Shoe sizes go $5, 5 \frac{1}{2}, 6, 6 \frac{1}{2}, \dots$ There is no shoe size $5.3$ on the scale, so shoe size is discrete even though some sizes are halves.
Test marks where half marks are allowed work the same way: $31, 31.5, 32, \dots$ The values go up in steps of $0.5$ with nothing in between, so the data are discrete.
An amount of money you pay is counted in the smallest unit of the currency, such as cents. After $3.47$ dollars the next possible amount is $3.48$ dollars, with no amount between them. So an amount of money paid is discrete, even though it has decimals.

Part 3 of 5

Continuous data: any value in a range

Continuous data can take any value within a range. Between any two possible values there is always another possible value.
Measuring produces continuous data: length, height, mass, time, temperature, volume, speed. None of these quantities comes in fixed steps.
Take two heights, $1.62$ m and $1.63$ m. A height of $1.625$ m lies between them. Between $1.62$ m and $1.625$ m lies $1.621$ m. You can keep going: the gap never closes, which is exactly what 'continuous' means.
On the number line, continuous values are not dots but a solid bar: every point along it is a possible value.
Every measurement you write down has been rounded, because no ruler or stopwatch is perfect. A height recorded as $162$ cm to the nearest cm stands for any height from $161.5$ cm up to $162.5$ cm.
$161.5 \leq h < 162.5$
So the record looks like a whole number, but the height it describes is still continuous. Rounding changes how the value is written, not what kind of quantity it is.

Part 4 of 5

The test to use, and the awkward cases

First test — counted or measured? If you get the value by counting, it is discrete. If you get it with a ruler, scale, stopwatch or thermometer, it is continuous.
Deciding test — is a value in between possible? Take two neighbouring possible values. If a value between them could really occur, the data are continuous. If it could not, they are discrete. Use this test whenever the first one feels unclear.
Shoe size against foot length. Both describe feet. Shoe size jumps from one size on the scale to the next, so it is discrete. Foot length is measured and can be any value, such as $24.37$ cm, so it is continuous. The question is about the variable, not the object.
Something that changes over time is not automatically continuous. The number of people in a shop changes all day, but at any moment it is a count — $14$ people, never $14.6$ — so it is discrete. 'Continuous' is about the possible values, not about change.
Recorded to the nearest second, all the times in a table may be whole numbers. They are still continuous, because time is measured and the true times lie anywhere in a range. Judge the variable, not the rounded numbers in the table.
When you have to give a reason, say why using the definition. For discrete: 'it is counted, so it can only take certain separate values'. For continuous: 'it is measured, so it can take any value in a range'. 'It is a number' is not a reason: both kinds are numbers.

Part 5 of 5

Why the type changes how you write groups

When you group data into a table (10.1) or estimate a mean from groups (10.4), the type of data decides how the classes are written.
Discrete data can use classes like $0$ – $4$ , $5$ – $9$ , $10$ – $14$ . There seems to be a gap between $4$ and $5$ , but no count can fall in it, so nothing is left out.
Continuous data cannot use those classes. A height of $154.6$ cm would fit neither $150$ – $154$ nor $155$ – $159$ : it falls in the gap, because continuous values fill the gap.
So write continuous classes with inequalities, each class starting exactly where the last one stops.
$150 \leq h < 155, 155 \leq h < 160, 160 \leq h < 165$
The $\leq$ on the left and $<$ on the right decide where a boundary value goes. A height of exactly $155$ cm satisfies $155 \leq h < 160$ but not $150 \leq h < 155$ , so it belongs to exactly one class. No gaps, no overlaps.

Worked examples

Example 1. For each variable, state whether the data are discrete or continuous, and give a reason. (a) The number of goals scored in a football match. (b) The time taken to run $400$ m. (c) The mass of a bag of apples. (d) The number of pages in a book. (e) The temperature at noon.

Show the working

For each one, ask first: is it counted or measured? Then check with the deciding test: is a value in between possible?
(a) Goals are counted. A team scores $2$ or $3$ goals, never $2.5$ . Discrete.
(b) Time is measured with a stopwatch. Between $58.3$ s and $58.4$ s a time such as $58.35$ s can occur. Continuous.
(c) Mass is measured on a scale and can be any value in a range, such as $1.237$ kg. Continuous.
(d) Pages are counted. A book has $212$ or $213$ pages, nothing between. Discrete.
(e) Temperature is measured with a thermometer and can be any value in a range, such as $24.36$ °C. Continuous.

Answer: (a) discrete — counted; (b) continuous — measured, any value in a range; (c) continuous — measured; (d) discrete — counted; (e) continuous — measured.

Example 2. Mia records the shoe size and the foot length of $20$ people. Shoe sizes on her list include $4$ , $4 \frac{1}{2}$ and $5$ . Foot lengths are measured to the nearest millimetre. State whether each variable is discrete or continuous, and explain why they are different types when both are about feet.

Show the working

Shoe size: the scale goes in steps of a half size. Between size $4$ and size $4 \frac{1}{2}$ no shoe size exists. So shoe size is discrete — even though it includes halves.
Foot length: measured with a ruler. A foot could be $23.4$ cm, $23.45$ cm or $23.452$ cm long. Any value in a range is possible, so foot length is continuous.
Rounding to the nearest millimetre does not change this. It only limits how the length is written down.
The difference: shoe size is a choice from a fixed list of sizes, while foot length is a measurement of something that can be any length.

Answer: Shoe size is discrete: it can only take the separate values on the size scale. Foot length is continuous: it is measured and can take any value in a range.

Example 3. Kofi records the time each person takes to solve a puzzle, to the nearest second. All his values are whole numbers, so he says that the data are discrete. Comment on Kofi's statement.

Show the working

Start with your verdict, so the reader knows your opinion: Kofi is wrong.
Then give the reason from the definition: time is measured, so it can take any value in a range.
Then explain why the values look like whole numbers: they have been rounded. A time recorded as $47$ s to the nearest second could be any time from $46.5$ s up to $47.5$ s.
$46.5 \leq t < 47.5$
Finish by tying it together: the times only look discrete because of how they were recorded. The data are continuous.

Answer: Kofi is wrong. Time is measured and can take any value in a range; the values are whole numbers only because they were rounded (a recorded $47$ s could be anything from $46.5$ s to $47.5$ s). The data are continuous.

Example 4. The lengths of $30$ leaves are measured. They lie between $3.0$ cm and $8.9$ cm. A student plans to group them in the classes $3$ – $4$ , $5$ – $6$ , $7$ – $8$ cm. Explain why these classes are not suitable, and write three classes of width $2$ cm that are suitable.

Show the working

Length is measured, so the data are continuous. That is the reason the classes matter.
Look for a value that fits no class. A leaf of $4.5$ cm is more than $4$ and less than $5$ , so it falls into the gap between $3$ – $4$ and $5$ – $6$ . A leaf of $8.9$ cm is longer than $8$ , so it fits no class either.
Write the classes with inequalities, starting at $3$ and going up in steps of $2$ , each starting where the last one stops.
$3 \leq l < 5, 5 \leq l < 7, 7 \leq l < 9$
Check: the longest leaf, $8.9$ cm, satisfies $7 \leq l < 9$ , and the shortest, $3.0$ cm, satisfies $3 \leq l < 5$ . Every length from $3$ to $9$ cm lands in exactly one class.

Answer: Length is continuous, so the classes leave gaps: a length such as $4.5$ cm fits no class. Suitable classes: $3 \leq l < 5$ , $5 \leq l < 7$ , $7 \leq l < 9$ .

Where people go wrong

Mistake: Thinking discrete means 'whole numbers' and continuous means 'has decimals'.

Instead: Discrete means only separate values are possible; continuous means any value in a range is possible.

Shoe sizes include halves and are discrete. Heights recorded to the nearest cm are whole numbers and are continuous. The decimals in the table do not decide the type.

Mistake: Calling rounded measurements discrete because every recorded value is a whole number.

Instead: Measured quantities are continuous however they are rounded.

A time of $47$ s to the nearest second stands for a range of possible times. Rounding changes how a value is written, not what kind of quantity it is.

Mistake: Calling data continuous because the quantity keeps changing, such as the number of people in a shop through the day.

Instead: Ask what values are possible at any moment. A count is discrete, however often it changes.

'Continuous' describes the possible values of the variable, not whether it changes over time.

Mistake: Giving 'because it is a number' as the reason.

Instead: Use the definition: 'counted, so only separate values are possible', or 'measured, so any value in a range is possible'.

Discrete and continuous data are both numerical, so 'it is a number' does not tell them apart.

Mistake: Writing classes for continuous data with gaps, such as $10$ – $19$ , $20$ – $29$ .

Instead: Use inequalities with no gaps: $10 \leq x < 20$ , $20 \leq x < 30$ .

A continuous value such as $19.6$ falls between $19$ and $20$ and would fit no class.

Check yourself

Is the number of cars in a car park discrete or continuous?
Answer
Discrete. Cars are counted, so only whole numbers of cars are possible.
Is the volume of water in a glass discrete or continuous?
Answer
Continuous. Volume is measured and can take any value in a range.
Shoe sizes include half sizes such as $6 \frac{1}{2}$ . Explain why shoe size is still discrete.
Answer
Shoe size can only take the separate values on the size scale. No size exists between $6$ and $6 \frac{1}{2}$ , so no value in between is possible.
A height is recorded as $158$ cm, to the nearest cm. Write down the range of possible true heights, and say whether height is discrete or continuous.
Answer
$157.5 \leq h < 158.5$ cm. Height is continuous: the record is rounded, but the true height can be any value in that range.
The number of people in a lift changes every few minutes. Does that make the data continuous?
Answer
No. At any moment the number of people is a count, such as $6$ , never $6.3$ . The data are discrete.
A test is marked out of $40$ , and half marks can be given. Are the marks discrete or continuous?
Answer
Discrete. The marks go up in steps of $0.5$ ( $31$ , $31.5$ , $32$ , …) and no mark is possible between two neighbouring steps.
Masses of parcels start at $40$ kg. Write the first two classes, each of width $10$ kg, in a form suitable for continuous data.
Answer
$40 \leq m < 50$ and $50 \leq m < 60$ .

Exam questions

Try each part first. Open the mark scheme after you have written your answer.

Question 1. A school survey records five variables for each student. A: number of pets B: mass of school bag C: time spent on homework last night D: shoe size E: number of lessons attended this week

(a) Write down the letters of all the variables that are discrete. 2 marks No calculator

Mark scheme

B2 A, D and E, and no others. B1 for two of A, D, E with no wrong letter, or for A, D and E with one wrong letter

(b) Explain why variable C is continuous. 1 mark No calculator

Mark scheme

B1 time is measured and can take any value (within a range) oe

Question 2. A bakery keeps records of its bread.

(a) State whether the number of loaves sold each day is discrete or continuous. 1 mark No calculator

Mark scheme

B1 discrete

(b) Each loaf is weighed, and its mass is recorded to the nearest gram. State whether these masses are discrete or continuous. Give a reason for your answer. 2 marks No calculator

Mark scheme

B1 continuous
B1 dep on continuous: mass is measured and can take any value (within a range) oe

(c) The baker says, 'Every mass in my records is a whole number of grams, so the masses are discrete data.' Comment on the baker's statement. 2 marks No calculator

Mark scheme

B1 the statement is wrong / the masses are continuous
B1 the masses have been rounded to the nearest gram, so a recorded mass stands for a range of possible masses, e.g. 802 g could be any mass from 801.5 g to 802.5 g oe

Question 3. A swimming coach times $40$ swimmers over $50$ m. Each time is measured to the nearest $0.1$ second. All the times are between $31.0$ s and $42.9$ s. The coach plans a table with the classes $31$ – $33$ , $34$ – $36$ , $37$ – $39$ and $40$ – $42$ seconds.

(a) Explain why these classes are not suitable for the times. 2 marks No calculator

Mark scheme

B1 time is continuous (measured, any value in a range) oe
B1 there are gaps between the classes, so some times fit no class, e.g. 33.5 s or 42.9 s oe

(b) Write a set of classes that are suitable for the times. Each class must have width $3$ seconds, and together the classes must cover every time from $31$ s up to $43$ s. 2 marks No calculator

Mark scheme

B1 classes with no gaps or overlaps, written as inequalities or in words, e.g. 31 ≤ t < 34, 34 ≤ t < 37, or '31 to less than 34' oe
B1 all four classes correct: 31 ≤ t < 34, 34 ≤ t < 37, 37 ≤ t < 40, 40 ≤ t < 43 oe (any letter for the variable; equivalent wording such as '40 to less than 43')

(c) A second coach counts the number of lengths each swimmer completes in a training session. She groups the counts as $0$ – $9$ , $10$ – $19$ , $20$ – $29$ . Explain why these classes are suitable for her data. 1 mark No calculator

Mark scheme

B1 the number of lengths is discrete (counted), so no value can lie between 9 and 10 or between 19 and 20 oe

How this is examined

Saying which type a variable is, and backing it up with the definition: a two-mark 'State whether … is discrete or continuous. Give a reason.' about one named variable One mark for the correct word, one for a reason that uses the definition. Continuous: 'it is measured and can take any value (in a range)'. Discrete: 'it is counted / it can only take certain separate values'. 'It is a number', 'it has decimals' or 'it is a whole number' does not earn the reason mark, because it does not tell the two types apart.
Why rounded measurements are still continuous: a two-mark Comment on a person's claim that the data are discrete because every recorded value is a whole number First give your verdict — the claim is wrong — then the reason: the quantity is measured and the values have been rounded, so a recorded value stands for a range. Quote one, for example 'a time recorded as 47 s could be anything from 46.5 s to 47.5 s'. A verdict with no reason is not a Comment.
Why classes with gaps do not work for continuous data: a two-mark Explain about a grouped table with classes such as 31–33, 34–36, followed by a Write that asks for better classes Say the data are continuous and name a value that fits no class, such as 33.5. For the rewrite, use inequalities with each class starting where the last stops: $31 \leq t < 34$ , $34 \leq t < 37$ . Classes with gaps, or classes that overlap at a boundary, do not earn the mark.
Sorting a list of variables into discrete and continuous: a one- or two-mark Write down: several variables are listed and you pick out the discrete ones (or the continuous ones) The correct letters or names, and no wrong ones. Write down asks for the answer without significant working, so no reasons are needed. Check each half-size or rounded case (shoe size, times to the nearest second) with the 'is a value in between possible?' test before you write.

Words to know

Variable: The thing being recorded for each person or item, such as height or number of pets.
Categorical data: Data that are words or categories, such as eye colour. Also called qualitative data.
Numerical data: Data that are numbers which count or measure something. Also called quantitative data. Numerical data are either discrete or continuous.
Discrete data: Numerical data that can take only particular, separate values, with no possible value in between. Counts are discrete.
Continuous data: Numerical data that can take any value within a range. Measurements such as length, mass and time are continuous.
Class: One group in a grouped table. For continuous data, write classes as inequalities, such as $150 \leq h < 155$ , so there are no gaps.

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