Mini-Lesson

Handling Data

9 missions. Guess, learn, then check. Your place is saved.

Mission 1A coin lands heads 7 times in a row. Is tails now "due"?
Mission 2Roll two dice and add them. Which total is most likely, and why?
Mission 3Roll a dice four times. Is at least one six better than evens?
Mission 41 in 100 are ill; a 90% accurate test says you are. Chance you're ill?
Mission 5Can asking 1,000 people tell you what a whole country thinks?
Mission 6Same data, two headlines: "soaring" and "flat". How?
Mission 7A firm's average pay is £60,000, yet 9 in 10 staff earn £25,000. How?
Mission 8Ice-cream sales and sunburn rise together. Does ice cream cause it?
Mission 9Two classes have the same median mark. Were their results the same?

The specification points and sources are listed under the lesson: see Sources.

Mission 1

A coin lands heads 7 times in a row. Is tails now "due"?

Make a guess: A coin lands heads 7 times in a row. Is tails now "due"?

Probability · basics and relative frequency

The probability scale

Probability measures how likely an event is, from 0 (impossible) to 1 (certain). Evens is 0.5.
Write probabilities as fractions, decimals or percentages, never as "1 in 4" or "1 : 4" in an answer.
Words along the scale: impossible, unlikely, evens, likely, certain.
A probability can never be below 0 or above 1. If you get 1.2, something has gone wrong.

Probability · basics and relative frequency

Equally likely outcomes

When outcomes are equally likely: $P (event) = \frac{number of outcomes in the event}{total number of outcomes}$ P(event) = (number of outcomes in the event)/(total number of outcomes)
A fair dice: $P (5 or 6) = \frac{2}{6} = \frac{1}{3}$ P(5 or 6) = 2/6 = 1/3.
$P (not A) = 1 - P (A)$ P(not A) = 1 − P(A). A fair dice: $P (not 6) = 1 - \frac{1}{6} = \frac{5}{6}$ P(not 6) = 1 − 1/6 = 5/6.
The probabilities of all the possible outcomes, when they cannot happen together (mutually exclusive), add up to 1.

Worked example Probability · basics and relative frequency

A missing probability

Question

A spinner lands on red, blue, green or yellow.

P (red) = 0.35

and

P (blue) = 0.2

. Yellow is twice as likely as green. Find

P (yellow)

.

1
The four outcomes are mutually exclusive, so their probabilities sum to 1: green and yellow share $1 - 0.35 - 0.2 = 0.45$ 1 − 0.35 − 0.2 = 0.45
2
Let $P (green) = x$ P(green) = x, so $P (yellow) = 2 x$ P(yellow) = 2x and $3 x = 0.45$ 3x = 0.45
3
Solve: $x = 0.15$ x = 0.15, so $P (yellow) = 2 \times 0.15$ P(yellow) = 2 × 0.15

Answer

P (yellow) = 0.3

⚠️ Watch out

halving 0.45 gives green and yellow equal chances; the question says yellow is twice as likely.

Calculate

?A bag holds 5 red, 7 blue and 8 green counters. One counter is taken at random. Find the probability that it is not blue. Give your answer as a decimal.

Probability · basics and relative frequency

Relative frequency and expected outcomes

From an experiment: relative frequency $= \frac{number of times the event happened}{number of trials}$ = (number of times the event happened)/(number of trials). Use it as an estimate of the probability.
The more trials, the closer relative frequency tends to get to the true probability. A few trials can wobble a lot. Each flip or spin is unaffected by earlier ones: a coin has no memory.
Expected number of outcomes $= probability \times number of trials$ = probability × number of trials. It is an estimate, not a promise.
A fair (unbiased) coin, dice or spinner has equally likely outcomes. A large experiment whose relative frequencies are far from these suggests bias.

Worked example Probability · basics and relative frequency

Estimating from an experiment

Question

A spinner numbered 1 to 5 is spun 50 times and lands on 5 eighteen times. Estimate how many times it will land on 5 in 400 spins.

1
Relative frequency of 5: $\frac{18}{50} = 0.36$ 18/50 = 0.36
2
Use it as the probability: expected number $= 0.36 \times 400$ = 0.36 × 400
3
Calculate: $0.36 \times 400 = 144$ 0.36 × 400 = 144

Answer

About 144 times.

⚠️ Watch out

assuming the spinner is fair (

\frac{1}{5} \times 400 = 80

) ignores the experiment, and the experiment suggests it is biased towards 5.

Calculate

?Exam-style. Ade thinks a dice is biased. He rolls it 250 times and gets a six 70 times. He rolls it 600 more times. Use his results to estimate how many sixes he will get in the 600 rolls.

Quick check

?Two groups test the same spinner. Group A: 9 reds in 20 spins. Group B: 124 reds in 400 spins. Which is the most reliable estimate of

P (red)

?

Mission check

?A fair coin has landed heads 7 times in a row. What is the probability that the next flip is tails?

Mission 2

Roll two dice and add them. Which total is most likely, and why?

Make a guess: Roll two dice and add them. Which total is most likely, and why?

Probability · sample spaces, two-way tables and Venn diagrams

Sample spaces

A sample space is the list of every possible outcome. List it systematically so you miss none and repeat none.
Two coins: HH, HT, TH, TT. HT and TH are different outcomes (first coin, second coin).
For two dice use a sample space diagram: a 6 by 6 grid with one cell for each of the 36 equally likely outcomes.
When outcomes are equally likely:

P(event) = \frac{number of outcomes in the event}{total number of outcomes}

Worked example Probability · sample spaces, two-way tables and Venn diagrams

Using a sample space diagram

Question

Two fair dice are rolled and the scores are added. Find the probability that the total is 10 or more.

1
Draw the 6 by 6 grid of totals: there are 36 equally likely outcomes
2
Count the cells with total 10, 11 or 12: $3 + 2 + 1 = 6$ 3 + 2 + 1 = 6 cells
3
Divide by the total number of outcomes: $\frac{6}{36}$ 6/36
4
Simplify: $\frac{6}{36} = \frac{1}{6}$ 6/36 = 1/6

Answer

\frac{1}{6}

⚠️ Watch out

there are 11 possible totals, but they are not equally likely, so the answer is not

\frac{3}{11}

.

Quick check

?Two fair coins are thrown. What is the probability of getting exactly one head and one tail?

Calculate

?A fair four-sided spinner numbered 1 to 4 is spun and a fair dice is rolled. The two scores are multiplied. Find the probability that the product is 12. Give your answer as a decimal.

Worked example Probability · sample spaces, two-way tables and Venn diagrams

Completing a two-way table

Question

80 students in Years 10 and 11 each chose school dinner, packed lunch or going home. In Year 10, 18 chose dinner and 12 chose packed lunch, out of 38 students. In Year 11, 15 chose packed lunch and 11 went home. 34 students chose dinner altogether. A student is picked at random. Find the probability that they are in Year 11 and chose school dinner.

1
Draw a two-way table: Year 10 and Year 11 as rows; dinner, packed lunch, home and total as columns
2
Use the dinner column total: Year 11 dinner $= 34 - 18 = 16$ = 34 − 18 = 16
3
Check the rows: Year 10 home $= 38 - 18 - 12 = 8$ = 38 − 18 − 12 = 8; Year 11 total $= 16 + 15 + 11 = 42 = 80 - 38$ = 16 + 15 + 11 = 42 = 80 − 38
4
"Picked at random from all 80", so divide by the grand total: $\frac{16}{80}$ 16/80

Answer

\frac{16}{80} = \frac{1}{5}

⚠️ Watch out

dividing by the Year 11 total (42) answers a different question; the student is chosen from all 80.

Calculate

?A cinema surveyed 200 people: 120 adults and 80 children. Each watched Film X, Y or Z. 45 adults watched X and 30 adults watched Z. 22 children watched Y and 24 children watched Z. One of the 200 people is chosen at random. Find the probability that the person is a child who watched Film X. Give your answer as a decimal.

Probability · sample spaces, two-way tables and Venn diagrams

Venn diagrams

A Venn diagram sorts items into overlapping circles inside a rectangle. The overlap is in both sets; outside the circles is in neither.
Fill in the overlap first, then the "only" parts, then outside. All regions sum to the total.
Probability = (number in your region) ÷ (total in the rectangle).
With three sets, fill in from the centre (in all three) outwards.

Calculate

?A sports club has 80 members. 45 swim, 38 cycle and 9 do neither. A member is chosen at random. Find the probability that the member both swims and cycles. Give your answer as a decimal.

Mission check

?Roll two dice and add them. Which total is most likely, and why?

Mission 3

Roll a dice four times. Is at least one six better than evens?

Make a guess: Roll a dice four times. Is at least one six better than evens?

Probability · combined events and tree diagrams

The AND rule and the OR rule

Events are independent if one happening does not change the probability of the other (two dice, a coin and a spinner, repeated spins).
AND for independent events: multiply. $P (A and B) = P (A) \times P (B)$ P(A and B) = P(A) × P(B)
OR for mutually exclusive events (they cannot both happen): add. $P (A or B) = P (A) + P (B)$ P(A or B) = P(A) + P(B)
$P (not A) = 1 - P (A)$ P(not A) = 1 − P(A), so P(at least one) $= 1 -$ = 1 − P(none).
Higher: for CCEA the AND and OR rules and tree diagrams are Higher tier (Units M7 and M8).

Quick check

?Two fair dice are rolled. What is the probability that both show a six?

Probability · combined events and tree diagrams

Tree diagrams (Higher)

A tree diagram shows each stage as a set of branches. Write the outcome at the end of each branch and its probability beside it.
The branches from any one point add to 1.
Multiply along a path to get the probability of that combined outcome.
Add the paths that give the event you want (the paths are mutually exclusive).
Check: all the end probabilities together add to 1.

Worked example Probability · combined events and tree diagrams

Exactly one of two events (Higher)

Question

The probability that a train is late on any day is 0.2, independently of other days. Find the probability that it is late on exactly one of two days.

1
Draw a tree: Day 1 then Day 2, each with Late 0.2 and On time 0.8
2
Find the paths with exactly one late: (Late, On time) and (On time, Late)
3
Multiply along each path: $0.2 \times 0.8 = 0.16$ 0.2 × 0.8 = 0.16 and $0.8 \times 0.2 = 0.16$ 0.8 × 0.2 = 0.16
4
Add the two paths: $0.16 + 0.16 = 0.32$ 0.16 + 0.16 = 0.32

Answer

0.32

⚠️ Watch out

stopping at 0.16 misses the second path; the late day could be either day.

Calculate

?The probability that a seed germinates is 0.7, independently of other seeds. Three seeds are planted. Find the probability that at least one of them germinates. Give your answer as a decimal.

Sort it

?Add or multiply? Sort each calculation.

Tap a statement, then tap the group it belongs in. Then press Check.

Probability · combined events and tree diagrams

Without replacement (Higher)

With replacement, the item goes back: the second-stage probabilities are the same as the first.
Without replacement, the events are dependent: the second branches change because there is one fewer item.
Take one away from the total on every second branch, and from the colour already taken on its own branch.
Then multiply along and add the paths as before.

Worked example Probability · combined events and tree diagrams

Two bulbs without replacement (Higher)

Question

A box has 10 bulbs and 3 are faulty. Two are taken at random without replacement. Find the probability that exactly one is faulty.

1
First pick: P(faulty) $= \frac{3}{10}$ = 3/10, P(good) $= \frac{7}{10}$ = 7/10
2
Second pick, 9 bulbs left: after a faulty one, $\frac{2}{9}$ 2/9 faulty and $\frac{7}{9}$ 7/9 good; after a good one, $\frac{3}{9}$ 3/9 faulty and $\frac{6}{9}$ 6/9 good
3
Paths with exactly one faulty: $\frac{3}{10} \times \frac{7}{9} = \frac{21}{90}$ 3/10 × 7/9 = 21/90 and $\frac{7}{10} \times \frac{3}{9} = \frac{21}{90}$ 7/10 × 3/9 = 21/90
4
Add: $\frac{21}{90} + \frac{21}{90} = \frac{42}{90}$ 21/90 + 21/90 = 42/90

Answer

\frac{7}{15}

⚠️ Watch out

keeping

\frac{3}{10}

on the second pick treats it as with replacement.

Calculate

?A bag has 5 red and 3 green counters. Two counters are taken at random without replacement. Find the probability that they are the same colour. Give your answer to 3 significant figures.

Mission check

?Roll a dice four times. Is at least one six better than evens?

Mission 4

1 in 100 are ill; a 90% accurate test says you are. Chance you're ill?

Make a guess: 1 in 100 are ill; a 90% accurate test says you are. Chance you're ill?

Probability · conditional probability

"Given that" shrinks the total (Higher)

A conditional probability is a probability given that something has already happened.
"Given B" means you only look at the outcomes in B. B's total becomes the new denominator.

P (A given B) = \frac{number in both A and B}{number in B}

Worked example Probability · conditional probability

Conditional probability from a two-way table (Higher)

Question

120 people at a gym chose a class. Yoga: 34 women and 14 men. Spin: 30 women and 42 men. A person is chosen at random from the spin class. Find the probability that the person is a woman.

1
"From the spin class" is the condition: total spin $= 30 + 42 = 72$ = 30 + 42 = 72
2
Count spin-class women: 30
3
Divide by the spin total, not by 120: $\frac{30}{72}$ 30/72
4
Simplify: $\frac{30}{72} = \frac{5}{12}$ 30/72 = 5/12

Answer

\frac{5}{12}

⚠️ Watch out

\frac{30}{120}

is P(woman and spin), and

\frac{30}{64}

is P(spin given woman); read which group the person comes from.

Quick check

?Use the gym table: yoga 34 women, 14 men; spin 30 women, 42 men. A man is chosen at random. What is the probability that he does spin?

Probability · conditional probability

Conditional probability from a Venn diagram (Higher)

"Given B": cover everything outside circle B. Circle B's total is the new denominator.
The numerator is the overlap (the part of B that is also in A).
If you are given set sizes and "neither", find the overlap first: overlap = A + B − (total − neither).

Calculate

?In a group of 50 students, 21 study history, 28 study geography and 8 study neither. A student who studies geography is chosen at random. Find the probability that this student also studies history. Give your answer as a decimal.

Probability · conditional probability

Reversing a tree with expected frequencies (Higher)

The second branches of a tree are already conditional: "late, given it rained".
To reverse the condition ("rained, given late"), imagine a round number, such as 100 or 1,000, going through the tree. These are expected frequencies.
Add up every path that ends in the given outcome: that is the new total.
Divide the path you want by that total.

Worked example Probability · conditional probability

Rain, given the bus was late (Higher)

Question

On 30% of days it rains. On a rainy day the probability the bus is late is 0.5; on a dry day it is 0.1. The bus was late. Find the probability that it was raining.

1
Imagine 100 days: 30 rainy, 70 dry
2
Late days: rainy $30 \times 0.5 = 15$ 30 × 0.5 = 15, dry $70 \times 0.1 = 7$ 70 × 0.1 = 7
3
The condition is "late": $15 + 7 = 22$ 15 + 7 = 22 late days in total
4
Of these, 15 were rainy: $\frac{15}{22}$ 15/22

Answer

\frac{15}{22} \approx 0.682

⚠️ Watch out

0.5 is P(late given rain), not P(rain given late); the condition is the other way round.

Calculate

?Machine A makes 60% of a factory's items and 5% of them are faulty. Machine B makes the rest and 10% of them are faulty. An item is chosen at random and is faulty. Find the probability that it was made by machine B. Give your answer to 3 significant figures.

Mission check

?1 in 100 people are ill. A 90% accurate test says you are. What is the chance you are ill?

Mission 5

Can asking 1,000 people tell you what a whole country thinks?

Make a guess: Can asking 1,000 people tell you what a whole country thinks?

Statistics · sampling and types of data

Populations and samples

The population is the whole group you want to know about. A sample is the part of it you actually collect data from.
Samples are quicker and cheaper; sometimes testing destroys the item (testing how long batteries last).
You use the sample to infer (estimate) facts about the population.
Limitations: a sample gives only an estimate, and a different sample would give slightly different results.
Higher: inferring from a sample is Higher tier for CCEA (Unit M3).

Statistics · sampling and types of data

Random samples and bias

In a simple random sample, every member of the population has an equal chance of being chosen.
Method: number every member, then use a random number generator (or names from a hat); ignore repeats.
A sample is biased if it is likely to misrepresent the population, e.g. asking people at a gym how often they exercise.
Larger random samples give more reliable estimates. A larger biased sample is still biased.

Sort it

?Is each sampling method random or likely to be biased?

Tap a statement, then tap the group it belongs in. Then press Check.

Worked example Statistics · sampling and types of data

Estimating a population from a sample (Higher)

Question

A wood has 900 trees. In a random sample of 40 of them, 14 are oaks. Estimate the number of oaks in the wood.

1
Find the proportion in the sample: $\frac{14}{40} = 0.35$ 14/40 = 0.35
2
Assume the sample represents the wood, so the same proportion applies to all 900 trees
3
Multiply: $0.35 \times 900 = 315$ 0.35 × 900 = 315

Answer

about 315 oaks

⚠️ Watch out

this is an estimate; it is only sensible because the sample is random and not too small.

Calculate

?A school has 1,200 students. In a random sample of 80 students, 28 cycle to school. Estimate the number of students at the school who cycle to school.

Statistics · sampling and types of data

Types of data

Qualitative (categorical) data are words or categories: favourite sport, eye colour.
Quantitative data are numbers. Discrete data are counted and take separate values (number of siblings). Continuous data are measured and can take any value in a range (height, time, mass).
The type of data decides which charts and averages make sense.
Primary data you collect yourself; secondary data were collected by someone else (a website, a census).

Sort it

?Classify each kind of data.

Tap a statement, then tap the group it belongs in. Then press Check.

Statistics · sampling and types of data

Designing a questionnaire

A good question is clear, not leading ("Don't you agree that…?") and not too personal.
Response boxes must cover every possible answer and not overlap. Give a time frame ("per week").
A survey tests a hypothesis, a statement data can check. The handling data cycle: plan, collect, process, interpret.
A data collection sheet has a row for each category or group, a tally column and a frequency column.

Worked example Statistics · sampling and types of data

Stratified sampling (Higher)

Question

A school has 1,200 students: 360 in Year 8, 300 in Year 9, 280 in Year 10 and 260 in Year 11. Choose a stratified sample of 60, in proportion to the size of each year group.

1
Find the sampling fraction: $\frac{60}{1200} = \frac{1}{20}$ 60/1200 = 1/20
2
Multiply each group by it: $360 \times \frac{1}{20} = 18$ 360 × 1/20 = 18, $300 \times \frac{1}{20} = 15$ 300 × 1/20 = 15
3
Continue: $280 \times \frac{1}{20} = 14$ 280 × 1/20 = 14, $260 \times \frac{1}{20} = 13$ 260 × 1/20 = 13
4
Check the total: $18 + 15 + 14 + 13 = 60$ 18 + 15 + 14 + 13 = 60

Answer

18, 15, 14 and 13 students from Years 8, 9, 10 and 11, then chosen at random within each year

⚠️ Watch out

15 from each year ignores the group sizes, so the large Year 8 would be under-represented.

Mission check

?Can asking 1,000 people tell you what a whole country thinks?

Mission 6

Same data, two headlines: "soaring" and "flat". How?

Make a guess: Same data, two headlines: "soaring" and "flat". How?

Statistics · charts and diagrams

Tally charts, frequency tables and pictograms

A tally chart records data as you collect it, in groups of five strokes.
A frequency table lists each category or value with its frequency (how many). A two-way table sorts by two features at once.
A pictogram uses a symbol for a fixed number of items, shown in a key: if one car symbol = 4 cars, then $3 \frac{1}{2}$ 31/2 symbols = 14 cars.
Part-symbols must be easy to read: halves or quarters.

Statistics · charts and diagrams

Bar charts: single, dual and composite

A bar chart shows categorical or discrete data: equal widths, gaps between bars, and an even frequency scale from 0.
A dual bar chart puts bars for two groups side by side, with a key.
A composite (stacked) bar chart stacks the groups in one bar: its total height is the total frequency.
A vertical line chart uses thin lines instead of bars, for ungrouped discrete data such as goals scored.
Check what each small square is worth first.

Calculate

?The dual bar chart shows the number of library books borrowed by Year 7 and Year 8 from Monday to Thursday. How many more books did Year 8 borrow than Year 7 altogether over the four days?

Worked example Statistics · charts and diagrams

Drawing a pie chart

Question

60 students chose their favourite sport: Football 21, Netball 15, Swimming 9, Tennis 15. Work out the angle for each sector of a pie chart.

1
A pie chart's angles total $36 0^{\circ}$ 360°, so each student gets $36 0^{\circ} \div 60 = 6^{\circ}$ 360° ÷ 60 = 6° (angle $= \frac{frequency}{total} \times 36 0^{\circ}$ = frequency/total × 360°)
2
Multiply each frequency by $6^{\circ}$ 6°: Football $21 \times 6 = 12 6^{\circ}$ 21 × 6 = 126°, Netball $15 \times 6 = 9 0^{\circ}$ 15 × 6 = 90°
3
Carry on: Swimming $9 \times 6 = 5 4^{\circ}$ 9 × 6 = 54°, Tennis $15 \times 6 = 9 0^{\circ}$ 15 × 6 = 90°
4
Check the total: $126 + 90 + 54 + 90 = 36 0^{\circ}$ 126 + 90 + 54 + 90 = 360°

Answer

Football

12 6^{\circ}

, Netball

9 0^{\circ}

, Swimming

5 4^{\circ}

, Tennis

9 0^{\circ}

⚠️ Watch out

using the frequencies as angles (a

2 1^{\circ}

sector) gives a pie that does not fill the circle. To read a pie chart, reverse it: frequency

= \frac{angle}{36 0 ^{\circ}} \times

total.

Calculate

?A pie chart shows how 180 visitors travelled to a museum. The sector for bus is

7 2^{\circ}

and the sector for car is

13 0^{\circ}

. How many more visitors came by car than by bus? (Non-calculator.)

Statistics · charts and diagrams

Line graphs: time series and frequency polygons

A time series graph plots a quantity against time (horizontal axis), with points joined by straight lines.
Look for the trend (general rise or fall) and any seasonal pattern that repeats each year. Here sales peak every third quarter.
A frequency polygon shows grouped data: plot each frequency at its class midpoint (5, 15, 25, … for 0–10, 10–20, …) and join with straight lines.
Two polygons on the same axes are easy to compare.

Worked example Statistics · charts and diagrams

Reading a stem-and-leaf diagram

Question

An ordered stem-and-leaf diagram shows the ages of 11 people. Key:

3 ∣ 4

means 34. Rows:

1 ∣ 255

,

2 ∣ 138

,

3 ∣ 0444

,

4 ∣ 1

. Find the mode and the range.

1
Use the key: the stem is the tens digit and each leaf a units digit, so the ages are 12, 15, 15, 21, 23, 28, 30, 34, 34, 34, 41
2
Mode: the leaf 4 appears three times on stem 3, so the mode is 34
3
Range: largest $-$ − smallest $= 41 - 12 = 29$ = 41 − 12 = 29

Answer

mode 34 years, range 29 years

⚠️ Watch out

a stem-and-leaf diagram needs ordered leaves and a key. Subtracting the stems only (

4 - 1 = 3

) ignores the leaves.

Statistics · charts and diagrams

How graphs can mislead

Read the vertical axis before the picture. If it does not start at 0 (and has no break mark), small differences look huge.
In chart B the April bar looks 7 times as tall as January's, yet April's 1,030 members are only 3% more than January's 1,000.
Other tricks: unequal steps on a scale, missing labels, pictogram symbols of different sizes, tilted 3-D pie charts.
To say what is misleading, name the feature and its effect.

Sort it

?Is each chart feature fair or misleading?

Tap a statement, then tap the group it belongs in. Then press Check.

Mission check

?Two papers plot the same members (1,000, 1,010, 1,020, 1,030). One says "soaring", the other "flat". How?

Mission 7

A firm's average pay is £60,000, yet 9 in 10 staff earn £25,000. How?

Make a guess: A firm's average pay is £60,000, yet 9 in 10 staff earn £25,000. How?

Statistics · averages and spread

Mean, median, mode and range

Mean: add all the values and divide by how many there are.
Median: put the values in order; the median is the middle one. With an even number of values it is halfway between the middle two. Its position is the $\frac{n + 1}{2}$ (n+1)/2th value.
Mode: the value that occurs most often (there can be more than one, or none).
Range $=$ = largest value $-$ − smallest value. It measures spread: how varied the data are.

Worked example Statistics · averages and spread

Four measures from a list

Question

8 students spent these times (minutes) on homework: 25, 30, 20, 30, 35, 30, 40, 110. Find the mean, median, mode and range.

1
Order the list: 20, 25, 30, 30, 30, 35, 40, 110
2
Mean: $320 \div 8 = 40$ 320 ÷ 8 = 40 minutes
3
Median: $n = 8$ n = 8, so halfway between the 4th and 5th values: $\frac{30 + 30}{2} = 30$ (30 + 30)/2 = 30 minutes
4
Mode: 30 minutes (three times). Range: $110 - 20 = 90$ 110 − 20 = 90 minutes

Answer

mean 40 min, median 30 min, mode 30 min, range 90 min

⚠️ Watch out

110 is an outlier. It lifts the mean to 40, above 6 of the 8 values. Without it the mean is 30. Finding the "middle" of the unordered list gives

\frac{30 + 35}{2} = 32.5

, which is wrong.

Statistics · averages and spread

Averages from a frequency table

In a frequency table each value $x$ x occurs $f$ f times, so each row totals $f x$ fx.

mean = \frac{\sum f x}{\sum f} = \frac{total of all values}{number of values}

Mode: the value with the highest frequency. Median: find its position, $\frac{n + 1}{2}$ (n+1)/2, then use running totals.
Pets owned by 25 people (0 pets: 7, 1: 9, 2: 6, 3: 2, 4: 1): the median is the 13th person, and running totals 7, 16, … show they own 1 pet. The mode is also 1.
Divide by $\sum f$ Σ f, not by the number of rows.

Calculate

?Use the pets table: 0 pets: 7 people, 1 pet: 9, 2 pets: 6, 3 pets: 2, 4 pets: 1. Work out the mean number of pets per person. Give your answer as a decimal.

Statistics · averages and spread

Estimated mean from grouped data

In grouped data you do not know the exact values, so use the midpoint of each class to stand for every value in it.
Estimated mean $= \frac{\sum ( midpoint \times f )}{\sum f}$ = (Σ (midpoint × f))/(Σ f). It is an estimate because the values are not all at the midpoint.
The modal class is the class with the highest frequency.
The median class is the class containing the $\frac{n + 1}{2}$ (n+1)/2th value.
The range can only be estimated: largest possible value minus smallest possible value.

Worked example Statistics · averages and spread

Estimating a mean from a grouped table

Question

40 students recorded their journey time to school,

t

minutes:

0 < t \leq 10

: 8,

10 < t \leq 20

: 14,

20 < t \leq 30

: 12,

30 < t \leq 40

: 6. Estimate the mean and state the modal class.

1
Midpoints: 5, 15, 25, 35
2
Midpoint × frequency: $5 \times 8 = 40$ 5 × 8 = 40, $15 \times 14 = 210$ 15 × 14 = 210, $25 \times 12 = 300$ 25 × 12 = 300, $35 \times 6 = 210$ 35 × 6 = 210
3
Add: $40 + 210 + 300 + 210 = 760$ 40 + 210 + 300 + 210 = 760
4
Divide by the total frequency: $760 \div 40 = 19$ 760 ÷ 40 = 19

Answer

estimated mean 19 minutes; modal class

10 < t \leq 20

⚠️ Watch out

using the upper class limits (10, 20, 30, 40) gives 24 minutes, and dividing by 4 classes instead of 40 students gives 190.

Calculate

?A courier weighs 50 parcels. Mass

m

kg:

0 < m \leq 2

: 9 parcels,

2 < m \leq 4

: 16,

4 < m \leq 6

: 15,

6 < m \leq 10

: 10. The courier claims the mean mass is under 4 kg. Work out an estimate of the mean mass. Give your answer to 2 decimal places.

kg

Statistics · averages and spread

Choosing an average and comparing data

Mean: uses every value, but an outlier (far from the rest) can drag it a long way.
Median: not affected by a few extreme values; best for data like pay or house prices.
Mode: the only average for categorical data, such as favourite colour.
To compare two data sets, compare an average and the range, in context: "P spent less time on average, and their times were more consistent."
An outlier may be an error: check it before removing it.

Quick check

?Homework times: Class P mean 32 minutes, range 15 minutes; Class Q mean 35 minutes, range 60 minutes. Which comparison is correct?

Mission check

?A firm says average pay is £60,000, yet 9 of its 10 staff earn £25,000. How?

Mission 8

Ice-cream sales and sunburn rise together. Does ice cream cause it?

Make a guess: Ice-cream sales and sunburn rise together. Does ice cream cause it?

Statistics · scatter graphs

Scatter graphs and correlation

A scatter graph plots pairs of data (two variables for each item) as points, usually crosses. Each axis has its own scale.
Positive correlation: as one variable increases, the other tends to increase.
Negative correlation: as one increases, the other tends to decrease.
No correlation: no clear pattern.
CCEA says little or no correlation.
Points close to a line show strong correlation; points loosely spread show weak correlation.

Sort it

?Which correlation would you expect?

Tap a statement, then tap the group it belongs in. Then press Check.

Statistics · scatter graphs

Lines of best fit

A line of best fit is a single straight ruled line, drawn by eye, that follows the trend.
It runs across the whole data set with roughly the same number of points on each side. It need not pass through any point or the origin.
Ignore any outlier (a point that does not fit the pattern).
To estimate, go from a known value to the line, then across to the other axis.

Worked example Statistics · scatter graphs

Estimating from a line of best fit

Question

The line of best fit for ice-cream sales passes through (15, 120) and (30, 300), where

x

is temperature in °C and

y

is sales. Estimate the sales on a day at 24 °C.

1
Find the gradient: $\frac{300 - 120}{30 - 15} = \frac{180}{15} = 12$ (300 − 120)/(30 − 15) = 180/15 = 12 more sales per °C
2
Count up from the known point (15, 120): 24 °C is 9 °C warmer
3
Add on: $120 + 9 \times 12 = 120 + 108 = 228$ 120 + 9 × 12 = 120 + 108 = 228

Answer

about 228 ice creams

⚠️ Watch out

on a paper you would read this straight off the line on the graph. Working out

24 \times 12 = 288

forgets that the line does not go through the origin.

Calculate

?Using the same line of best fit, estimate the temperature on a day when 192 ice creams were sold.

°C

Statistics · scatter graphs

Interpolation, extrapolation and outliers

Interpolation: estimating inside the range of the data. Usually reliable if the correlation is strong.
Extrapolation: estimating outside the range of the data. Risky: the trend may not continue.
An outlier may be an error or an unusual case (a day the shop shut early). Leave it out of the line.
A small sample or weak correlation gives only rough estimates.

Quick check

?The ice-cream data covers 12 °C to 31 °C. A student uses the line to predict 480 sales at 45 °C. How reliable is this?

Calculate

?A teacher plots hours of revision (

x

) against test mark (

y

) for 20 students who revised between 1 and 11 hours. Her line of best fit passes through (2, 40) and (10, 80). Estimate the mark for a student who revised for 7 hours.

Statistics · scatter graphs

Correlation does not prove causation

Correlation shows that two variables change together. It does not show that one causes the other.
Often a third factor drives both. Example: over a summer, sales of electric fans and of cold drinks rise together; hot weather causes both.
Sometimes the link is coincidence, especially with small data sets.
To show cause you need more than a scatter graph, such as a controlled experiment. In an exam, say "correlation does not mean causation" and name a possible third factor.

Mission check

?Ice-cream sales and sunburn cases show strong positive correlation. Does ice cream cause sunburn?

Mission 9

Two classes have the same median mark. Were their results the same?

Make a guess: Two classes have the same median mark. Were their results the same?

Statistics · cumulative frequency and histograms

Quartiles and the interquartile range (Higher)

In ordered data the median splits it in half; the lower and upper quartiles (LQ, UQ) split it into quarters.
For $n$ n listed values: LQ is the $\frac{n + 1}{4}$ (n+1)/4th value, median the $\frac{n + 1}{2}$ (n+1)/2th, UQ the $\frac{3 ( n + 1 )}{4}$ (3(n+1))/4th.
Interquartile range (IQR) $=$ = UQ $-$ − LQ: the spread of the middle half, unaffected by outliers.
For 3, 4, 5, 7, 8, 9, 12: LQ is the 2nd value (4), median the 4th (7), UQ the 6th (9): IQR $= 5$ = 5.

Calculate

?The marks of 11 students are 23, 15, 30, 18, 12, 28, 20, 35, 17, 25, 22. Find the interquartile range.

Statistics · cumulative frequency and histograms

Cumulative frequency graphs (Higher)

Cumulative frequency is a running total: how many values are up to each upper class boundary.
Plot each total at the upper class boundary, start with 0 at the lowest boundary, and join with a smooth curve.
For $n$ n values, read across from $\frac{n}{2}$ n/2 (median), $\frac{n}{4}$ n/4 (LQ) and $\frac{3 n}{4}$ 3n/4 (UQ), then down.
For how many are more than a value, read up from it and subtract from $n$ n.

Worked example Statistics · cumulative frequency and histograms

Median and IQR from a cumulative frequency graph (Higher)

Question

The journey times

t

(minutes) of 80 commuters:

0 < t \leq 10

: 6,

10 < t \leq 20

: 14,

20 < t \leq 30

: 20,

30 < t \leq 40

: 20,

40 < t \leq 50

: 20. Draw a cumulative frequency graph and estimate the median and IQR.

1
Running totals: 6, 20, 40, 60, 80
2
Plot (0, 0), (10, 6), (20, 20), (30, 40), (40, 60), (50, 80) and join with a smooth curve
3
Median: read across from $80 \div 2 = 40$ 80 ÷ 2 = 40 to the curve, then down: 30 minutes
4
LQ from $80 \div 4 = 20$ 80 ÷ 4 = 20: 20 minutes. UQ from $\frac{3}{4} \times 80 = 60$ 3/4 × 80 = 60: 40 minutes
5
IQR $= 40 - 20 = 20$ = 40 − 20 = 20 minutes

Answer

median 30 minutes, IQR 20 minutes

⚠️ Watch out

plotting at the class midpoints (5, 15, 25, …) shifts the whole curve left by 5 minutes. Plot at the upper boundaries.

Statistics · cumulative frequency and histograms

Comparing distributions (Higher)

Two data sets can share a median yet be very different. Compare an average (the median) and a spread (the IQR).
A smaller IQR means the middle half of the data is more consistent (closer together).
Write each comparison in context: "The median time for Group A was lower, so on average Group A was quicker. Group A's IQR was smaller, so their times were more consistent."
Use like for like: median with IQR, or mean with range.

Worked example Statistics · cumulative frequency and histograms

Drawing a box plot (Higher)

Question

For the 80 commuters, the shortest journey was 3 minutes and the longest 49 minutes. Draw a box plot.

1
Collect the five values: lowest 3, LQ 20, median 30, UQ 40, highest 49
2
Draw a scale from 0 to 50 minutes
3
Draw the box from the LQ (20) to the UQ (40), with a line at the median (30)
4
Draw whiskers from the box out to the lowest (3) and highest (49) values

Answer

a box plot showing 3, 20, 30, 40, 49 on the scale

⚠️ Watch out

the box length is the IQR (20 minutes), not the range (46 minutes). To compare two groups, draw both box plots on the same scale.

Quick check

?Journey times: Group A median 34 minutes, IQR 12 minutes; Group B median 41 minutes, IQR 20 minutes. Which comparison is correct?

Statistics · cumulative frequency and histograms

Histograms and frequency density (Higher)

A histogram shows continuous data with no gaps between bars. When class widths are unequal, the vertical axis is frequency density.

frequency density = \frac{frequency}{class width}

The area of each bar represents the frequency: $frequency = frequency density \times class width$ frequency = frequency density × class width.
Example: the class $20 < h \leq 30$ 20 < h ≤ 30 has frequency 15 and width 10, so its bar has height $15 \div 10 = 1.5$ 15 ÷ 10 = 1.5.

Calculate

?The histogram shows the heights

h

cm of some plants. The frequency densities are:

0 < h \leq 10

: 1.2,

10 < h \leq 15

: 3.6,

15 < h \leq 20

: 4,

20 < h \leq 30

: 1.5,

30 < h \leq 50

: 0.5. How many plants are taller than 15 cm?

Mission check

?Class X: median 62, LQ 55, UQ 68. Class Y: median 62, LQ 44, UQ 76. Were their results the same?

Recap

Big ideas

Probability runs from 0 (impossible) to 1 (certain); $P (not A) = 1 - P (A)$ P(not A) = 1 − P(A); all mutually exclusive outcomes sum to 1.
Relative frequency estimates probability, and gets more reliable with more trials. Expected number = probability × trials.
A fair coin has no memory: past results never make an outcome "due".
List outcomes systematically; for two events use a grid. P(event) = favourable ÷ total, only when the outcomes are equally likely.
In a two-way table or Venn diagram, find the missing values from the totals first, then divide by the total you are choosing from.
AND (independent events): multiply. OR (mutually exclusive events): add. P(at least one) $= 1 -$ = 1 − P(none).
On a tree diagram, multiply along the branches and add the paths you want; each set of branches adds to 1.
Without replacement, the second-stage probabilities change: one fewer in the total and in the group already taken.
"Given B" means choose only from B: B's total is the denominator.
From a two-way table or Venn diagram, divide the overlap by the total of the given group; to reverse a tree, use expected frequencies of a round number.
A sample is used to estimate facts about a population; the estimate is only as good as the sample.
Random samples avoid bias; bigger random samples are more reliable, but size never cures bias.
Data are qualitative or quantitative; quantitative data are discrete (counted) or continuous (measured).
Stratified samples take each group in proportion to its size.
Choose the chart for the data: pictogram, bar chart (dual or composite) or pie chart for categories; read every value against the scale and key.
Pie chart angle = frequency ÷ total × 360°; reverse it to find a frequency.
A vertical axis that does not start at 0 can make a tiny change look dramatic.
Time series show trends and seasonal patterns; frequency polygons plot frequencies at class midpoints.
Stem-and-leaf diagrams keep every value: ordered leaves and a key.
Mean $= \frac{total}{number of values}$ = total/(number of values); median = middle value of the ordered list; mode = most common; range = largest $-$ − smallest.
From a table, mean $= \frac{\sum f x}{\sum f}$ = (Σ fx)/(Σ f); for grouped data use midpoints, and the answer is an estimate.
Outliers drag the mean, not the median: choose the average that fits the data, and compare an average and a spread in context.
Scatter graphs show positive, negative or no correlation.
Describe the strength too: strong or weak.
A line of best fit is a straight ruled line through the trend; use it to estimate, trusting interpolation more than extrapolation.
Correlation does not mean causation: look for a third factor.
Quartiles split ordered data into quarters; IQR = UQ − LQ measures the spread of the middle half.
Cumulative frequency: plot running totals at upper class boundaries; read across from $\frac{n}{2}$ n/2, $\frac{n}{4}$ n/4, $\frac{3 n}{4}$ 3n/4.
A box plot shows lowest, LQ, median, UQ, highest on a scale; compare box plots on the same scale.
Histograms: frequency density = frequency ÷ class width, and bar area gives frequency.

🏆

Mini-lesson complete!

⭐⭐⭐

You've worked through Handling Data for CCEA GCSE Maths. 🎉

Mission 1A coin lands heads 7 times in a row. Is tails now "due"?
Mission 2Roll two dice and add them. Which total is most likely, and why?
Mission 3Roll a dice four times. Is at least one six better than evens?
Mission 41 in 100 are ill; a 90% accurate test says you are. Chance you're ill?
Mission 5Can asking 1,000 people tell you what a whole country thinks?
Mission 6Same data, two headlines: "soaring" and "flat". How?
Mission 7A firm's average pay is £60,000, yet 9 in 10 staff earn £25,000. How?
Mission 8Ice-cream sales and sunburn rise together. Does ice cream cause it?
Mission 9Two classes have the same median mark. Were their results the same?

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