Module 30 of 46Lesson 1 of 13 · AA 4.1 — Population, sample and sampling

Course map · Topic 4: Statistics and Probability

4.1 Population, sample and sampling

SL & HL13 worked questions

Statistics starts with data, and data is only useful if it was collected fairly. This subtopic covers the language of statistics (population, sample, random sample, discrete and continuous data), the five sampling techniques, bias and the reliability of data sources, and how to recognise and interpret outliers. Most of it is tested on Paper 1 in words and short calculations.

✏️ Start with the questions. Try each one on paper, then open the solution and compare.

Stuck? Open the notes ↓

✏️ Practice questions

Try each question on paper first. Then open the solution and compare every step.

Question 1●○○ Warm-up✍️ By hand · Paper 1 · no calculator[3 marks]

Kabo collects information about the cattle posts near Serowe. For each of the following, state whether the data is discrete or continuous.
(i) the number of cattle at each post (ii) the mass of each calf at birth (iii) the distance from each post to the nearest borehole (iv) the number of boreholes on each farm (v) the time taken to walk the cattle to water (vi) the shoe size of each herder.

Show the full step-by-step solution

Step 1🧠 ThinkThe one question to ask

Ask yourself: do I count it or do I measure it?
• If you count it (1, 2, 3, …) or it can only take certain values, it is discrete.
• If you measure it with a scale, ruler or clock, it can be any value (like 3.47 kg), so it is continuous.

Step 2✍️ By handGo through the list one by one

(i) number of cattle — counted → discrete

(ii) mass of a calf — measured on a scale → continuous

(iii) distance — measured → continuousA1

(iv) number of boreholes — counted → discrete

(v) time — measured → continuousA1

(vi) shoe size — only sizes like 7, 7½, 8 exist → discreteA1

💡 Tricky one Shoe size is discrete even though it contains halves: there is no shoe size 7.38. Only certain values are possible.

Final answer Discrete: (i), (iv), (vi). Continuous: (ii), (iii), (v).

⚠️ Common mistake Thinking “has a decimal or fraction ⇒ continuous”. Shoe sizes and money (P 12.50) have decimals but are still discrete.

Question 2●○○ Warm-up✍️ By hand · Paper 1 · no calculator[4 marks]

A school in Gaborone has 1 350 students. The head of school wants to know how long students spend on homework each night. She gives a questionnaire to 90 students chosen at random from the school register.

(a) Describe the population and the sample.[2]
(b) Explain what “chosen at random” means here.[1]
(c) Give one reason why she uses a sample instead of asking all 1 350 students.[1]

Show the full step-by-step solution

Part (a)

Step 1🧠 ThinkWho do we want to know about? Who did we actually ask?

The population is the whole group she wants to learn about. The sample is the smaller group she actually asks.

Step 2✍️ By handWrite both clearly, with numbers

Population: all 1 350 students at the schoolA1

Sample: the 90 students who receive the questionnaireA1

Part (b)

Step 1✍️ By handRandom = equal chance

Every student on the register has the same chance of being chosen.A1

Part (c)

Step 1✍️ By handWhy not ask everyone?

Asking all 1 350 students (a census) would take too much time and cost too much; a sample is quicker and cheaper.R1

Final answer (a) Population: all 1 350 students; sample: the 90 chosen students. (b) Each student has an equal chance of being selected. (c) A census would be too slow/expensive — a sample is quicker and cheaper.

Question 3●○○ Warm-up✍️ By hand · Paper 1 · no calculator[5 marks]

Name the sampling technique used in each case.
(a) Neo stands at the entrance of a mall in Francistown and interviews the first 40 people who walk past.
(b) A list of 600 members of a sports club is numbered 1–600. A computer chooses 50 different numbers at random.
(c) A factory checks every 25th bottle that comes off the production line, starting with bottle number 11.
(d) A researcher must interview 30 men and 30 women. She stops people in the street until she has 30 of each.
(e) A university has 2 000 science students and 3 000 arts students. A random sample of 20 science students and 30 arts students is taken.

Show the full step-by-step solution

Step 1🧠 ThinkLook for the key clue in each description

• “Easiest people / whoever walks past” → convenience.
• “Every member equally likely, random numbers” → simple random.
• “Every

k

-th one from a list or a line” → systematic.
• “Fixed number from each group, but not chosen randomly” → quota.
• “Groups sampled in proportion and randomly” → stratified.

Step 2✍️ By handMatch each description

(a) Convenience — the first people who happen to walk pastA1

(b) Simple random — every member has an equal chanceA1

(d) Quota — 30 of each group, but whoever she meetsA1

(e) Stratified —

\dfrac{2000}{5000}\times50 = 20

and

\dfrac{3000}{5000}\times50 = 30

, chosen randomlyA1

✅ Check it In (e) the proportions match the population: science is

40\%

of the students and

40\%

of the sample. That is what makes it stratified.

Final answer (a) convenience (b) simple random (c) systematic (d) quota (e) stratified.

⚠️ Common mistake Writing “stratified” for (d). Quota sampling has groups too, but the people inside each group are not chosen at random.

Question 4●●○ Standard✍️ By hand · Paper 1 · no calculator[4 marks]

A mobile network company has 480 employees in four offices.

Office	Gaborone	Francistown	Maun	Kasane
Employees	216	132	84	48

The company wants a stratified sample of 40 employees.

(a) Find the number of employees that should be chosen from each office.[3]
(b) Describe how the employees from the Maun office should be chosen.[1]

Show the full step-by-step solution

Part (a)

Step 1🧠 ThinkWhat does “stratified” ask us to do?

Each office should get the same share of the sample as it has of the company. Gaborone has

\dfrac{216}{480}

of all the workers, so it gets

\dfrac{216}{480}

of the 40 places.

Step 2✍️ By handUse (stratum ÷ population) × sample size for each office

Gaborone:

\dfrac{216}{480}\times40 = 18

Francistown:

\dfrac{132}{480}\times40 = 11

Maun:

\dfrac{84}{480}\times40 = 7

Kasane:

\dfrac{48}{480}\times40 = 4

✅ Check it

18 + 11 + 7 + 4 = 40

✔ — the strata add up to the full sample.

Part (b)

Step 1✍️ By handRandom inside the stratum

Number the 84 Maun employees 1–84 and use random numbers (e.g. randInt) to pick 7 different ones.R1

Final answer (a) Gaborone 18, Francistown 11, Maun 7, Kasane 4. (b) Choose 7 of the 84 Maun employees at random (number them and use a random number generator).

Question 5●●○ Standard✍️ By hand · Paper 1 · no calculator[4 marks]

A supermarket in Maun has a list of the 840 customers who used its loyalty card last month. The manager wants a systematic sample of 35 customers.

(a) Find the sampling interval.[1]
(b) The first customer chosen is number 9 on the list. Write down the numbers of the next four customers chosen, and the number of the last customer chosen.[2]
(c) State one advantage of systematic sampling.[1]

Show the full step-by-step solution

Part (a)

Step 1🧠 ThinkThe interval tells you how far to jump each time

We want 35 people spread evenly through a list of 840. Divide to find the jump size

k

Step 2✍️ By hand

k = \dfrac{840}{35} = 24

Part (b)

Step 1🧠 ThinkStart at 9 and keep adding 24

The start (9) should be chosen at random from 1 to 24. After that, every 24th customer is taken.

Step 2✍️ By hand

9,\ 33,\ 57,\ 81,\ 105,\ \ldots

Last one: the 35th number is

9 + 34\times24

= 9 + 816 = 825

✅ Check it

825 \le 840

✔ and the next jump,

849

, would be past the end of the list — so exactly 35 customers are chosen.

Part (c)

Step 1✍️ By hand

It is quick and easy, and spreads the sample evenly through the whole list.R1

Final answer (a)

24

(b) 33, 57, 81, 105; last customer is number 825. (c) Quick/simple and spreads the sample evenly across the list.

Question 6●●○ Standard✍️ By hand · Paper 1 · no calculator[4 marks]

A junior school in Molepolole has

x

boys and

360

girls. A stratified sample of

50

learners, by gender, contains

20

boys.

(a) Write down the number of girls in the sample.[1]
(b) Find the value of $x$ .[3]

Show the full step-by-step solution

Part (a)

Step 1✍️ By hand

50 - 20 = 30

girlsA1

Part (b)

Step 1🧠 ThinkStratified means “same share”

In a stratified sample each group has the same fraction of the sample as it has of the school. The girls are

\dfrac{30}{50}

of the sample, so they must also be

\dfrac{30}{50}

of the whole school, which has

x + 360

learners.

Step 2✍️ By handMake an equation and solve it

\dfrac{360}{x+360} = \dfrac{30}{50}

30(x+360) = 50\times360

30x + 10\,800 = 18\,000

30x = 7\,200

x = 240

✅ Check it School:

240 + 360 = 600

. Boys' share:

\dfrac{240}{600}\times50 = 20

✔

Final answer (a) 30 girls. (b)

x = 240

⚠️ Common mistake Writing

\dfrac{20}{50} = \dfrac{x}{360}

. The fraction must be out of the whole school,

x + 360

, not out of the girls.

Question 7●●○ Standard✍️ By hand · Paper 1 · no calculator[5 marks]

A Gaborone radio station wants to know what adults in Botswana think about a new plan to increase taxi fares. During a morning show it asks listeners to phone in. Of the 250 callers, 82% are against the plan.

(a) Name the type of sampling used.[1]
(b) Give two reasons why this sample may be biased.[2]
(c) Suggest a better sampling method and describe how it could be carried out.[2]

Show the full step-by-step solution

Part (a)

Step 1🧠 ThinkWho ended up in the sample?

Only people who happened to be listening and chose to phone. Nobody was picked at random — the sample is simply whoever was easiest to get.

Step 2✍️ By hand

Convenience sampling (a self-selected, volunteer sample)A1

Part (b)

Step 1🧠 ThinkBias = the sample is not like the population

Think about the kind of person who phones a radio show at that time. Are they typical of all adults in Botswana?

Step 2✍️ By hand

People with strong feelings (often people who are angry about the plan) are more likely to call.R1

Only listeners of that one station, at that time of day, in that area can take part — e.g. people at work, or outside the broadcast area, are left out.R1

Part (c)

Step 1✍️ By handMake it random and representative

Stratified sampling: split adults by district (or age group) and take a sample from each group in proportion to its population,M1

choosing people inside each group at random (e.g. random numbers from the voters' roll).A1

💡 Remember A large sample does not fix bias. 250 callers sounds like a lot, but if they are the wrong kind of people the result is still unreliable.

Final answer (a) Convenience (self-selected). (b) e.g. callers have strong opinions; only listeners of one station at one time are included. (c) e.g. a stratified random sample by district/age, chosen randomly from a full list of adults.

Question 8●●○ Standard✍️ By hand · Paper 1 · no calculator[5 marks]

The numbers of hours that 11 students spent on social media in one week are:

12,\ 15,\ 9,\ 14,\ 18,\ 11,\ 13,\ 16,\ 35,\ 10,\ 14

(a) Find the median and the interquartile range.[3]
(b) Show that the data contains exactly one outlier.[2]

Show the full step-by-step solution

Part (a)

Step 1🧠 ThinkFirst put the numbers in order

Quartiles only make sense when the data is in order from smallest to largest. There are 11 values, so the median is the

\dfrac{11+1}{2} = 6

th value.

Step 2✍️ By handOrder, then split into halves

9,\ 10,\ 11,\ 12,\ 13,\ \boxed{14},\ 14,\ 15,\ 16,\ 18,\ 35

Median

= 14

Lower half:

9, 10, \underline{11}, 12, 13

→

Q_1 = 11

Upper half:

14, 15, \underline{16}, 18, 35

→

Q_3 = 16

\text{IQR} = 16 - 11 = 5

M1A1

✅ Check it At home you can check this on the GDC with One-Variable Statistics (Q₁X = 11, MedianX = 14, Q₃X = 16). In Paper 1 the ordering by hand is the method.

Part (b)

Step 1🧠 ThinkBuild the two “fences”

A value is an outlier if it is more than

1.5\times\text{IQR}

beyond the box. So we find a lower fence and an upper fence and see if any value lies outside them.

Step 2✍️ By hand

1.5\times5 = 7.5

Lower fence:

11 - 7.5 = 3.5

— no value is below

3.5

Upper fence:

16 + 7.5 = 23.5

35 \gt 23.5

, and every other value is

\le 18

, so

35

is the only outlierA1

Final answer (a) Median

= 14

hours, IQR

= 5

hours. (b) Fences are 3.5 and 23.5; only

35 \gt 23.5

, so 35 is the single outlier.

⚠️ Common mistake Including the median in both halves when

n

is odd. IB (and the GDC) leaves the median out of the halves.

Question 9●●○ Standard🖩 GDC · Paper 2[4 marks]

Mr Kgosi's class has 32 students, numbered 1 to 32 on the class list. He wants to choose a simple random sample of 6 students to test a new app.

(a) Describe how he could use his GDC to choose the sample.[2]
(b) His GDC gives 17, 4, 29, 11, 4, 23. Explain what he should do next.[1]
(c) Write down the probability that a particular student, Onalenna, is in the sample.[1]

Show the full step-by-step solution

Part (a)

Step 1🧠 ThinkRandom numbers = equal chance for everyone

Each student already has a number. If the calculator picks numbers at random, every student has the same chance — that is exactly a simple random sample.

Step 2🖩 On the GDCrandInt(lowest, highest, how many)

randInt(1,32,6)›enter💻 Enter

📍 Where to find it on the TI-Nspire

randInt(is inmenu💻 click menu on keypad›5: Probability›4: Random›2: Integer

On a Calculator page type randInt(1,32,6) (or find it under menu → Probability → Random). It gives 6 random whole numbers from 1 to 32. Your numbers will be different — they are random.

One possible output

Step 3✍️ By hand

Number the students 1–32, generate random integers from 1 to 32,M1

and choose the students with those numbers until 6 different students are chosen.A1

Part (b)

Step 1✍️ By handA repeat is not a new student

Student 4 appears twice, so ignore the repeat and generate another number until there are 6 different students.R1

Step 2🖩 On the GDCGet one more number

Another random number (8 here) completes the sample

Part (c)

Step 1✍️ By hand

P(\text{Onalenna chosen}) = \dfrac{6}{32} = \dfrac{3}{16}

Final answer (a) Number students 1–32 and use randInt(1,32,6) to pick 6 different numbers. (b) Discard the repeated 4 and generate another number. (c)

\dfrac{3}{16}\ (=0.1875)

Question 10●●● Exam-level✍️ By hand · Paper 2[7 marks]

A senior school has 1 150 students in Forms 4 and 5.

	Boys	Girls
Form 4	312	298
Form 5	271	269

The school wants a stratified sample of 80 students, using the four groups in the table as strata.

(a) Calculate the exact (unrounded) number of students that should come from each group.[2]
(b) Show that rounding each value to the nearest whole number does not give a sample of 80.[2]
(c) Hence suggest a suitable number of students from each group, giving a reason.[2]
(d) Explain why a stratified sample may be better than a simple random sample here.[1]

Show the full step-by-step solution

Part (a)

Step 1🧠 ThinkSame rule as always — but now the answers are not whole numbers

Each group gets

\dfrac{\text{group size}}{1150}\times80

places. Keep the decimals for now.

Step 2✍️ By hand

Form 4 boys:

\dfrac{312}{1150}\times80 = 21.70\ldots

Form 4 girls:

\dfrac{298}{1150}\times80 = 20.73\ldots

Form 5 boys:

\dfrac{271}{1150}\times80 = 18.85\ldots

Form 5 girls:

\dfrac{269}{1150}\times80 = 18.71\ldots

Part (b)

Step 1✍️ By hand

Rounded:

22,\ 21,\ 19,\ 19

22 + 21 + 19 + 19 = 81 \ne 80

Part (c)

Step 1🧠 ThinkWe need to take one place away — from which group?

Take it from the group whose number was rounded up the most unfairly — the one with the smallest decimal part among those rounded up. 21.70 was pushed up by 0.30, more than 20.73 (+0.27), 18.85 (+0.15) or 18.71 (+0.29).

Step 2✍️ By hand

Form 4 boys

21.70 \to 21

(largest rounding-up error, 0.30)R1

Sample: Form 4 boys 21, Form 4 girls 21, Form 5 boys 19, Form 5 girls 19 (total 80)A1

✅ Check it

21+21+19+19 = 80

✔. Each number is within 1 of its exact value, so the sample is still proportional.

Part (d)

Step 1✍️ By hand

It guarantees every form and gender is represented in the right proportion; a simple random sample could, by chance, contain too many of one group.R1

Final answer (a) 21.70, 20.73, 18.85, 18.71. (b) 22 + 21 + 19 + 19 = 81. (c) e.g. 21, 21, 19, 19 — reduce Form 4 boys because 21.70 had the largest rounding-up error. (d) Every group is represented in proportion; SRS may over/under-represent a group by chance.

⚠️ Common mistake Just writing 22, 21, 19, 19 without checking the total. Always add your strata back up.

Question 11●●● Exam-level✍️ + 🖩 · Paper 2[9 marks]

Twelve students in Francistown recorded how much they spent on airtime last month, in pula:

45,\ 52,\ 38,\ 60,\ 49,\ 55,\ 41,\ 47,\ 58,\ 50,\ 44,\ 210

(a) Find the mean, the median and the standard deviation of the amounts.[3]
(b) Show that P 210 is an outlier.[2]
(c) The student who spent P 210 also buys airtime to resell it. The survey is about personal spending, so this value is removed. Find the new mean and standard deviation.[2]
(d) Comment on the effect of the outlier on the mean, the median and the standard deviation.[2]

Show the full step-by-step solution

Part (a)

Step 1🧠 ThinkTwelve values — let the GDC do the arithmetic

Type the amounts into a list called a and run One-Variable Statistics. In IB the standard deviation is σx (not sx).

Step 2🖩 On the GDCEnter the data and run One-Variable Statistics

ctrl+doc💻 Ctrl+I›a›enter💻 Enter›45›enter💻 Enter›52›enter💻 Enter›…›ctrl+doc💻 Ctrl+I›menu💻 click menu on keypad›6: Statistics›1: Stat Calculations›1: One-Variable Statistics…›enter💻 Enter

Num of Lists = 1, X1 List = a, Frequency List = 1. Scroll down with ▼ for the quartiles.

Results, part 1

Results, part 2 (scroll down)

Step 3✍️ By hand

Mean

= 62.4

, median

= 49.5

\sigma = 44.9

(pula, 3 s.f.)A1A1A1

Part (b)

Step 1✍️ By handUse the quartiles from the screen

\text{IQR} = 56.5 - 44.5 = 12

Upper fence:

56.5 + 1.5\times12 = 74.5

210 \gt 74.5

, so 210 is an outlierA1

Part (c)

Step 1🖩 On the GDCDelete 210 from the list and run the statistics again

In Lists & Spreadsheet, move to the cell holding 210 and press del. Then repeat One-Variable Statistics.

Without 210: n = 11

Step 2✍️ By hand

New mean

= \text{P }49

, new

\sigma = \text{P }6.59

A1A1

Part (d)

Step 1✍️ By handCompare before and after

The outlier raised the mean from 49 to 62.4 and made the standard deviation about seven times bigger (6.59 → 44.9),R1

but the median hardly changed (49 → 49.5). So the median is the better average when there is an outlier.R1

💡 Remember One extreme value is added into the mean and its distance from the mean is squared in the standard deviation — that is why these two react so strongly. The median only looks at the middle.

Final answer (a) Mean

\approx \text{P }62.4

, median

= \text{P }49.5

\sigma \approx \text{P }44.9

. (b) IQR

= 12

, upper fence

74.5 \lt 210

. (c) Mean

= \text{P }49

\sigma \approx \text{P }6.59

. (d) The outlier greatly increases the mean and the standard deviation; the median is almost unaffected.

⚠️ Common mistake Using sx = 46.9 instead of σx = 44.9. IB uses σx for the standard deviation of a data set.

Question 12●●● Exam-level🖩 GDC · Paper 2[8 marks]

A poultry farmer near Lobatse weighs 16 eggs (in grams):

58.2,\ 61.5,\ 60.1,\ 57.8,\ 63.0,\ 59.4,\ 44.6,\ 60.8,

62.3,\ 58.9,\ 61.1,\ 59.7,\ 74.5,\ 60.4,\ 57.2,\ 62.8

(a) Write down the lower quartile, median and upper quartile.[2]
(b) Determine which values are outliers.[3]
(c) The farmer finds that the value 44.6 was a recording error and should be 64.6. The 74.5 g egg was a genuine double-yolk egg. With the corrected data, determine whether 74.5 is still an outlier, and state whether it should be removed from the data.[3]

Show the full step-by-step solution

Part (a)

Step 1🧠 Think16 values — let the GDC do the ordering

With this many decimals, ordering by hand is slow and risky. Type the data into a list and use One-Variable Statistics.

Step 2🖩 On the GDCEnter the data and run One-Variable Statistics

ctrl+doc💻 Ctrl+I›m›enter💻 Enter›58.2›enter💻 Enter›…›ctrl+doc💻 Ctrl+I›menu💻 click menu on keypad›6: Statistics›1: Stat Calculations›1: One-Variable Statistics…›enter💻 Enter

Name column A m, type all 16 masses, then on a Calculator page choose One-Variable Statistics with X1 List = m, Frequency List = 1.

Q₁ = 58.55, median = 60.25, Q₃ = 61.9

Step 3✍️ By hand

Q_1 = 58.55\text{ g},\ \ \text{median} = 60.25\text{ g},\ \ Q_3 = 61.9\text{ g}

A1A1

Part (b)

Step 1✍️ By handFences

\text{IQR} = 61.9 - 58.55 = 3.35

Lower fence:

58.55 - 1.5\times3.35 = 53.525

Upper fence:

61.9 + 1.5\times3.35 = 66.925

44.6 \lt 53.525

and

74.5 \gt 66.925

: both are outliersA1A1

✅ Check it The next values in from the ends are 57.2 and 63.0 — both inside the fences, so there are exactly two outliers.

Part (c)

Step 1🧠 ThinkFix the error, then test again

Change 44.6 to 64.6 in the list. The quartiles change, so the fences change too — we must recalculate, not reuse the old fences.

Step 2🖩 On the GDCEdit the list and run One-Variable Statistics again

Corrected data: Q₁ = 59.15, Q₃ = 62.55

Step 3✍️ By hand

\text{IQR} = 62.55 - 59.15 = 3.4

; upper fence

= 62.55 + 1.5\times3.4 = 67.65

74.5 \gt 67.65

, so 74.5 is still an outlierA1

It is a genuine value (a real double-yolk egg), so it should stay in the data.R1

Final answer (a)

Q_1 = 58.55

, median

= 60.25

Q_3 = 61.9

(grams). (b) Fences 53.525 and 66.925 ⇒ 44.6 g and 74.5 g are outliers. (c) New fence 67.65; 74.5 is still an outlier but is genuine, so keep it.

⚠️ Common mistake Deleting an outlier just because it is an outlier. Errors are corrected or removed; genuine unusual values are kept.

Question 13●●● Exam-level✍️ By hand · Paper 2[8 marks]

A water utility studies the daily water use (litres) of households in a ward of Maun. A researcher records the use of 40 households whose houses are next to the main tarred road. The summary of her data is:

Minimum	$Q_1$	Median	$Q_3$	Maximum
45	120	160	210	480

(a) State whether daily water use is discrete or continuous.[1]
(b) Determine whether the minimum and the maximum are outliers.[3]
(c) Explain why the researcher's sample may be biased.[1]
(d) The ward has 600 houses on a numbered list. Describe how a systematic sample of 40 houses could be chosen.[3]

Show the full step-by-step solution

Part (a)

Step 1✍️ By hand

Water is measured (with a meter), so it is continuous.A1

Part (b)

Step 1🧠 ThinkWe do not need the raw data — only Q₁ and Q₃

The outlier test only uses the quartiles. Work out the IQR, then the two fences, then compare the minimum and the maximum.

Step 2✍️ By hand

\text{IQR} = 210 - 120 = 90

\ 1.5\times90 = 135

Lower fence:

120 - 135 = -15

. Minimum

45 \gt -15

→ not an outlier

Upper fence:

210 + 135 = 345

. Maximum

480 \gt 345

→ is an outlierA1A1

💡 Remember A negative lower fence just means no low value can ever be an outlier here — water use cannot be negative.

Part (c)

Step 1✍️ By hand

She only chose houses next to the main road (convenience); these may be larger or better-connected homes and not typical of the whole ward.R1

Part (d)

Step 1✍️ By handInterval, random start, then jump

Interval

k = \dfrac{600}{40} = 15

Choose a random start between 1 and 15, e.g. house 6,A1

then take every 15th house: 6, 21, 36, …, up to 591 (40 houses).A1

✅ Check it

6 + 39\times15 = 591 \le 600

✔ — exactly 40 houses.

Final answer (a) Continuous. (b) Fences −15 and 345: the minimum (45) is not an outlier; the maximum (480) is an outlier. (c) Only houses by the main road were chosen (convenience) — not representative of the ward. (d)

k = 15

: random start from 1–15, then every 15th house on the list.

🆘 Stuck on a question? Open the notes, GDC toolkit and common mistakes

💻 Using the TI-Nspire CX Student Software on a laptop

The screens in these notes are drawn exactly as the Student Software shows them. The keypad on the left of the software works like the handheld — click its keys with the mouse.
menu: click menu on the on-screen keypad (or Documents Toolbox ▸ 🔧). The menu numbers and letters are the same as in these notes.
enter = Enter · tab = Tab (next box) · esc = Esc · del = Backspace.
Powers: type ^ (Shift+6), the power, then press → to come back down before typing more. x² = ^2 then →.
Fraction: type / · √: type sqrt( · π: type pi · eˣ: type exp( · store: type :=. For EE (×10ⁿ) and i, click them on the keypad.
New page: Ctrl+I (or the Insert menu) · change page: Ctrl+← / Ctrl+→ · graph entry line: Ctrl+G · var: Ctrl+L.
Radian / Degree: click Settings in the status bar at the bottom of the window. The mode (RAD or DEG) shows in the top bar.
IB exam mode (Press-to-Test): everything in these notes works on the non-CAS TI-Nspire with the IB exam settings. In exam mode there are no dot-product, cross-product, unit-vector or eigenvalue keys and no Diff Eq graphs — the notes show the allowed way (type the components, use polyRoots on the characteristic equation, do Euler steps on a Calculator page).
If you type nDeriv(, the software shows it as centralDiff( — it is the same command.

🎯 What you must know

Population = every member of the group you want to know about. Sample = the part you actually collect data from. A random sample gives every member of the population an equal chance of being chosen.
Discrete data is counted or can only take certain values (number of goats, shoe size). Continuous data is measured and can take any value in a range (height, mass, time).
Simple random: equal chance for everyone (random numbers). Systematic: every $k$ -th member of a list, $k = \dfrac{\text{population size}}{\text{sample size}}$ , with a random start.
Stratified: split into groups (strata) and sample each group in proportion, at random: $\dfrac{\text{stratum size}}{\text{population size}}\times\text{sample size}$ .
Quota: fixed numbers from each group, but not chosen at random. Convenience: whoever is easiest to reach — quick, but likely to be biased.
Bias: the sample does not fairly represent the population. Reliability of a source: ask who collected it, how, how big the sample was and whether they had a reason to want a particular result.
Outlier: a value below $Q_1 - 1.5\times\text{IQR}$ or above $Q_3 + 1.5\times\text{IQR}$ . Correct or remove it only if it is an error; genuine unusual values stay in the data.
An outlier pulls the mean and the standard deviation a lot, but hardly moves the median or the IQR.

Interquartile range

\text{IQR} = Q_3 - Q_1

📘 In the formula booklet

Outlier test

x \lt Q_1 - 1.5\times\text{IQR}\

\ x \gt Q_3 + 1.5\times\text{IQR}

🧠 Learn this one

Stratum sample size

\dfrac{\text{stratum size}}{\text{population size}}\times\text{sample size}

🧠 Learn this one

Systematic sampling interval

k = \dfrac{\text{population size}}{\text{sample size}}

🧠 Learn this one

🖩 Your GDC toolkit for this subtopic

Follow the steps on your own TI-Nspire CX II. The screens show what you should see.

Type data into a list (Lists & Spreadsheet page): name the column, then enter values

ctrl+doc💻 Ctrl+I›x›enter💻 Enter

Press ctrl then doc to add a new page.
Choose x.
Press enter.

Quartiles, mean, standard deviation of a list (One-Variable Statistics)

menu💻 click menu on keypad›6: Statistics›1: Stat Calculations›1: One-Variable Statistics…

Start on a Calculator page (ctrl doc adds one).
Press menu.
Choose 6: Statistics.
Choose 1: Stat Calculations.
Choose 1: One-Variable Statistics….
A box opens. Fill in each field (press tab to move between fields), then press enter. The results appear on the screen.

The menu path

Results, part 1

Results, part 2 (scroll down)

Worked example: Question 11 — Enter the data and run One-Variable Statistics

Num of Lists = 1, X1 List = a, Frequency List = 1. Scroll down with ▼ for the quartiles.

Random whole numbers for a simple random sample (Calculator page)

randInt(1,32,6)›enter💻 Enter

📍 Where to find it on the TI-Nspire

randInt(is inmenu💻 click menu on keypad›5: Probability›4: Random›2: Integer

Type randInt(1,32,6).
Press enter.

One possible output

Worked example: Question 9 — randInt(lowest, highest, how many)

On a Calculator page type randInt(1,32,6) (or find it under menu → Probability → Random). It gives 6 random whole numbers from 1 to 32. Your numbers will be different — they are random.

⚠️ Where students lose marks

Calling quota sampling “stratified”. Both use groups, but only stratified sampling picks people at random inside each group.
Stratum sizes that do not add up to the sample size after rounding. Always add them back up and adjust.
Saying a value is an outlier because it “looks big”. Show $Q_3 + 1.5\times\text{IQR}$ (or $Q_1 - 1.5\times\text{IQR}$ ) and compare.
Calling shoe size or age in whole years continuous. Counted / only certain values → discrete.
Thinking a large sample removes bias. A big biased sample is still biased.

IB DP Mathematics: Analysis and Approaches (AA) — SL and HL AA 4.1 — Population, sample and sampling

IB DP Mathematics: Analysis and Approaches (AA) — SL and HL

4.1 Population, sample and sampling

✏️ Practice questions

🎯 What you must know

🖩 Your GDC toolkit for this subtopic

⚠️ Where students lose marks