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Statistics sampling distributions and scatter - Mathematics Year 9

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Introduction

Can two samples have the same centre but tell different stories? Compare distributions, inspect who was sampled and distinguish association from a causal explanation.

Build a complete comparison of two invented waiting-time samples. Then critique a sample design and use paired numerical data to interpret a scatter pattern cautiously.

Compare centres, spreads and unusual values

A population is the complete group a question concerns; a sample is the part actually observed. A sample statistic summarises its observations. It may estimate a population feature, but a calculation does not automatically make an unrepresentative sample representative. Define the quantity and units before collecting or comparing it.

Consider two invented samples of waiting times in minutes. Sample A is 4, 5, 5, 6, 7 and B is 2, 4, 5, 8, 8. Both are already ordered. The median is the third value of five, so both medians are 5 minutes. Range is maximum minus minimum: A has 7−4=3 minutes, while B has 8−2=6 minutes.

Centre and spread together. The third row deliberately changes one value from the original B sample; it is a separate sensitivity experiment, not an additional real observation.

Sample	Ordered minutes	Total	Mean minutes	Median minutes	Range minutes
A	4, 5, 5, 6, 7	27	5.4	5	3
B	2, 4, 5, 8, 8	27	5.4	5	6
B with 8 replaced by 80	2, 4, 5, 8, 80	99	19.8	5	78

Both original totals are 27, so both means are 27/5=5.4 minutes as well. A complete comparison says that the samples have the same median 5 and mean 5.4, but B has a larger range 6 versus 3. B is more spread out by this measure. Saying “the samples are the same” from equal medians alone discards important variation.

For an independent median check, there are two observations no greater than the middle position before it, and two after it in each ordered five-item list. For a mean check, multiply 5.4 by 5 to recover 27. For range, keep the original endpoints visible instead of subtracting the first two values or the largest frequency.

Replacing one 8 in B by 80 gives 2, 4, 5, 8, 80. The total becomes 99, mean 19.8 and range 78, but median remains 5. The mean and range respond strongly to this large value, while this particular median does not change. Do not claim that medians can never change: changing enough values, or a value that crosses the middle position, can change one.

Choosing a summary depends on the question. Mean retains every numerical value and supports total-per-item reasoning; median may better describe a typical wait when a very long wait dominates the mean. Range uses only two extremes and does not describe all internal clustering. A short comparison should name both a centre and a spread measure, with units and a limitation.

These five observations are illustrative, not a study proving that one real service is faster or more consistent. Equal sample size makes this simple comparison transparent, but does not remove sampling bias or establish a population difference.

Evaluate who was sampled and what was asked

Choose a sample for a stated question. Suppose a fictional library wants to estimate which opening time its 600 eligible members prefer. Asking the first 30 volunteers near the entrance at lunchtime is convenient, but may miss members who cannot attend at lunchtime. More precise arithmetic on those 30 replies cannot repair this coverage problem.

A stronger plan starts with a complete eligible-member list, uses a documented random selection and offers the same neutral question through accessible response routes. Each eligible member needs a known chance of selection. Record who did not respond without publishing identities. Random selection can reduce selection bias, but nonresponse can still make respondents differ from the selected sample.

If age groups or membership categories matter, a stratified random design could sample within those groups using stated allocations. Do not confuse deliberately choosing a mix of convenient volunteers with random selection inside each group. A larger convenience sample is not automatically better than a smaller well-designed sample; size and representativeness are different issues.

Inspect the question as well as the sample. “Would you support our excellent new evening opening?” invites approval. A neutral alternative supplies clear options, including no preference, and a defined reference period. Avoid combining two questions such as longer hours and a new fee into one yes/no response. Pilot unclear wording before interpreting percentages.

Suppose 18 of 30 actual respondents prefer an evening slot. The sample proportion 18/30=60% describes those replies. It does not prove that exactly 60% of all 600 members agree or that precisely 360 members would attend. A projection 360 is a model-based estimate under assumptions about representativeness and response behaviour, not a counted population fact.

Compare two limitations separately. Selection bias concerns how people entered the sample; sampling variation concerns how different random samples can differ by chance. Even a genuine random sample can give a different percentage next time. This course does not invent a confidence interval or claim that a small sample identifies an exact population parameter.

Retain an audit note: target population, frame, selection method, question wording, response count and plausible missing groups. Keep invented examples distinct from any actual school survey. Obtain appropriate real-world permissions before collecting identifiable information; the mathematical activity needs only synthetic or anonymous aggregate counts.

Interpret paired data without inventing causes

A scatter graph displays paired numerical values, one observation per point. Put height in centimetres on the horizontal axis and arm span in centimetres on the vertical axis. The invented observations below are for learning the method, not real records or a biological study.

Observation	Height cm	Arm span cm
A	150	150
B	160	165
C	170	170
D	180	185
E	190	190

Arm span cm
190 | .  .  .  .  E
185 | .  .  .  D  .
180 | .  .  .  .  .
175 | .  .  .  .  .
170 | .  .  C  .  .
165 | .  B  .  .  .
160 | .  .  .  .  .
155 | .  .  .  .  .
150 | A  .  .  .  .
    +---------------
      150 160 170 180 190 Height cm
Vertical step5cm; horizontal step10cm.

As height increases in this sample, arm span generally increases: a positive association. The two axes deliberately have different units-per-grid-step, so visual steepness is not a scale-free numerical slope. Read coordinates and labelled increments rather than estimating an angle on the screen.

A rough illustrative trend estimate for height 175 cm, between observed 170/170 and 180/185, is arm span 177.5 cm by linear interpolation between those two neighbours. This is a local estimate, not an exact prediction or a formally fitted line. Actual individuals may vary. Predicting at 230 cm is extrapolation far outside the observed 150–190 cm range and is much less justified.

The original Corbettmaths preview plots paired observations and draws a positive trend line. Identify labelled axes and the overall pattern before making an estimate. Its observations are separate from our five invented height/arm-span pairs. Original: corbettmaths, Scatter Graphs - Corbettmaths. The written explanations and native tasks work without sound or the external player.

Association alone does not prove height causes a longer arm span. Shared developmental factors, measurement choices and selection may explain the pattern. Reversing a graph's axes also does not establish a causal direction. A line drawn through a scatter cloud summarises a model rather than supplying evidence of a controlled experiment.

Write a conclusion that joins evidence to limits: equal source medians 5 and means 5.4, larger B range 6 versus 3; sample design and unusual values affect interpretation. For paired data, state the positive association, observed range, interpolation assumptions and absence of a causal claim. Recheck an actual later comparison independently rather than copying the worked conclusion.

Interactive Tasks

Quiz: Test Your Knowledge

Source A median? (5 minutes) (!5.4 minutes) (!3 minutes) (!27 minutes)

Source B range? (6 minutes) (!3 minutes) (!8 minutes) (!5 minutes)

Which comparison is justified? (Same centre summaries but different ranges) (!Identical distributions) (!B has smaller range) (!A proves the population is faster)

Changing one B value 8 to 80 gives mean? (19.8 minutes) (!5.4 minutes) (!5 minutes) (!78 minutes)

A population is? (The complete group the question concerns) (!Only those who reply) (!Always thirty people) (!Every number on a chart)

A larger convenience sample automatically removes bias? (No) (!Yes in every case) (!Only with a bar chart) (!Only at lunchtime)

18 of 30 respondents is? (60 percent of those replies) (!Exactly 360 of 600 people counted) (!An exact population parameter) (!A guaranteed future attendance count)

Our height/arm-span pairs show? (Positive association in this sample) (!Proof of a causal direction) (!No variation in either variable) (!A guaranteed prediction for everyone)

Estimating at 175 cm within observed 150–190 is? (Interpolation) (!Extrapolation) (!Random selection) (!Proof of cause)

Predicting at 230 cm from those pairs is? (Extrapolation with added uncertainty) (!An observed result) (!A measured population mean) (!Always exact)

Memory Game

Population	Complete target group
Sample	Part of the population actually observed
Median	Middle position in an ordered list
Range	Maximum minus minimum
Association	Pattern linking paired observations
Extrapolation	Estimate beyond observed data range

Drag and Drop

5	Both original sample medians
5.4	Both original sample means
3	Original A range
6	Original B range
19.8	Mean after replacing one 8 by 80

Crossword Puzzle

SAMPLE	Observed part of a population
POPULATION	Complete target group
MEDIAN	Middle ordered value
RANGE	Maximum minus minimum
ASSOCIATION	Pattern between paired variables
INTERPOLATION	Estimation within the observed range

LearningApps

Optional extra practice. Choose an appropriate activity with your teacher. If the external collection is unavailable, use the native activities and independent tasks in this course.

Cloze Text

Open Tasks

Easy

Order both source samples: Locate the middle value and endpoints.
Compute two means: Rebuild total 27 from 5.4 and five values.
Identify the target population: Separate 600 eligible members from 30 volunteers.
Read a paired point: Explain observationB as height 160/arm span 165 cm.

Standard

Write the full comparison: Include median 5, mean 5.4 and ranges 3/6 with units.
Critique the library sample: Identify lunchtime coverage and nonresponse separately.
Plot the five pairs: Label unequal axis increments and describe association.
Estimate carefully: Interpolate at 175 cm and state why 230 cm is riskier.

Advanced

Investigate the unusual value: Explain mean 19.8 and range 78 while median stays 5 here.
Improve a survey design: Specify frame, random selection, neutral wording and missing-response record.
Challenge a causal claim: Offer a plausible shared factor without claiming it is proven.
Retain a comparison portfolio: Keep summaries, sampling limits, paired-plot reasoning, error repair and later retrieval.

Learning Check

Source distributions: Compute centre and spread with an independent check.
Unusual-value sensitivity: Replace one 8 by 80 and recalculate.
Sampling limitation: Evaluate 30 lunchtime volunteers from 600 eligible members.
Percentage interpretation: Interpret 18 of 30 replies.
Scatter and estimates: Use the five invented pairs and labelled units.
Complete evidence: Retain a justified conclusion and first-error repair.

Worked checks and assessment guidance

A 4, 5, 5, 6, 7 and B 2, 4, 5, 8, 8 both total 27, mean 5.4, median 5. Ranges 3 and 6 differ; B is more spread out by range. Mean×5 rebuilds 27. Equal centres do not imply identical samples.
New B 2, 4, 5, 8, 80 totals 99, mean 19.8, median 5, range 78. Mean and range change strongly; this particular median does not. Do not claim medians can never change.
Convenience and self-selection may miss unavailable groups. A complete frame, documented random selection, neutral question and response tracking improve the plan; nonresponse and random variation remain.
18/30=60% of respondents. A projected 360 of 600 is conditional on a representative model, not 360 counted supporters or guaranteed attendance.
Points 150/150, 160/165, 170/170, 180/185, 190/190 show positive association. Interpolation 175→177.5 uses a local linear assumption; 230 cm extrapolates beyond the observed range. Association does not prove causation.
Compare both centre and spread, identify sample and question limitations, read axes and unusual values, distinguish estimate from observation and repair the first causal or equal-centre error. Actual later work must be recorded rather than invented.

Learning evidence

Retain a full two-distribution comparison, a sensitivity check, a sample-design critique, a labelled five-point scatter model, cautious estimates, first-error correction and actual later retrieval.

OER and Wikipedia

Department for Education: England mathematics programme of study, Key Stage 3, Source area: Key stage 3 > Subject content > Statistics. Fluency, reasoning and problem solving. Year 9 allocation is editorial within KS3. Numerical models, tasks and feedback are original; images retain actual Commons attribution and videos name their original publisher. Reviewed 12 November 2026.

Simple bar chart.svg, Masur, Public domain. Unchanged original; caption and learning activities are original.

2D Cartesian Coordinates.svg, H Padleckas, Public domain. Unchanged original; caption and learning activities are original.