Module 27 of 46Lesson 2 of 13 · Core Statistics 1 · Populations, samples & types of data
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Core Statistics · Essential understanding 1

Populations, samples & types of data

IB AA SL 4.1with Prof. Kitso
💻 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.
The big idea: We collect data from a small sample so we can learn about a big population. Data is either discrete (counted) or continuous (measured).
By the end of this lesson you can:
  • explain the difference between a population and a sample
  • decide whether data is discrete or continuous
  • use a sample result to estimate something about the population
  • write the interval for a rounded continuous value
💡 AA Paper 1 has no calculator
In AA, Paper 1 is calculator-free, so the by-hand working is what earns the marks there. Use the TI-Nspire steps for Paper 2, and to check your hand answers.
🧑🏾 Imagine this…
You want to know the favourite food of every student at a school of 450 students. Asking all 450 would take days. Instead you ask 50 students and use their answers to make a good guess about everyone. The 450 students are the population. The 50 you asked are the sample.
Population
the WHOLE group you want to know about (e.g. all 450 students at the school).
Sample
the SMALLER group you actually collect data from (e.g. 50 of those students).
Census
collecting data from EVERY member of the population.
Data
the information collected: numbers or words.
Discrete data
data you COUNT. Only certain values are possible: 0, 1, 2, 3… (number of brothers, goals, cars).
Continuous data
data you MEASURE. Any value in a range is possible: 1.62 m, 1.625 m… (height, mass, time).
★ Key points
  • We use a sample because studying the whole population is usually too slow, too expensive or impossible.
  • A good sample lets us estimate a population value (e.g. the population mean).
  • We can only generalise (apply the result to the population) if the sample represents the population.
  • Quick test: Can I count it? → discrete. Do I measure it with a ruler, scale or clock? → continuous.
⚠️ Tricky ones
Shoe size (37, 37.5, 38…) is discrete: only certain sizes exist. Age in whole years is usually treated as discrete, but true age is continuous. Money is discrete (the smallest unit is 1 thebe) but is often treated as continuous because the steps are tiny.
🧪 Try it yourselfDiscrete or continuous?
Sort each one. Ask yourself: do I count it or measure it?

Worked examples

Example 1.1Population or sample?
The head teacher of a school wants to know how many hours per week students spend on homework. There are 450 students in the school. She asks 60 students from Form 5.
  1. (a) Identify the population and the sample.
  2. (b) What percentage of the population is in the sample?
🧠 Think it through
  1. The population is everyone she wants to know about → all 450 students at the school.
  2. The sample is the people she actually asked → the 60 Form 5 students.
  3. Percentage .
🔢 On your TI-Nspire CX II
  1. Press ⌂ on💻 click ⌂ on keypad, 1 and choose 1: Add Calculator.
  2. Type 60/450*100 and press ctrl💻 Ctrl enter ↵💻 Enter. (This gives a decimal; plain enter ↵💻 Enter may give the fraction .)
  3. Write your answer to 3 significant figures.
1.1*DocRAD60450·10013.3333
13.3333 → 13.3% (3 s.f.)
✅ Answer (a) Population: all 450 students. Sample: the 60 Form 5 students. (b) 13.3%
Do you think only asking Form 5 is fair? Form 5 students probably do MORE homework than Form 1 students. We come back to this in Lesson 2 (bias).
Example 1.2Discrete or continuous?
(a) Classify each as discrete or continuous: (i) number of goals in a match (ii) height of a student (iii) shoe size (iv) time to run 100 m (v) number of taxis at a rank (vi) mass of a watermelon.
(b) Six netball matches had these numbers of goals by a school team: 2, 0, 1, 3, 1, 2. Find the total and the mean number of goals.
🧠 Think it through
  1. Ask: count or measure? Goals, shoe sizes, taxis → counted → discrete.
  2. Height, time, mass → measured → continuous.
  3. Total = add them all. Mean .
🔢 On your TI-Nspire CX II
  1. On a Calculator page, type a list using curly brackets: ctrl💻 Ctrl ( gives { }.
  2. Type sum({2,0,1,3,1,2}) enter ↵💻 Enter.
    📍 Where to find it on the TI-Nspire
    sum( is in menu💻 click menu on keypad▸6: Statistics▸3: List Math▸5: Sum of Elements
  3. Type mean({2,0,1,3,1,2}) enter ↵💻 Enter. (Tip: press ▲ to copy the last line and edit it.)
    📍 Where to find it on the TI-Nspire
    mean( is in menu💻 click menu on keypad▸6: Statistics▸3: List Math▸3: Mean
1.1*DocRADsum({2,0,1,3,1,2})9mean({2,0,1,3,1,2})1.5
sum( ) and mean( ) work on any list
✅ Answer (a) Discrete: (i), (iii), (v). Continuous: (ii), (iv), (vi). (b) Total = 9 goals; mean = 1.5 goals per match.
Example 1.3Can we generalise?
A report says: “492 employees from one company in Gaborone took part in a study. Those on flexible hours increased productivity by 5.3%. So every worker in Botswana should have flexible hours.”
  1. (a) Identify the sample and the population.
  2. (b) Is it reasonable to generalise?
  3. (c) A worker produced 80 items per day before. Using the 5.3% increase, how many would they produce now?
🧠 Think it through
  1. Sample = the 492 employees who took part. Population = all workers in Botswana (who the claim is about).
  2. One company is not like all workplaces (farms, shops, mines, schools…) → not representative.
  3. A 5.3% increase means multiply by 1 + 0.053 = 1.053.
🔢 On your TI-Nspire CX II
  1. On a Calculator page type 80*1.053 enter ↵💻 Enter.
  2. Round sensibly: items are counted, so about 84 items.
1.1*DocRAD80·1.05384.2480·0.0534.24
An increase of 4.24 items
✅ Answer (a) Sample: 492 employees of one company; population: all workers in Botswana. (b) No: one company is not representative of all workers. (c) 84.24 ≈ 84 items per day.
Example 1.4Using a sample to estimate the population
A town has 18 000 households. A survey of 450 randomly chosen households found that 117 have solar panels.
  1. (a) Why use a sample and not a census?
  2. (b) What percentage of the sample has solar panels?
  3. (c) Estimate how many households in the town have solar panels.
🧠 Think it through
  1. A census of 18 000 homes would take a long time and cost a lot.
  2. Sample percentage .
  3. If the sample is representative, the same proportion applies to the town: .
🔢 On your TI-Nspire CX II
  1. Type 117/450*100 enter ↵💻 Enter → 26.
  2. Type 18000*117/450 enter ↵💻 Enter → 4680.
1.1*DocRAD117450·1002618000·1174504680
Same proportion → estimate for the population
✅ Answer (a) Quicker and cheaper. (b) 26%. (c) About 4680 households.
Example 1.5Continuous data that has been rounded
The lengths of 5 leaves were measured to the nearest cm: 7, 9, 8, 10, 6.
  1. (a) A leaf is recorded as 7 cm. Between which values does its true length lie?
  2. (b) Find the smallest and largest possible total length of the 5 leaves.
🧠 Think it through
  1. “Nearest cm” means the true value is within 0.5 cm: 6.5 ≤ L < 7.5.
  2. Smallest total: every leaf is 0.5 cm shorter → total − 5 × 0.5.
  3. Largest total: every leaf is 0.5 cm longer → total + 5 × 0.5.
🔢 On your TI-Nspire CX II
  1. Type sum({7,9,8,10,6}) enter ↵💻 Enter → 40.
    📍 Where to find it on the TI-Nspire
    sum( is in menu💻 click menu on keypad▸6: Statistics▸3: List Math▸5: Sum of Elements
  2. Type 40-5*0.5 and 40+5*0.5.
1.1*DocRADsum({7,9,8,10,6})4040−5·0.537.540+5·0.542.5
✅ Answer (a) 6.5 ≤ L < 7.5 cm. (b) Smallest 37.5 cm, largest 42.5 cm.
Continuous data is ALWAYS rounded when we record it. That is why we say the true value lies in an interval.

Challenge questions

⭐ Challenge 1.6School reading surveyPaper 1 style
There are 800 students at a school. To find out how many books students read per month, a teacher asks 40 students who are in the library on a Friday afternoon. The mean for these 40 students is 2.4 books per month.
  1. (a) State the population.
  2. (b) State whether “number of books read” is discrete or continuous.
  3. (c) Use the sample to estimate the total number of books read per month by all 800 students.
  4. (d) Give one reason why this estimate may be unreliable.
✏️ Try it on paper and on your calculator first. Then check.
Show the full solution
🧠 Think it through
  1. Population = all 800 students.
  2. Books are counted → discrete.
  3. Estimate of total = population size × sample mean = 800 × 2.4.
  4. Who goes to the library on a Friday? Students who like reading! They probably read more than average.
🔢 On your TI-Nspire CX II
  1. Type 800*2.4 enter ↵💻 Enter.
1.1*DocRAD800·2.41920
✅ Answer (a) All 800 students. (b) Discrete. (c) 1920 books. (d) Library users are likely to read more than other students, so the sample is not representative: the estimate is probably too high.
⭐ Challenge 1.7Masses of newborn calvesPaper 2 style
A farmer weighs 8 newborn calves and records their masses to the nearest kg: 31, 28, 35, 30, 33, 29, 34, 32.
  1. (a) Is mass discrete or continuous?
  2. (b) Use your GDC to find the mean mass.
  3. (c) Write down the interval in which the true mass of the calf recorded as 31 kg lies.
  4. (d) Find the upper bound for the total mass of the 8 calves.
  5. (e) The farmer has 120 calves. Estimate their total mass and state one assumption you made.
✏️ Try it on paper and on your calculator first. Then check.
Show the full solution
🧠 Think it through
  1. Mass is measured → continuous.
  2. Mean from the GDC (x̄).
  3. Nearest kg → ±0.5 kg.
  4. Upper bound of total = Σx + 8 × 0.5.
  5. Estimate = 120 × mean. Assumption: the 8 calves are representative of all 120.
🔢 On your TI-Nspire CX II
  1. On page 1.1 Lists & Spreadsheet, move to the very top cell of column A, type the name calf and press enter ↵💻 Enter.
  2. Type the 8 masses down column A, pressing enter ↵💻 Enter after each number.
    1.1*DocRAD◆Acalf3128353033293432BCD12345678A131
  3. Add a Calculator page: ctrl💻 Ctrl doc ▾💻 File menu then 1 (Add Calculator).
  4. Go to menu💻 click menu on keypad▸6: Statistics▸1: Stat Calculations▸1: One-Variable Statistics, press enter ↵💻 Enter. Leave Num of Lists = 1 and press enter ↵💻 Enter.
  5. Set X1 List to calf and Frequency List to 1 (use tab💻 Tab to move between boxes). Press enter ↵💻 Enter. Read off x̄ and Σx.
    1.1*DocRADOneVarcalf,1:stat.results"Title""One-Variable Statistics""x̄"31.5"Σx"252."n"8."MinX"28."MaxX"35.
    Read the values you need
  6. For (d) and (e), type 252+8*0.5 and 120*31.5 on the Calculator page.
    1.1*DocRAD252+8·0.5256120·31.53780
✅ Answer (a) Continuous. (b) x̄ = 31.5 kg. (c) 30.5 ≤ m < 31.5 kg. (d) 256 kg. (e) About 3780 kg, assuming the 8 calves are representative (a random sample) of all 120.
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