Module 33 of 47Lesson 1 of 8 · HL · AI 4.12 — Designing data collection, reliability and validity

Course map · Topic 4: Statistics and Probability

4.12 Designing data collection, reliability and validity

HL only11 worked questions

Good statistics start before any calculation: with fair questions, sensible variables, sensible categories and measurements you can trust. This subtopic is about planning data collection and judging whether a test or questionnaire is reliable (consistent) and valid (measures what it claims to).

✏️ 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[6 marks]

Mpho is writing a questionnaire for students at a school in Gaborone about school transport. For each question below, state what is wrong with it and write an improved version.

(a) “Don't you agree that the school bus is too expensive?”[2]
(b) “How long and how safe is your journey to school?”[2]
(c) “Do you travel to school by combi often?”[2]

Show the full step-by-step solution

Step 1🧠 ThinkWhat makes a good question?

A good question is fair (does not push you to an answer), asks one thing, and uses words that mean the same to everyone. Tick-box answers are easiest to count.

Part (a)

Step 1✍️ By handLeading question

“Don't you agree…” and “too expensive” push the student towards saying yes. It is a leading question.R1

Better: “How would you rate the price of the school bus? □ Very cheap □ Cheap □ Fair □ Expensive □ Very expensive”A1

Part (b)

Step 1✍️ By handTwo questions in one

It asks about time and safety together (double-barrelled). A student cannot answer both with one answer.R1

Better: split it. “How many minutes does your journey to school usually take? □ 0–15 □ 16–30 □ 31–45 □ more than 45” and a separate safety question.A1

Part (c)

Step 1✍️ By handVague word

“Often” means different things to different people (twice a week? every day?).R1

Better: “On how many school days last week did you travel to school by combi? □ 0 □ 1 □ 2 □ 3 □ 4 □ 5”A1

💡 Remember Notice the improved versions give non-overlapping tick boxes that cover every possible answer. That makes the data easy to put into a table later.

Final answer (a) Leading — use a neutral rating scale. (b) Double-barrelled — split into a time question and a safety question. (c) Vague (“often”) — ask for a number of days in a fixed week, with tick boxes.

Question 2●○○ Warm-up✍️ By hand · Paper 1[5 marks]

Define the terms reliability and validity. Then, for each situation, state whether the measurement is reliable, valid, both or neither.
(i) A bathroom scale always shows a mass exactly 2 kg more than the true mass.
(ii) A stopwatch gives the correct time for a 100 m race every time it is used.
(iii) A “maths ability” test only contains spelling questions, and students' scores change a lot when they sit it twice.

(a) Define reliability and validity.[2]
(b) Classify situations (i), (ii) and (iii).[3]

Show the full step-by-step solution

Part (a)

Step 1✍️ By handDefinitions

Reliability: the measurement is consistent — repeating it under the same conditions gives the same (or very similar) results.A1

Validity: the measurement actually measures what it is supposed to measure.A1

Part (b)

Step 1🧠 ThinkAsk two separate questions

For each case ask: (1) Does it give the same answer every time? → reliable. (2) Is that answer the right thing? → valid.

Step 2✍️ By handClassify

(i) Same reading every time, but always wrong by 2 kg → reliable, not valid.A1

(ii) Same and correct every time → reliable and valid.A1

(iii) Wrong content and inconsistent scores → neither.A1

Final answer (a) Reliability = consistent results when repeated; validity = measures what it claims to. (b) (i) reliable, not valid; (ii) both; (iii) neither.

Question 3●○○ Warm-up✍️ By hand · Paper 1[4 marks]

State which of the following is being checked in each situation: test-retest reliability, parallel forms reliability, content validity or criterion-related validity.
(i) A teacher writes Version A and Version B of a fractions quiz and gives both versions to the same class on the same day.
(ii) A group of IB teachers checks that a revision test has questions on every subtopic of Topic 4.
(iii) Neo's class sits the same personality questionnaire in March and again in April.
(iv) A new 5-minute fitness test is compared with a full laboratory fitness test that is known to be accurate.

Show the full step-by-step solution

Step 1🧠 ThinkKey words to look for

Same test, two times → test-retest. Two versions → parallel forms. Covers all the content → content validity. Compared with a trusted measure → criterion-related validity.

Step 2✍️ By handMatch each situation

(i) two equivalent versions, same students → parallel forms reliabilityA1

(ii) checking coverage of the syllabus → content validityA1

(iii) same questionnaire repeated later → test-retest reliabilityA1

(iv) compared with an accepted accurate test → criterion-related validityA1

💡 Remember In test-retest, the time gap matters: too short and students remember their answers; too long and the thing being measured may really have changed.

Final answer (i) parallel forms reliability; (ii) content validity; (iii) test-retest reliability; (iv) criterion-related validity.

Question 4●●○ Standard✍️ By hand · Paper 1[6 marks]

Onalenna wants to investigate whether the number of hours students spend on homework affects their end-of-year maths mark. The school gives her a spreadsheet with these columns for 600 students: student ID, name, gender, age, home village, hours of homework per week, number of siblings, maths mark, favourite subject. About 40 students have no homework value recorded.

(a) State the two variables she should select, identifying the independent variable.[2]
(b) Suggest what she should do with the 40 incomplete records, and why.[2]
(c) She also considers collecting extra data with an online poll on the school's social media page. Explain why this data may not be suitable.[2]

Show the full step-by-step solution

Step 1🧠 ThinkOnly take what answers the question

The question is about homework hours and maths marks. Names, ID numbers, villages and favourite subjects do not help answer it — and names are private.

Part (a)

Step 1✍️ By handRelevant variables

Independent variable: hours of homework per weekA1

Dependent variable: maths markA1

Part (b)

Step 1✍️ By handMissing data

Leave the 40 records out of the analysis (she cannot use a point with only one coordinate).A1

560 complete records is still a large sample, but she should mention it and check the missing students are not all from one group (that could bias the results).R1

Part (c)

Step 1✍️ By handSelf-selected sample

Only students who follow the page and choose to answer will reply (self-selection), so the sample is not random/representative.R1

Students could also answer more than once or exaggerate their homework hours, so the data may not be reliable.R1

Final answer (a) Independent: hours of homework per week; dependent: maths mark. (b) Exclude the 40 incomplete records (still 560 left) and check they are not from one group. (c) Self-selected, non-representative sample; possible repeat or dishonest answers.

Question 5●●○ Standard🖩 GDC · Paper 2[4 marks]

To check the test-retest reliability of a 20-mark mental arithmetic test, Kabo gave it to 8 students, and then gave the same test to the same students two weeks later.

Student	A	B	C	D	E	F	G	H
1st score	12	15	9	18	14	7	16	11
2nd score	13	14	10	17	15	8	17	10

(a) Find Pearson's product-moment correlation coefficient $r$ between the two sets of scores.[2]
(b) Comment on the reliability of the test.[2]

Show the full step-by-step solution

Step 1🧠 ThinkHow do we measure “consistent”?

If the test is reliable, a student who scored high the first time scores high again, and a low scorer stays low. That is a strong positive correlation between the two lists. So we find

r

Part (a)

Step 1🖩 On the GDCEnter the lists and run Linear Regression (mx+b)

menu💻 click menu on keypad›4: Statistics›1: Stat Calculations›3: Linear Regression (mx+b)…

On a Lists & Spreadsheet page name column A t1 and column B t2. In the dialog choose X List: t1, Y List: t2. We only need the value of

r

1st and 2nd scores

r = 0.961

Step 2✍️ By handWrite down r

r = 0.961

(3 s.f.)A2

Part (b)

Step 1✍️ By handInterpret

r = 0.961

is a strong positive correlation: students scored almost the same on both occasions.R1

So the test appears to be reliable (test-retest).A1

💡 Remember Reliability says nothing about validity: the test could consistently measure the wrong thing.

Final answer (a)

r = 0.961

(b) Strong positive correlation, so the test is reliable (consistent results on retesting).

Question 6●●○ Standard🖩 GDC · Paper 2[4 marks]

A language school wrote two versions, Form A and Form B, of a 50-mark Setswana vocabulary test. To check parallel forms reliability, 10 learners sat both forms on the same day.

Learner	1	2	3	4	5	6	7	8	9	10
Form A	34	28	41	22	37	30	25	39	33	27
Form B	30	35	38	29	31	24	33	36	27	32

(a) Find the correlation coefficient $r$ .[2]
(b) Hence state, with a reason, whether the two forms can be used as equivalent tests.[2]

Show the full step-by-step solution

Step 1🧠 ThinkWhat should happen if the forms are equivalent?

Two equally good versions should rank the learners in the same way, so we expect a strong positive

r

. If

r

is weak, a learner's mark depends on which version they happened to get — not fair.

Part (a)

Step 1🖩 On the GDCLists A and B, then Linear Regression (mx+b)

menu💻 click menu on keypad›4: Statistics›1: Stat Calculations›3: Linear Regression (mx+b)…

rows 1–9 shown (row 10: 27, 32)

r = 0.389

Step 2✍️ By handWrite down r

r = 0.389

(3 s.f.)A2

Part (b)

Step 1✍️ By handInterpret

r = 0.389

is only a weak positive correlation.R1

The forms do not give consistent results, so they are not reliable as parallel forms and should not be used as equivalent tests.A1

Final answer (a)

r = 0.389

(b) Weak correlation, so the forms are not reliable as parallel forms; they are not equivalent.

⚠️ Common mistake Saying “positive, so reliable”. The correlation must be strong positive.

Question 7●●○ Standard✍️ + 🖩 · Paper 2[6 marks]

A clinic in Maun designs a quick 10-point eye-sight check. To test its criterion-related validity, 10 patients take the quick check and also a full optometrist examination (scored out of 100), which is accepted as accurate.

Quick check	6	9	4	8	7	10	5	8	3	9
Full exam	58	77	41	70	66	85	49	72	35	80

(a) Find the value of $r$ for these data.[2]
(b) Comment on the criterion-related validity of the quick check.[2]
(c) Describe how the clinic could check the content validity of the quick check.[2]

Show the full step-by-step solution

Step 1🧠 ThinkWhat is the “criterion”?

The criterion is the trusted measurement — the full examination. If the quick check is valid, its scores should agree closely with the full exam: strong positive correlation.

Part (a)

Step 1🖩 On the GDCQuick check as x, full exam as y

menu💻 click menu on keypad›4: Statistics›1: Stat Calculations›3: Linear Regression (mx+b)…

rows 1–9 shown (row 10: 9, 80)

r = 0.997

Step 2✍️ By handWrite down r

r = 0.997

Part (b)

Step 1✍️ By handInterpret

Very strong positive correlation with the accepted measure,R1

so the quick check has high criterion-related validity.A1

Part (c)

Step 1✍️ By handContent validity = expert check of what is covered

Ask experts (optometrists) to inspect the items of the quick check,M1

and confirm that it covers all the aspects of eye-sight it claims to measure (e.g. distance, near vision, both eyes).A1

Final answer (a)

r = 0.997

(b) very strong agreement with the trusted exam → valid (criterion-related) (c) experts check that the items cover every aspect of eye-sight it claims to test.

Question 8●●○ Standard✍️ By hand · Paper 1[6 marks]

Write down the number of degrees of freedom for each χ² test.

(a) A die is tested for fairness using the 6 faces.[1]
(b) Data are tested against a Poisson distribution with given mean 2. After combining, there are 7 classes.[1]
(c) Data are tested against a Poisson distribution whose mean is estimated from the data. After combining, there are 5 classes.[1]
(d) Data are tested against a normal distribution with μ and σ both estimated from the data, using 8 classes.[1]
(e) Data are tested against B(4, $p$ ), with $p$ estimated from the data, using the 5 classes 0, 1, 2, 3, 4.[1]
(f) A test for independence uses a contingency table with 3 rows and 4 columns.[1]

Show the full step-by-step solution

Step 1🧠 ThinkThe rule, in words

Start with the number of classes. Take away 1 because the totals must match. Then take away 1 more for every number you had to estimate from the data to get the expected frequencies.

Step 2✍️ By handApply $\nu = k - 1 - m$

(a)

6 - 1 - 0 = 5

(b)

7 - 1 - 0 = 6

(the mean was given)A1

(c)

5 - 1 - 1 = 3

(λ estimated)A1

(d)

8 - 1 - 2 = 5

(μ and σ estimated)A1

(e)

5 - 1 - 1 = 3

(

p

estimated;

n = 4

is known)A1

(f)

(3-1)(4-1) = 6

💡 Remember Always count classes after combining small expected frequencies, not before.

Final answer (a) 5 (b) 6 (c) 3 (d) 5 (e) 3 (f) 6

Question 9●●● Exam-level✍️ + 🖩 · Paper 2[10 marks]

Thato recorded the daily phone screen time (hours) of 45 Form 4 and 50 Form 5 students. He wants to test, at the 5% level, whether screen time is independent of form.

Hours $h$	$0\le h\lt 1$	$1\le h\lt 2$	$2\le h\lt 3$	$3\le h\lt 4$	$4\le h\lt 5$	$5\le h\lt 6$	$6\le h\lt 8$
Form 4	3	9	14	10	6	2	1
Form 5	1	4	10	13	11	6	5

(a) Show that the expected frequency for Form 4 students with $0\le h\lt 1$ is less than 5.[2]
(b) Thato combines the classes into 0–2, 2–3, 3–4, 4–5 and 5–8 hours. Justify this choice.[2]
(c) Write down the hypotheses and the number of degrees of freedom.[2]
(d) Using the combined table, find the p-value and state the conclusion of the test.[4]

Show the full step-by-step solution

Part (a)

Step 1🧠 ThinkExpected frequency of one cell

Row total for Form 4 = 45. Column total for

0\le h\lt 1

3+1 = 4

. Grand total = 95.

Step 2✍️ By handRow × column ÷ total

E = \dfrac{45\times4}{95}

E = 1.89 \lt 5

Part (b)

Step 1🧠 ThinkWhy combine — and why these classes?

Expected frequencies below 5 make the χ² test unreliable. The small ones are at the two ends (very little and very much screen time), so we join each end class to its neighbour: neighbouring classes keep the order of the numbers.

Step 2✍️ By handJustify with the new expected values

Combined observed table: Form 4: 12, 14, 10, 6, 3. Form 5: 5, 10, 13, 11, 11. Column totals 17, 24, 23, 17, 14.

Smallest expected frequency now:

\dfrac{45\times14}{95} = 6.63 \ge 5

Only neighbouring (adjacent) classes were joined, so the categories still make sense as ranges of hours.R1

Part (c)

Step 1✍️ By handHypotheses and ν

H_0

: screen time is independent of form.

H_1

: screen time is not independent of form.A1

\nu = (2-1)(5-1) = 4

Part (d)

Step 1🖩 On the GDCEnter the matrix and run the χ² 2-way Test

menu💻 click menu on keypad›6: Statistics›7: Stat Tests›8: χ² 2-way Test…

On a Calculator page store the combined table as a matrix: type o:=[[12,14,10,6,3][5,10,13,11,11]] (or use the 2×5 matrix template). Then choose χ² 2-way Test and set Observed Matrix: o.

the combined observed table

p = 0.0449

Step 2✍️ By handCompare with 5%

p-value

= 0.0449

0.0449 \lt 0.05

, so reject

H_0

.R1

There is evidence, at the 5% level, that screen time is not independent of form.A1

✅ Check it df on the screen is 4 — matching our

(2-1)(5-1)

. ✔

Final answer (a)

E = 1.89 \lt 5

(b) merging adjacent end classes makes every expected frequency ≥ 5 (smallest 6.63) while keeping meaningful ranges (c)

H_0

: independent;

\nu = 4

(d)

p = 0.0449 \lt 0.05

: reject

H_0

; screen time depends on form.

⚠️ Common mistake Running the test on the original 2×7 table — the GDC will happily give an answer, but it is not valid because several expected frequencies are below 5.

Question 10●●● Exam-level✍️ + 🖩 · Paper 2[11 marks]

Kagiso counts the number of cars

X

arriving at a filling station in Francistown in each of 100 one-minute periods.

$x$	0	1	2	3	4	5	$\ge 6$
Frequency	12	25	28	18	10	5	2

He tests, at the 5% significance level, whether

X

can be modelled by a Poisson distribution.

(a) Assuming that the two periods in the “ $\ge 6$ ” class had exactly 6 cars, show that the mean number of cars per minute is 2.12.[2]
(b) Using $X\sim\text{Po}(2.12)$ , find the expected frequencies for $x = 0, 1, 2, 3, 4$ and for $x \ge 5$ .[3]
(c) Explain why the classes $x = 5$ and $x \ge 6$ were combined, and find the number of degrees of freedom.[3]
(d) Find the p-value and state the conclusion.[3]

Show the full step-by-step solution

Part (a)

Step 1✍️ By handMean from a frequency table

\bar{x} = \dfrac{0\times12+1\times25+2\times28+3\times18+4\times10+5\times5+6\times2}{100}

= \dfrac{212}{100} = 2.12

Part (b)

Step 1🧠 ThinkExpected = 100 × probability

For

x = 0

to 4 use poissPdf. The last class “5 or more” is everything that is left:

100 - (\text{the other five})

, i.e.

100\times(1-\text{poissCdf}(2.12,0,4))

Step 2🖩 On the GDCBuild the expected list with a formula

100×poissPdf(2.12,{0,1,2,3,4})›enter💻 Enter

📍 Where to find it on the TI-Nspire

poissPdf(is inmenu💻 click menu on keypad›5: Probability›5: Distributions›J: Poisson Pdf…

On a Calculator page this gives the first five expected values in one list. Then type 100×(1−poissCdf(2.12,0,4)) for the last class.

expected frequencies

Step 3✍️ By handExpected frequencies

12.0, 25.4, 27.0, 19.1, 10.1 and 6.41 (for

x\ge5

)A1A1A1

Part (c)

Step 1✍️ By handWhy combine, and ν

Separately,

E(x=5) = 4.28

and

E(x\ge6) = 2.13

are both below 5, so they are combined into

x\ge5

(expected 6.41 ≥ 5).R1

6 classes; 1 parameter (λ) estimated from the data:M1

\nu = 6 - 1 - 1 = 4

Part (d)

Step 1🖩 On the GDCχ² GOF with observed and expected lists

menu💻 click menu on keypad›4: Statistics›4: Stat Tests›7: χ² GOF…

Put the observed values 12, 25, 28, 18, 10, 7 in list obs and the expected values in list exp. In the dialog: Observed List: obs, Expected List: exp, Deg of Freedom, df: 4 (the GDC does not know you estimated λ — you must type it).

observed and expected

p = 0.997

Step 2✍️ By handConclusion

p-value

= 0.997

0.997 \gt 0.05

, so do not reject

H_0

.R1

The Poisson distribution is a suitable model for the number of cars per minute.A1

Final answer (a)

\bar{x} = \dfrac{212}{100} = 2.12

(b) 12.0, 25.4, 27.0, 19.1, 10.1, 6.41 (c) expected 4.28 and 2.13 are below 5;

\nu = 4

(d)

p = 0.997 \gt 0.05

: Poisson model is suitable.

⚠️ Common mistake Using df = 6 − 1 = 5. One extra degree of freedom is lost because λ was calculated from the same data.

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

Boitumelo asked 175 people at a mall in Gaborone how they prefer to spend free time.

Age group	Sport	Music	Reading	Total
Under 18	18	12	4	34
18–30	22	16	6	44
31–50	25	30	3	58
Over 50	15	22	2	39
Total	80	80	15	175

(a) Find the expected frequency of people over 50 who prefer reading, and explain why the Reading column must be combined with another column.[3]
(b) Suggest, with a reason, which column Reading should be combined with.[1]
(c) After combining, test at the 5% level whether preference is independent of age group.[4]
(d) Comment on one limitation of the way the data were collected.[1]

Show the full step-by-step solution

Part (a)

Step 1✍️ By handExpected frequency

E = \dfrac{39\times15}{175}

E = 3.34

All four Reading expected values are below 5 (2.91, 3.77, 4.97, 3.34), so the test would not be valid unless Reading is combined.R1

Part (b)

Step 1✍️ By handA sensible partner

Combine Reading with Music to make “Indoor / non-sport activities” — both are quiet, non-sport hobbies, so the new category still has a clear meaning.R1

Part (c)

Step 1🧠 ThinkNew table: Sport vs Music or reading

Under 18: 18, 16. 18–30: 22, 22. 31–50: 25, 33. Over 50: 15, 24. It is a 4×2 table, so

\nu = (4-1)(2-1) = 3

Step 2🖩 On the GDCχ² 2-way Test on the 4×2 matrix

menu💻 click menu on keypad›6: Statistics›7: Stat Tests›8: χ² 2-way Test…

p = 0.567

Step 3✍️ By handHypotheses and conclusion

H_0

: preference is independent of age group;

H_1

: it is not independent.A1

p-value

= 0.567

0.567 \gt 0.05

: do not reject

H_0

.R1

There is not enough evidence that free-time preference depends on age group.A1

Part (d)

Step 1✍️ By handSampling limitation

Only people at a mall were asked (convenience sample), so it may not represent everyone — people who prefer sport may be less likely to be at a mall.R1

Final answer (a)

E = 3.34

; all Reading expected values are below 5 (b) with Music (both non-sport, indoor activities) (c)

\chi^2 = 2.03

\nu = 3

p = 0.567 \gt 0.05

: do not reject

H_0

— no evidence that preference depends on age group (d) convenience sample at a mall, not representative.

🆘 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

Good survey questions are clear, neutral and ask one thing at a time. Avoid leading questions, double-barrelled questions (two questions in one), vague words (“often”, “recently”) and unnecessary sensitive questions. Closed questions with tick boxes are easier to analyse.
Choose relevant variables: only collect and analyse the variables that answer your question; check the data are suitable (enough of them, no bias in who answered, no missing values).
Categorising numerical data for a χ² test: group the data into classes so that every expected frequency is at least 5. If one is below 5, combine it with a neighbouring class (or a sensible similar category) and say why.
Degrees of freedom in a χ² goodness of fit test: $\nu = (\text{number of classes after combining}) - 1 - (\text{number of parameters estimated from the data})$ . Independence test: $\nu = (r-1)(c-1)$ .
Reliability = consistency: the same thing measured again gives (nearly) the same result. Tests: test-retest (same test, same people, two times) and parallel forms (two equivalent versions of the test, same people). Find the correlation $r$ between the two sets of scores: strong positive $r$ means reliable.
Validity = it measures what it is supposed to measure. Content validity: the test covers all the parts of the topic it claims to test (judged by experts / the syllabus). Criterion-related validity: scores agree with an accepted, trusted measure (high correlation with the trusted test).
A measurement can be reliable but not valid (a scale that is always 2 kg too heavy). It cannot be truly valid if it is not reliable.

Degrees of freedom, χ² goodness of fit

\nu = k - 1 - m

(

k

classes,

m

parameters estimated from data)

🧠 Learn this one

Degrees of freedom, χ² test for independence

\nu = (r-1)(c-1)

🧠 Learn this one

Expected frequency (independence)

E = \dfrac{\text{row total}\times\text{column total}}{\text{grand total}}

🧠 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.

Correlation r for reliability / validity (two lists of scores)

menu💻 click menu on keypad›4: Statistics›1: Stat Calculations›3: Linear Regression (mx+b)…

Start on a Lists & Spreadsheet page (ctrl doc adds one).
Press menu.
Choose 4: Statistics.
Choose 1: Stat Calculations.
Choose 3: Linear Regression (mx+b)….
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

1st and 2nd scores

r = 0.961

Worked example: Question 5 — Enter the lists and run Linear Regression (mx+b)

On a Lists & Spreadsheet page name column A t1 and column B t2. In the dialog choose X List: t1, Y List: t2. We only need the value of

r

χ² goodness of fit test (observed and expected lists)

menu💻 click menu on keypad›4: Statistics›4: Stat Tests›7: χ² GOF…

Start on a Lists & Spreadsheet page (ctrl doc adds one).
Press menu.
Choose 4: Statistics.
Choose 4: Stat Tests.
Choose 7: χ² GOF….
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

observed and expected

p = 0.997

Worked example: Question 10 — χ² GOF with observed and expected lists

χ² test for independence (observed matrix on a Calculator page)

menu💻 click menu on keypad›6: Statistics›7: Stat Tests›8: χ² 2-way Test…

Start on a Calculator page (ctrl doc adds one).
Press menu.
Choose 6: Statistics.
Choose 7: Stat Tests.
Choose 8: χ² 2-way Test….
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

the combined observed table

p = 0.0449

Worked example: Question 9 — Enter the matrix and run the χ² 2-way Test

On a Calculator page store the combined table as a matrix: type o:=[[12,14,10,6,3][5,10,13,11,11]] (or use the 2×5 matrix template). Then choose χ² 2-way Test and set Observed Matrix: o.

⚠️ Where students lose marks

Combining classes because the observed frequency is small. The rule is about expected frequencies.
Forgetting to subtract 1 for each parameter (λ, $p$ , μ, σ) estimated from the data in a goodness of fit test.
Mixing up reliability and validity: “gives the same answer twice” is reliability; “measures the right thing” is validity.
Calling a question “biased” without saying how it pushes the answer, and without writing an improved version.
Using a weak or moderate $r$ as proof of reliability. You need a strong positive correlation.

IB DP Mathematics: Applications and Interpretation (AI) — SL and HL HL · AI 4.12 — Designing data collection, reliability and validity

IB DP Mathematics: Applications and Interpretation (AI) — SL and HL

4.12 Designing data collection, reliability and validity

✏️ Practice questions

🎯 What you must know

🖩 Your GDC toolkit for this subtopic

⚠️ Where students lose marks