Maths · 10 Statistics

10.1 Classifying statistical data

Before you start

Adding a column of whole numbers accurately, and checking the sum by adding it again in a different order.
Reading the inequality signs $<$ , $\leq$ , $>$ and $\geq$ : for example, $40 \leq t < 50$ means $t$ is 40 or more but less than 50.
Writing one number as a fraction of another and simplifying it, e.g. $\frac{12}{30} = \frac{2}{5}$ .
Changing a fraction to a percentage by multiplying by 100.

Syllabus objectives covered (1)

Classify and tabulate statistical data.

Part 1 of 6

Two families of data: categorical and numerical

Data is the information you collect, one value for each person or thing. The thing being recorded, such as eye colour or height, is called a variable. Before you put data in a table, decide what kind it is, because the kind decides what the table looks like.
Categorical data (also called qualitative data) is a word or a label that puts each item into a group: eye colour, favourite sport, type of car, country of birth. You can count how many items are in each group, but you cannot do arithmetic with the values themselves.
Numerical data (also called quantitative data) is a number that you get by counting or by measuring: the number of brothers and sisters, the mass of a parcel, the time to run 100 m.
The trap is a number that is only a label. A shirt number, a bus route number or a locker number is written in digits, but it is categorical. Test it: would adding two values, or finding their mean, tell you anything? Shirt 7 plus shirt 4 gives 11, and 11 tells you nothing about either player. So shirt numbers are categorical.

Part 2 of 6

Numerical data: discrete or continuous

Numerical data splits into two kinds. Discrete data can take only certain separate values, with gaps between them. Counts are discrete: a team can score 2 goals or 3 goals, but never 2.4 goals.
Continuous data can take any value in a range. It comes from measuring: length, mass, time, temperature. Between any two heights, such as 162 cm and 163 cm, there is always another possible height, such as 162.5 cm.
Rounding does not change the kind of data. A height written as 162 cm to the nearest centimetre is still continuous, because the real height can be any value from 161.5 cm up to 162.5 cm. Ask how the value was obtained, not how it was written down.
Discrete does not mean 'whole numbers only'. Shoe sizes go 5, $5 \frac{1}{2}$ , 6, and so on. They are discrete, because they come in fixed steps and there is no size between 5 and $5 \frac{1}{2}$ .
A quick test for numerical data: a value from counting is discrete, and a value from measuring on a scale (a ruler, a balance, a stopwatch, a thermometer) is continuous. Values that come only in fixed steps, such as shoe sizes, are also discrete.
If a question asks you to Explain the kind of data, give the reason as well as the name. 'Mass is continuous because it is measured and can take any value in a range' answers the question. 'Continuous' on its own does not.

Part 3 of 6

Tally charts and frequency tables

A frequency table lists each possible value, or each category, once. Next to it is the frequency: the number of times that value occurs in the data.
To build one from a list, first write every possible value in the first column, in order. Then go through the list once, from start to finish, and make one tally mark for each item. Do not search the list for all the 1s, then all the 2s: then you read the list many times, and it is easy to miss a value.
Tally marks are grouped in fives so that they are quick to count. Draw four upright strokes, then draw the fifth stroke across those four. This bundle of five is called a gate. So 7 is one gate and two more strokes, and 12 is two gates and two more strokes.
graph diagram — not drawn yet
Tally marks drawn above the numerals they stand for: 1, 2, 3 and 4 as one, two, three and four upright strokes, 5 as four upright strokes with a fifth stroke drawn diagonally across them, then 7 as one crossed gate of five followed by two upright strokes, and 12 as two crossed gates followed by two upright strokes.
The fifth stroke closes a gate, so you count the gates in fives and then add the loose strokes.
Put a light line through each value in the list as you tally it. If you stop part of the way through, you can see exactly where to start again.
Count each tally and write the number in the frequency column. The tally is your working. The frequency is the number that you use in the rest of the question.
Add up the frequency column. The total must equal the number of items in the list. If it does not, you have missed an item or counted one twice, so tally again before you go on.
Read the words of the question exactly. 'More than 2' means 3, 4, 5, and so on: it does not include 2. 'At least 2' means 2 or more, so it does include 2.
A frequency counts items, not the size of each item. If 10 cars each carry 2 people, the frequency is 10 cars, but the cars hold $2 \times 10 = 20$ people. To find the total of the values, multiply each value by its frequency, then add the results.

Part 4 of 6

Grouped frequency tables

When the data has many different values, a row for every value gives a long table full of frequencies of 0 and 1. Instead, put the values into groups called class intervals, and count how many values fall in each class.
Continuous data is grouped with inequalities, because it can take any value. The classes must touch each other, with no gaps and no overlaps:
$20 \leq t < 30 30 \leq t < 40 40 \leq t < 50$
Read each class as a sentence. $30 \leq t < 40$ means ' $t$ is 30 or more, but less than 40'. So a value of exactly 30 belongs in this class. A value of exactly 40 does not: it goes in $40 \leq t < 50$ .
Some tables put the 'or equal to' at the top end instead. $30 < t \leq 40$ means 'more than 30, up to and including 40'. Then 40 belongs in this class, and 30 belongs in the class below. Before you place a value that is on a boundary, look at which end of the class has the $\leq$ .
Discrete data can be grouped with ranges such as 0–4, 5–9 and 10–14. There is a gap between 4 and 5, and that is correct, because no count can lie between 4 and 5.
The trap is classes that share an end, such as 10–20 and 20–30. A value of 20 fits in both, so the table does not tell you where to put it. For discrete data, use 10–19 and 20–29. For continuous data, use inequalities.
Grouping loses detail. When a value is in a class, the table no longer shows its exact size. From the class $50 \leq t < 60$ you know that a time is at least 50 seconds and less than 60 seconds, but not what it was. So you cannot read the exact largest value from a grouped table.

Part 5 of 6

Two-way tables

A two-way table sorts each item by two things at the same time: one along the rows and one down the columns. Each cell counts the items that belong to both its row and its column. This table shows the favourite pet, cat or dog, of 33 students:
$\begin{matrix} Cat & Dog & Total \\ Boys & 7 & 9 & 16 \\ Girls & 11 & 6 & 17 \\ Total & 18 & 15 & 33 \end{matrix}$
The 6 is in the Girls row and the Dog column, so 6 girls chose a dog. Each student is counted in exactly one cell.
Each row adds up to its row total, and each column adds up to its column total. The row totals add up to the grand total in the bottom-right corner, and so do the column totals:
$16 + 17 = 33 18 + 15 = 33$
To complete a table that has gaps, look for a row or a column with exactly one missing number. Find that number by subtracting the known numbers from the total, or by adding the cells to make the total. The new number may leave another row or column with only one gap, so repeat until the table is full.
If the information is given in sentences, draw the empty table first, with a Total row and a Total column. Put each fact in its cell before you calculate anything. '64 students are in Year 10' is a row total. '18 of the students who walk are in Year 10' is one cell.
Finish with a check. The row totals must add up to the grand total, and so must the column totals. When the two routes give the same number, you have good evidence that the cells are correct.

Part 6 of 6

Taking a fraction from a two-way table

A fraction from a two-way table is one cell divided by one total. The difficult part is choosing the total, and the words of the question tell you which one to use.
The group named after 'of the' gives the denominator. 'What fraction of the girls chose a dog?' compares with all the girls, so divide by the girls' row total, 17:
$\frac{6}{17}$
'What fraction of the students who chose a dog are girls?' compares with everyone who chose a dog, so divide by the Dog column total, 15:
$\frac{6}{15} = \frac{2}{5}$
'What fraction of all the students are girls who chose a dog?' compares with everyone, so divide by the grand total, 33:
$\frac{6}{33} = \frac{2}{11}$
The same cell, 6, gave three different answers. Before you divide, say the denominator in words, for example 'out of all the girls'. Then find that total in the table.
For a percentage, find the fraction in the same way and multiply it by 100. If the answer is not exact, give it to 3 significant figures.

Worked examples

Example 1. For each variable, state whether the data is categorical, discrete or continuous. (a) The number of pets each student in a class owns. (b) The time each runner takes to finish a 5 km race. (c) The favourite fruit of each person at a party. (d) The mass of each egg in a box. (e) The number on each locker in a corridor. (f) The shoe size of each player in a team.

Show the working

For each one, first ask: is the value a word or a label, or is it a real amount? Only a real amount is numerical. Then ask whether it was counted or measured.
(a) The number of pets is counted, and it can only be 0, 1, 2, and so on. It is discrete.
(b) Time is measured with a stopwatch and can be any value in a range. It is continuous, even if the times are written to the nearest second.
(c) A fruit is a word, not an amount. It is categorical.
(d) Mass is measured on a balance and can be any value in a range. It is continuous.
(e) A locker number is a label. Adding two locker numbers gives nothing useful. It is categorical, although it is written in digits.
(f) Shoe sizes come in fixed steps, with no size between two neighbouring sizes. They are discrete.

Answer: (a) discrete (b) continuous (c) categorical (d) continuous (e) categorical (f) discrete

Example 2. The number of people in each of 30 cars that passed a school gate was recorded: $2, 1, 1, 3, 2, 1, 4, 2, 1, 1, 3, 2, 5, 1, 2, 2, 1, 3, 1, 4, 2, 1, 1, 2, 3, 1, 2, 5, 1, 2$ . (a) Make a frequency table for the data. (b) Find the number of cars that had more than 2 people in them. (c) Find the total number of people in the 30 cars.

Show the working

The smallest value is 1 and the largest is 5. Write 1, 2, 3, 4 and 5 in the first column before you start to tally.
Go through the list once, in order, and make a tally mark for each value. The first three values are 2, 1 and 1, so they give one stroke for 2 and two strokes for 1. Put a light line through each value as you tally it.
Count each tally. For 1 there are two gates and two more strokes, which is 12. Write the frequencies:
$\begin{matrix} People in car & Frequency \\ 1 & 12 \\ 2 & 10 \\ 3 & 4 \\ 4 & 2 \\ 5 & 2 \\ Total & 30 \end{matrix}$
Check the total: $12 + 10 + 4 + 2 + 2 = 30$ . This is the number of cars in the list, so no value was missed or counted twice.
(b) 'More than 2' means 3, 4 or 5 people. Do not include the cars with exactly 2 people.
$4 + 2 + 2 = 8$
(c) The frequencies count cars, not people. To count the people, multiply each number of people by the number of cars with that many people, then add.
$1 \times 12 + 2 \times 10 + 3 \times 4 + 4 \times 2 + 5 \times 2 = 12 + 20 + 12 + 8 + 10 = 62$

Answer: (a) Frequencies 12, 10, 4, 2, 2 for 1, 2, 3, 4, 5 people (total 30) (b) 8 cars (c) 62 people

Example 3. The times, in seconds, that 20 students took to solve a puzzle are: $34.2, 41.5, 28.7, 30.0, 55.1, 47.3, 39.9, 40.0, 22.4, 36.8, 44.0, 50.0, 26.1, 29.5, 42.2, 38.1, 57.6, 45.9, 33.3, 48.4$ . Make a grouped frequency table with the classes $20 \leq t < 30$ , $30 \leq t < 40$ , $40 \leq t < 50$ and $50 \leq t < 60$ . Then find how many students took less than 40 seconds.

Show the working

Time is measured, so the data is continuous, and the classes are inequalities. Each class includes its lower end and excludes its upper end.
Find the boundary values first, because that is where mistakes happen. The list has 30.0, 40.0 and 50.0.
30.0 satisfies $30 \leq t < 40$ , so it goes in the second class, not the first. In the same way, 40.0 goes in $40 \leq t < 50$ , and 50.0 goes in $50 \leq t < 60$ .
39.9 is less than 40, so it stays in $30 \leq t < 40$ . A value that is close to a boundary does not move across it.
Tally the other values in one pass through the list. The frequencies are:
$\begin{matrix} Time, t seconds & Frequency \\ 20 \leq t < 30 & 4 \\ 30 \leq t < 40 & 6 \\ 40 \leq t < 50 & 7 \\ 50 \leq t < 60 & 3 \\ Total & 20 \end{matrix}$
Check: $4 + 6 + 7 + 3 = 20$ , the number of students.
'Less than 40 seconds' is the first two classes, because every time in $30 \leq t < 40$ is below 40.
$4 + 6 = 10$

Answer: Frequencies 4, 6, 7, 3 for the four classes (total 20); 10 students took less than 40 seconds.

Example 4. There are 120 students in Years 10 and 11 at a school. Each student walks to school, takes the bus or comes by car. 64 of the students are in Year 10. 35 students walk, and 18 of them are in Year 10. 25 of the Year 11 students take the bus. 30 students come by car. (a) Complete a two-way table for this information. (b) Find the fraction of the Year 11 students who come by car. (c) Find the fraction of the students who come by car that are in Year 11.

Show the working

Draw the empty table: rows for Year 10, Year 11 and Total, and columns for Walk, Bus, Car and Total. Put in each fact. 120 is the grand total. 64 is the Year 10 row total. 35 is the Walk column total. 18 is the cell for Year 10 and Walk. 25 is the cell for Year 11 and Bus. 30 is the Car column total.
Find a row or a column with one gap. In the Total column, the Year 11 total is missing:
$120 - 64 = 56$
In the Walk column, the Year 11 walkers are missing:
$35 - 18 = 17$
In the Total row, the Bus total is missing:
$120 - 35 - 30 = 55$
Now the Bus column has one gap, Year 10:
$55 - 25 = 30$
Now the Year 10 row has one gap, Car:
$64 - 18 - 30 = 16$
Last, the Car column gives Year 11 by car: $30 - 16 = 14$ . Check it with the Year 11 row: $17 + 25 + 14 = 56$ , which agrees.
$\begin{matrix} Walk & Bus & Car & Total \\ Year 10 & 18 & 30 & 16 & 64 \\ Year 11 & 17 & 25 & 14 & 56 \\ Total & 35 & 55 & 30 & 120 \end{matrix}$
(b) 'Of the Year 11 students' means the denominator is the Year 11 row total, 56.
$\frac{14}{56} = \frac{1}{4}$
(c) 'Of the students who come by car' means the denominator is the Car column total, 30.
$\frac{14}{30} = \frac{7}{15}$

Answer: (a) Year 10: Walk 18, Bus 30, Car 16, Total 64. Year 11: Walk 17, Bus 25, Car 14, Total 56. Totals: Walk 35, Bus 55, Car 30, grand total 120. (b) $\frac{1}{4}$ (c) $\frac{7}{15}$

Where people go wrong

Mistake: Calling a height or a time discrete because it is written as a whole number, such as 162 cm or 48 seconds.

Instead: It is continuous. Height and time are measured, and the recorded value is a rounded form of a value that could be anything in a range.

The kind of data depends on how the value is obtained, not on how many decimal places it is written to.

Mistake: Treating a number that is only a label, such as a shirt number or a bus route number, as numerical data.

Instead: It is categorical. Ask whether adding the values or finding their mean would tell you anything. For a label, it does not.

The digits look like an amount, but the number only names a group.

Mistake: Using classes that share an end, such as 10–20 and 20–30, or putting a time of exactly 40 seconds in the class $30 \leq t < 40$ .

Instead: Read the inequality: $30 \leq t < 40$ does not include 40, so 40 goes in $40 \leq t < 50$ . For grouped discrete data, use ranges that do not share a value, such as 10–19 and 20–29.

Every value must belong to exactly one class. If not, the frequencies do not add up to the number of items.

Mistake: Dividing by the grand total whatever the question asks. For example, answering 'what fraction of the girls chose a dog?' with $\frac{6}{33}$ .

Instead: Divide by the total of the group named after 'of the'. Here that is the girls' row total, so the answer is $\frac{6}{17}$ .

A fraction 'of the girls' compares with all the girls, not with everyone in the table.

Mistake: Adding the frequencies to find a total of the values, for example saying that 30 cars carried 30 people.

Instead: Multiply each value by its frequency and add the results. In Worked example 2, the 30 cars carried 62 people.

A frequency counts how many items there are, not the size of each item.

Check yourself

Is the number of text messages you sent yesterday discrete or continuous? Give a reason.
Answer
Discrete. It is counted, and it can only be a whole number: you cannot send 3.5 messages.
A nurse records the mass of each baby to the nearest 10 g. Is this data discrete or continuous?
Answer
Continuous. Mass is measured. Rounding to the nearest 10 g changes how the value is written, not what kind of data it is.
A survey records the number of the bus route that each person takes, for example route 12 or route 47. What kind of data is this?
Answer
Categorical. The route number is a label. Adding route numbers or finding their mean gives nothing meaningful.
A tally shows three complete gates and four more strokes. What is the frequency?
Answer
$3 \times 5 + 4 = 19$
The classes in a table are $10 < m \leq 20$ , $20 < m \leq 30$ and $30 < m \leq 40$ . In which class does a mass of exactly 20 go? In which class does 30.1 go?
Answer
20 goes in $10 < m \leq 20$ , because that class includes 20 and the next class starts above 20. 30.1 goes in $30 < m \leq 40$ .
Ben groups the number of books that students read as 0–5, 5–10 and 10–15. What is wrong, and how can he fix it?
Answer
The values 5 and 10 each fit in two classes. He should use classes that do not share a value, such as 0–4, 5–9 and 10–14.
Complete this two-way table. $\begin{matrix} Pass & Fail & Total \\ Group A & 22 & \dots & 30 \\ Group B & \dots & 5 & \dots \\ Total & 40 & \dots & \dots \end{matrix}$
Answer
Group A Fail $= 30 - 22 = 8$ . Group B Pass $= 40 - 22 = 18$ . Group B Total $= 18 + 5 = 23$ . Fail total $= 8 + 5 = 13$ . Grand total $= 30 + 23 = 53$ , and $40 + 13 = 53$ agrees.
Use your completed table from the last question. What fraction of the students who failed were in Group B?
Answer
The denominator is the Fail total, 13, so the fraction is $\frac{5}{13}$ .

Exam questions

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

Question 1. Amira collects five pieces of information about each student in her class. A: the number of brothers and sisters. B: height. C: eye colour. D: the time taken to travel to school. E: the number on the student's locker.

(a) Write down the letters of the two variables that are continuous. 1 mark No calculator

Mark scheme

B1 B and D, and no others

(b) Write down the letters of the two variables that are categorical. 1 mark No calculator

Mark scheme

B1 C and E, and no others

(c) Explain why variable A is discrete data. 1 mark No calculator

Mark scheme

B1 it is counted, or it can only take whole-number values / certain separate values, oe

Question 2. The number of goals that a hockey team scored in each of 24 matches is listed below. $1, 0, 2, 3, 2, 1, 0, 2, 4, 1, 2, 0, 3, 3, 2, 0, 1, 5, 2, 1, 0, 3, 1, 2$ $\begin{matrix} Goals & Tally & Frequency \\ 0 \\ 1 \\ 2 \\ 3 \\ 4 \\ 5 \end{matrix}$

(a) Complete the frequency table. 2 marks No calculator

Mark scheme

B2 frequencies 5, 6, 7, 4, 1, 1 for 0, 1, 2, 3, 4, 5 goals; if B2 is not scored, B1 for four or five correct frequencies

(b) Find the number of matches in which the team scored at least 2 goals. 1 mark No calculator

Mark scheme

B1 13; FT the sum of their frequencies for 2, 3, 4 and 5 goals

(c) Work out the percentage of the matches in which the team scored exactly 1 goal. 2 marks No calculator

Mark scheme

M1 6/24 × 100 oe; FT their frequency for 1 goal
A1 25 (%) cao

Question 3. The lengths, in centimetres, of 20 leaves are listed below. $5.2, 7.8, 9.1, 6.0, 10.4, 8.0, 7.3, 11.6, 6.9, 9.8, 4.7, 8.6, 10.0, 7.1, 9.5, 6.4, 8.3, 11.2, 7.7, 9.0$ $\begin{matrix} Length, l cm & Frequency \\ 4 < l \leq 6 \\ 6 < l \leq 8 \\ 8 < l \leq 10 \\ 10 < l \leq 12 \end{matrix}$

(a) Complete the frequency table. 2 marks No calculator

Mark scheme

B2 3, 7, 7, 3 in order; if B2 is not scored, B1 for two or three correct frequencies

(b) Write down the number of leaves that are longer than 8 cm. 1 mark No calculator

Mark scheme

B1 10; FT the sum of their last two frequencies

(c) Explain why the table cannot be used to find the length of the longest leaf. 1 mark No calculator

Mark scheme

B1 the data is grouped, so the table only shows that the longest leaf is in the class 10 < l ≤ 12, not its exact length, oe

(d) Kofi says that the classes 4–6, 6–8, 8–10 and 10–12 would be just as good. Explain why Kofi is wrong. 1 mark No calculator

Mark scheme

B1 a boundary length such as 6.0, 8.0 or 10.0 would belong to two classes, oe

Question 4. 200 people at a sports centre were asked whether they came to swim, to use the gym or to join a class. Each person gave one answer. The two-way table shows some of the results. $\begin{matrix} Swim & Gym & Class & Total \\ Under 18 & 34 & 12 & \dots & 62 \\ 18 and over & \dots & 58 & \dots & \dots \\ Total & 61 & \dots & \dots & 200 \end{matrix}$

(a) Complete the two-way table. 3 marks

Mark scheme

B3 all six correct: Under 18 Class 16; 18 and over Swim 27, Class 53, Total 138; Total Gym 70, Total Class 69. If B3 is not scored, B2 for four or five correct, or B1 for two or three correct

(b) Find the fraction of the people under 18 who came to use the gym. Give your answer in its simplest form. 2 marks

Mark scheme

M1 12/62 oe, may be implied by a correct answer
A1 6/31 cao

(c) Work out the percentage of the people who came to use the gym who were 18 and over. 2 marks

Mark scheme

M1 58 ÷ their 70 × 100 oe
A1 awrt 82.9 (%)

How this is examined

Completing a two-way table from given numbers or from sentences: A table with row and column totals and several blank cells, or a set of facts to place in an empty table, with the instruction Complete; then one or two parts that ask for a number, a fraction or a percentage from the table Every blank filled so that each row adds up to its row total and each column adds up to its column total. Write each number in its cell, not only in your working. Check that the row totals and the column totals both add up to the grand total.
Putting raw data into a grouped frequency table, including values that are exactly on a class boundary: A list of measurements and a table of class intervals written as inequalities, with the instruction to complete the frequency column The correct frequency in every class. A boundary value goes in the class that has $\leq$ at that end: in $30 \leq t < 40$ , 30 is in the class and 40 is not. The frequencies add up to the number of values in the list.
Making a tally and frequency table from a list, then reading values from it: A list of whole-number results and a table with Tally and Frequency columns to complete, then a short Find or Work out part such as how many results were at least 2 The correct frequency for every value; the tally is the working. 'More than 2' does not include 2, and 'at least 2' does. A percentage is frequency $\div$ total $\times 100$ .
Choosing the correct total when you take a fraction or a percentage from a two-way table: A Find or Work out part after a two-way table: 'what fraction of the …' or 'what percentage of the …' The M mark is for the correct cell over the correct total, for example $\frac{12}{62}$ when the question says 'of the people under 18'. The A mark is for the correct value. An equivalent fraction that is not simplified is accepted unless the question asks for the simplest form. Give a percentage that is not exact to 3 significant figures.
Saying whether data is categorical, discrete or continuous, with a reason: A list of variables with Write down the ones that are continuous, or a one-mark Explain why a variable is discrete The correct kind, and for Explain a reason: 'it is counted' or 'it can only take certain separate values' for discrete; 'it is measured' or 'it can take any value in a range' for continuous; 'it is a word or a label' for categorical. The name alone does not answer an Explain.

Words to know

Data: The information you collect, one value for each person or thing.
Variable: The thing that is recorded for each item, such as height or eye colour.
Categorical data: Data that is a word or a label, such as colour or a shirt number. Also called qualitative data.
Numerical data: Data that is an amount, from counting or measuring. Also called quantitative data.
Discrete data: Numerical data that can take only certain separate values, such as counts or shoe sizes.
Continuous data: Numerical data that can take any value in a range, such as length, mass or time. It comes from measuring.
Tally: A mark for each item as you count. Marks are grouped in gates of five: four strokes crossed by a fifth.
Frequency: The number of times a value or a category occurs in the data.
Frequency table: A table that lists each value or category with its frequency.
Class interval: A group of values in a grouped frequency table, such as $30 \leq t < 40$ or 10–19.
Two-way table: A table that sorts items by two things at the same time, one in the rows and one in the columns, usually with row and column totals.
Grand total: The number in the bottom-right corner of a two-way table: the total of all the items.

All Maths lessons Next: 10.2 Interpreting statistical data