Population vs. Sample & The Census Decision
In statistics, we seek to understand properties of an entire group without necessarily inspecting every single individual member.
- Population ($N$): The entire collection of individuals, objects, or measurements about which information is desired. Parameters describing a population include $\mu$ (mean) and $\sigma$ (standard deviation).
- Sample ($n$): A selected representative subset of the population used to estimate population characteristics. Statistics describing a sample include $\bar{x}$ (mean) and $s$ (standard deviation).
- Sampling Frame: A comprehensive, up-to-date numbered list of all members of the population from which the sample is drawn.
- Census: Surveying every member of the population ($n = N$).
Pros: Complete accuracy, no sampling error. Cons: Extremely expensive, slow, logistically difficult. - When a Sample is Mandatory:
- Destructive Testing: Testing battery lifespan, light bulb durability, or automotive crash safety destroys the item. Testing every item would destroy all inventory!
- Infinite/Uncountable Populations: Quality testing river water, air pollutants, or wild fish populations.
- Time & Budget Constraints: Election polling across millions of voters.
Probability Sampling Techniques
In a probability sample, every member of the population has a known, non-zero probability of being selected, eliminating researcher bias.
Every individual and every combination of $n$ individuals has an equal chance of being chosen.
Method: Assign every member a number from $1$ to $N$, then use a random number generator to select $n$ unique numbers.
Selecting members at regular fixed intervals from an ordered list: $$\text{Sampling Interval: } k = \frac{\text{Population Size } N}{\text{Sample Size } n}$$ Select a random starting number between $1$ and $k$, then select every $k\text{th}$ person thereafter: $$\text{Sample indices: } r, \; r + k, \; r + 2k, \; r + 3k, \dots$$
Used when the population contains distinct non-overlapping sub-groups (strata) that differ in the variable being studied (e.g., age groups, grade levels, departments). $$\text{Stratum Sample Size } n_i = \frac{\text{Stratum Size } N_i}{\text{Total Population } N} \times \text{Total Sample } n$$ Within each stratum, individuals are selected via Simple Random Sampling. This guarantees fair representation of minority sub-groups.
The population is divided into naturally occurring heterogeneous groups (clusters), such as classrooms or geographical postal codes. A random sample of entire clusters is chosen, and all members within the selected clusters are surveyed.
Non-Probability Sampling Methods
Non-probability methods do not use random chance, making it impossible to determine the probability of selection and introducing substantial risk of bias.
- Convenience Sampling: Selecting subjects who are easiest to access (e.g., interviewing friends or passers-by outside a shopping mall). Highly prone to selection bias.
- Quota Sampling: Setting non-random targets for sub-categories (e.g., interview 20 men and 20 women), but the interviewer chooses whoever is available until the quota is filled.
- Voluntary Response (Self-Selection): Putting up an open invitation (e.g., internet polls, call-in radio shows). Suffer from severe self-selection bias because people with strong negative or extreme feelings are far more likely to participate.
Sources of Bias & Evaluation (Criterion D)
Bias occurs when a study systematically favors certain outcomes over others, undermining statistical validity.
- Selection / Undercoverage Bias: Some members of the population are systematically excluded from the sampling frame (e.g., phone directory surveys excluding cell-phone-only users).
- Non-Response Bias: Selected participants refuse to respond or cannot be contacted, and their opinions systematically differ from those who respond.
- Response / Social Desirability Bias: Participants give false or inaccurate answers to sensitive questions (e.g., illegal behavior, hygiene) to present a favorable image.
- Questionnaire / Wording Bias: Loaded or leading phrasing that nudges the participant toward a specific point of view.
Worked Example 1: Stratified Sampling Allocation
An international school has 800 secondary students distributed across four grade levels:
Grade 9: $240$ students; Grade 10: $200$ students; Grade 11: $180$ students; Grade 12: $180$ students.
The school principal wants to survey a representative sample of $120$ students regarding the school cafeteria.
a) Calculate the number of students to be sampled from each grade using proportional stratified sampling.
b) Describe how the students in Grade 9 should be chosen once their quota is determined.
Step-by-step Solution:
Part a: Stratified Allocation:
Total population $N = 800$, Total sample $n = 120$.
Sampling fraction $= \frac{n}{N} = \frac{120}{800} = \frac{3}{20} = 0.15$ ($15\%$). [M1]
- Grade 9: $\frac{240}{800} \times 120 = 0.15 \times 240 = 36\text{ students}$. [A1]
- Grade 10: $\frac{200}{800} \times 120 = 0.15 \times 200 = 30\text{ students}$. [A1]
- Grade 11: $\frac{180}{800} \times 120 = 0.15 \times 180 = 27\text{ students}$. [A1]
- Grade 12: $\frac{180}{800} \times 120 = 0.15 \times 180 = 27\text{ students}$. [A1]
Part b: Selection of Grade 9 students:
To avoid researcher bias, obtain a numbered list (sampling frame) of all 240 Grade 9 students from $1$ to $240$. Use a random number generator to pick 36 unique numbers without replacement. The students corresponding to these numbers form the sample. [Criterion C: Clear Methodology]
Worked Example 2: Systematic Sampling & Periodicity Risk
A manufacturing plant produces $4500$ smart bulbs per day. A quality engineer inspects a systematic sample of $150$ bulbs.
a) Find the sampling interval $k$.
b) If the random start number chosen is $18$, state the identification numbers of the first three bulbs tested.
c) Evaluate a potential hidden risk (periodicity) of systematic sampling on an assembly line.
Step-by-step Solution:
Part a: Sampling Interval:
$k = \frac{N}{n} = \frac{4500}{150} = 30$. [A1]
Part b: First three tested bulbs:
Start $r = 18$.
1st bulb: $18$.
2nd bulb: $18 + 30 = 48$.
3rd bulb: $48 + 30 = 78$. [A1]
Part c: Periodicity Risk:
If the manufacturing assembly line has a machine with a recurring cycle matching the sampling interval (e.g. a molding head with 30 nozzles where nozzle #18 is defective, or calibration occurs every 30th item), systematic sampling may either catch every single defective item or completely miss all defects. This cyclical pattern (periodicity) produces severely biased results. [Criterion D: Real-World Evaluation]
Tiered Practice Exercises
10 problems advancing from census decisions and systematic intervals to proportional stratified rounding, cluster comparisons, and survey bias critiques.
For each scenario, state whether a census or a sample is appropriate, giving a brief reason:
a) Testing the crash safety impact rating of newly manufactured cars.
b) Determining the final exam grades of all 32 students in an IB Chemistry class.
c) Estimating the proportion of national television viewers watching an international football match.
View Solution & Markscheme
Method & Marks:
a) Sample: Destructive testing (a census would destroy all cars manufactured). [A1]
b) Census: Small population ($N = 32$) where individual precision and reporting is legally required for every student. [A1]
c) Sample: Enormous, constantly changing population where surveying every citizen is logistically impossible and cost-prohibitive. [A1]
A company has 1200 employees. An auditor wishes to take a systematic sample of 40 employees.
a) Calculate the sampling interval $k$.
b) If the random start number is 14, write down the numbers of the first 4 employees selected.
View Solution & Markscheme
Method & Marks:
a) $k = \frac{1200}{40} = 30$. [A1]
b) Start $= 14$. Next $= 14 + 30 = 44$. Next $= 44 + 30 = 74$. Next $= 74 + 30 = 104$.
First 4 employees: $14, 44, 74, 104$. [A1]
A university faculty has 300 undergraduate students and 100 postgraduate students. In a sample of 60 students, calculate how many undergraduates and postgraduates should be selected using stratified sampling.
View Solution & Markscheme
Method & Marks:
Total population $N = 300 + 100 = 400$. Sample $n = 60$. [M1]
Undergraduates: $\frac{300}{400} \times 60 = 45\text{ students}$. [A1]
Postgraduates: $\frac{100}{400} \times 60 = 15\text{ students}$. [A1]
A hospital employs staff across three shifts: Morning (420), Evening (280), Night (140). A committee of 24 staff members is formed via stratified sampling.
a) Determine the number of representatives from each shift.
b) Explain why a simple random sample might fail to represent the Night shift fairly.
View Solution & Markscheme
Method & Marks:
a) Total $N = 420 + 280 + 140 = 840$. Sample $n = 24$.
Fraction $= \frac{24}{840} = \frac{1}{35}$. [M1]
Morning: $\frac{420}{35} = 12$ staff.
Evening: $\frac{280}{35} = 8$ staff.
Night: $\frac{140}{35} = 4$ staff. [A1]
b) In Simple Random Sampling, because Night staff make up only $\frac{1}{6}$ of the hospital, random variation could easily result in zero or only one Night staff member being selected, leaving their distinct working conditions unrepresented. Stratified sampling guarantees their exact proportional quota of 4 members. [R1: Criterion C]
A sports analytics firm wants to survey 50 spectators at a football stadium with 40,000 fans. The researcher interviews the first 50 fans entering Gate A.
a) Identify this sampling method.
b) Explain two reasons why this sampling method is likely to produce biased data.
View Solution & Markscheme
Method & Marks:
a) Convenience sampling (or accidental sampling). [A1]
b) Sources of bias:
1. Location bias: Gate A serves only a specific section of the stadium (e.g. VIP boxes, ultra-fan supporters, or visitors), which may not reflect general fans seated elsewhere.
2. Time bias: The first 50 fans arrive very early and likely have much higher dedication or different demographics compared to fans arriving just before kickoff. [R1: Criterion D]
A high school with 650 students has 150 Freshmen, 180 Sophomores, 170 Juniors, and 150 Seniors. A sample of 50 students is planned using stratified sampling.
a) Calculate the raw proportional quota for each grade.
b) Round the quotas to whole numbers such that the sum is exactly 50.
View Solution & Markscheme
Method & Marks:
a) Raw quotas ($\times \frac{50}{650}$): [M1]
Freshmen: $\frac{150}{650} \times 50 \approx 11.538$
Sophomores: $\frac{180}{650} \times 50 \approx 13.846$
Juniors: $\frac{170}{650} \times 50 \approx 13.077$
Seniors: $\frac{150}{650} \times 50 \approx 11.538$ [A1]
b) Integer rounding:
Rounding Sophomores to $14$, Juniors to $13$.
For Freshmen and Seniors ($11.538$ each), rounding one to $12$ and one to $11$ ensures sum $= 12 + 14 + 13 + 11 = 50$. (Or $11, 14, 13, 12$). [A1]
Explain the difference between stratified sampling and cluster sampling in terms of how groups are selected and how individuals within groups are sampled.
View Solution & Markscheme
Method & Marks:
Stratified Sampling: The population is divided into homogeneous strata. A random sample of individuals is taken from EVERY stratum. [A1]
Cluster Sampling: The population is divided into heterogeneous clusters. A random sample of CLUSTERS is selected, and then ALL individuals (or a sample) within the chosen clusters are surveyed. Non-chosen clusters are completely ignored. [A1][R1]
A wildlife researcher estimates the population of trout in a lake using the capture-recapture method.
1. On Monday, 200 trout are caught, tagged, and released back into the lake.
2. On Friday, a second sample of 150 trout is caught, of which 24 are found to have tags.
a) Use the proportion $\frac{\text{Tagged in Sample}}{\text{Total in Sample}} = \frac{\text{Tagged in Population}}{\text{Total Population } N}$ to estimate the total trout population $N$.
b) State two mathematical assumptions required for this estimate to be valid.
View Solution & Markscheme
Method & Marks:
a) Lincoln-Petersen Estimator: [M1]
$\frac{24}{150} = \frac{200}{N} \implies 24N = 150 \times 200 = 30{,}000$
$N = \frac{30{,}000}{24} = 1250\text{ trout}$. [A1]
b) Key Assumptions:
1. The tagged fish mix thoroughly and randomly throughout the entire population before the second catch.
2. No births, deaths, emigration, or immigration occurred during the week (closed population).
3. The tags do not fall off or make the fish more vulnerable to predators or recapturing. [R1: Criterion D Evaluation]
A company conducts a telephone survey on weekend satisfaction. Phone calls are placed to landlines on weekdays between 10:00 AM and 3:00 PM.
a) Identify the sampling frame used.
b) Analyze two major demographic groups that are systematically underrepresented by this study design.
c) Classify the types of bias produced.
View Solution & Markscheme
Method & Marks:
a) The sampling frame is the list of landline telephone subscribers who are home during weekday business hours. [A1]
b) Underrepresented groups:
1. Full-time employed workers and students who are away from home during 10 AM - 3 PM.
2. Younger generations who exclusively use mobile smartphones and do not own landlines. [A1]
c) Bias classification:
Undercoverage / Selection bias (excluding non-landline owners) and Non-response bias (working adults cannot answer). The sample will heavily over-represent retired elderly or unemployed individuals. [R1]
Criterion C & D Investigation on Experimental Controls: In testing a new sleep-aid supplement, a pharmaceutical company recruits 100 volunteers suffering from insomnia.
a) Explain why simply measuring their sleep quality after taking the supplement for a month is an invalid study design.
b) Describe how a double-blind randomized controlled trial solves this problem.
c) Explain the role of a placebo in eliminating the placebo effect as a confounding variable.
View Solution & Markscheme
Method & Marks:
Part a: Without a control group, any improvement could be caused by the placebo effect (psychological expectation of sleep), natural recovery over time, or seasonal changes, rather than the chemical efficacy of the drug. [R1: Criterion C]
Part b: Double-blind RCT design:
1. Random Assignment: Randomly allocate the 100 participants into two groups of 50 (treatment and control).
2. Double-Blind: Neither the patients nor the doctors evaluating sleep quality know who received the real drug versus the placebo, eliminating observer bias and confirmation bias. [A1]
Part c: Placebo control: Giving the control group an identical dummy pill accounts for the psychological placebo effect. If the treatment group shows statistically significant improvement above the placebo baseline, the effect can be legitimately attributed to the drug. [R1: Criterion D Reflection]
Sampling Formulas & Methods Reference
Systematic Interval
$$k = \frac{N}{n}$$
Population $N$ divided by sample $n$.Stratified Allocation
$$n_i = \frac{N_i}{N} \times n$$
Proportional stratum quota.Capture-Recapture
$$\frac{M}{N} = \frac{m}{n} \implies N = \frac{Mn}{m}$$
Lincoln-Petersen index.Simple Random (SRS)
$$\text{Equal Probability: } P = \frac{1}{N}$$
Random number generation.Cluster Sampling
$$\text{Sample whole groups}$$
Census within chosen clusters.Sampling Bias
$$\text{Systematic error in selection}$$
Destroys external validity.Top 5 Common MYP Sampling Pitfalls
- Confusing Random Sampling with Convenience: Asking people on the street is NOT random sampling. It is convenience sampling. "Random" in mathematics has a strict definition: every member must have an equal known chance of selection via a randomized mechanism.
- Believing a Large Sample Eliminates Bias: A sample of $100{,}000$ voluntary online respondents is far less statistically reliable than a rigorous simple random sample of $500$ people. Bias cannot be cured by sample size!
- Forgetting to Describe the Random Mechanism: When answering MYP Criterion C questions, saying "take a stratified sample" is not enough. You must calculate the exact numbers per stratum AND state how individuals within each stratum are chosen (e.g. random number generator).
- Ignoring Periodicity in Systematic Sampling: Always check whether the population list has a recurring pattern that matches the interval $k$.
- Confusing Parameters with Statistics: Population characteristics are parameters ($\mu, \sigma$); sample characteristics are statistics ($\bar{x}, s$).
Self-Grading Mini Test
5 quiz questions on sampling methodologies, stratified calculations, and bias identification.
In a school of 600 students, 150 are in Grade 9. If a stratified sample of 80 students is chosen, how many Grade 9 students should be selected?
Explanation: $n_i = \frac{150}{600} \times 80 = \frac{1}{4} \times 80 = 20\text{ students}$.
Why must light bulb manufacturers test lifespan using a sample rather than a census?
Explanation: Testing a bulb until it burns out destroys the item. Testing every item in a census would leave zero inventory to sell.
A population has $N = 2500$ members. A researcher requires a systematic sample of $n = 50$. What is the interval $k$?
Explanation: $k = \frac{N}{n} = \frac{2500}{50} = 50$.
An online news poll asks: "Should taxes be raised?" What major bias invalidates this study?
Explanation: Online open polls attract only volunteers, particularly individuals with strong emotional or extreme views, which misrepresents the broader population.
A researcher randomly selects 5 entire schools in a city and surveys every student in those 5 schools. This is an example of:
Explanation: Because whole natural groupings (schools) are randomly chosen and all members within them are surveyed, it is cluster sampling.