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Statistics sampling distributions and scatter graphs - Mathematics Year 10

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Introduction

A summary is useful only when it answers the question for the right data. Calculate averages and spread, then inspect how the data were collected before generalising.

Compare two small distributions, design a proportionate sample, and use a scatter model cautiously. Separate exact descriptive results from uncertain claims about a wider population or a causal effect.

Compare centre spread and exact frequencies

For numerical observations, the mean is the total divided by the number of observations. The median is the middle of an ordered list; the mode is a most frequent value. The range is the maximum minus the minimum. These describe different features: there is no rule that one average is automatically best for every purpose.

The source example is A=2, 3, 4, 4, 17. Its total is 30, so the mean is 6. The ordered middle value is 4, the mode is 4, and the range is 17−2=15. The unusually large 17 raises the mean well above most observations. This describes its influence; calling it a formally identified outlier would require a stated rule, not just a judgement from appearance.

Compare B=4, 5, 6, 6, 9. Its total is also 30 and its mean 6, but its median and mode are 6 and its range 5. Equal means do not imply equal distributions. If you want a typical observation for A, the median 4 may communicate the central cluster more clearly, provided you also retain the high value and report spread rather than deleting an inconvenient observation.

Distribution	Ordered data	Total	Mean	Median	Mode	Range
A	2, 3, 4, 4, 17	30	6	4	4	15
B	4, 5, 6, 6, 9	30	6	6	6	5

An exact frequency table compresses repeated values without losing their count. For A, the value 4 has frequency 2; it must contribute twice to the total. The mean is $\bar{x} = \frac{\sum f x}{\sum f} = \frac{30}{5} = 6.$ Here f is frequency and x is the observed value, not an interval midpoint.

Observed value x	Frequency f	Product fx
2	1	2
3	1	3
4	2	8
17	1	17

Check the frequency method by expanding the table back to 2, 3, 4, 4, 17. Averaging the four distinct value labels gives 26/4=6.5 and answers a different question: it discards the repeated 4. Keep the denominator equal to the total frequency 5, not the number of different labels 4.

Data type controls which summaries are meaningful. A most common transport category can be a mode, but the arithmetic mean of labels such as bus, walk and cycle is undefined. For grouped numerical intervals, midpoints can produce an estimated mean because the exact observations inside each interval are unknown. Do not present such an estimate as the exact mean of the original data.

For a fair comparison, write a sentence about centre and a sentence about spread in the same units. In our synthetic examples A and B have equal means 6, but A has lower median 4 versus 6 and larger range 15 versus 5. Neither short list alone establishes the pattern in an entire real school.

Design and evaluate a sample

A population is the whole group named by the question; a sample is the subset actually observed. Begin with a precise question, an eligible population and a sampling frame. A sample average describes the collected observations exactly, but it does not automatically equal the population average.

Suppose a fictional school wants to describe travel-to-school patterns among its 400 pupils in Years 7–11. Asking only pupils at a lunchtime football club is a convenience sample. Club membership, attendance and availability may be related to the question, so the results need not represent everyone. A larger sample from that same restricted group does not by itself remove the selection problem.

A proportionate stratified design could select 40 pupils using year groups as strata. The numbers below are fictional planning counts, not a real school roster. Within each year, use an appropriate random selection from the eligible list rather than choosing friends or the first people to volunteer.

Stratum	Population count	Planned sample count
Year 7	80	8
Year 8	120	12
Year 9	100	10
Year 10	60	6
Year 11	40	4

The allocation rule is $n_{h} = 40 \frac{N_{h}}{400} .$ For Year 8, the count is 40×120/400=12. The five allocations 8+12+10+6+4 sum to 40, and each is one tenth of its stratum’s population. This exact example needs no rounding; other totals may require an explicit rounding rule which preserves the final sample size.

Random selection within the stated strata addresses that part of the selection design; it does not guarantee a perfectly representative realised sample. Missing pupils, incomplete frames, non-response, ambiguous questions and recording errors can still affect the results. Record how many were invited and responded, and avoid claiming that every selected pupil answered.

Compare two hypothetical claims. “The forty respondents have a mean journey time of 18 minutes” is a descriptive statement if those measurements and calculation are correct. “All four hundred pupils have mean 18 minutes” adds an unsupported exact population claim. Generalisation needs a defensible sampling process and an account of uncertainty, not merely a numerical average.

Question wording matters too. “How wonderful is cycling to school?” encourages a judgement rather than neutrally asking for the main transport mode. Offer clear, appropriate response categories and specify the reference period, such as the main mode on the most recent usual school day. Avoid collecting names or sensitive information for a classroom practice example.

An honest report names the target population, frame, selection method, planned and achieved sample sizes, and limitations. Do not promise that stratification removes every bias or that a random sample must reproduce each population characteristic exactly. For this course, use the supplied synthetic counts and proposed questions; no real pupil records or actual survey competence are implied.

Interpret scatter models without overclaiming

A scatter graph represents paired numerical observations, one point for each pair. Here x is fictional study time in hours and y is a fictional quiz score out of 20. The five supplied pairs show a generally positive association, with variation rather than a perfectly exact linear relationship.

Study hours x	Quiz score y out of 20
1	5
2	8
3	8
4	12
5	13

14 |                    
13 |                  * 
12 |              *     
11 |                    
10 |                    
 9 |                    
 8 |      *   *         
 7 |                    
 6 |                    
 5 |  *                 
 4 |                    
 3 |                    
 2 |                    
 1 |                    
 0 |                    
   +--------------------
      1   2   3   4   5  x

Text alternative: the plotted points are(1, 5), (2, 8), (3, 8), (4, 12), (5, 13). The vertical coordinate is score, the horizontal coordinate is study hours, and an asterisk marks one pair. Both the table and the plot preserve the same observations; the plot is an original schematic, not the Commons grid’s original content.

A simple illustrative trend line is y=2 x+3. It is a guide for this small example, not a claimed least-squares regression or a law. The point at x 2 has score 8 versus line prediction 7; the point at x 3 has score 8 versus prediction 9. A line need not pass through every point, and joining observations in time order would show a different type of graph.

At x 3.5, the line predicts 10 marks. This is interpolation because 3.5 lies within the observed x range 1 to 5. It is still an estimate for an individual pupil, not a guaranteed score. At x 12, the same formula gives 27, which exceeds the quiz maximum 20. This extrapolation is both outside the observed range and incompatible with the bounded score scale, so the linear guide should not be extended that far.

The actual Corbettmaths Scatter Graphs preview shows a drawn line through a positive cloud of plotted observations. Inspect whether a point lies above or below the line. Our five fictional pairs and guide equation have separate specified values; the video is not evidence that study causes a particular score. Original: corbettmaths, Scatter Graphs - Corbettmaths. The written explanations and native tasks work without sound or the external player.

Correlation alone does not establish causation. Prior knowledge, task choice, teaching, measurement differences or another factor could help explain an observed association. This tiny invented data set is for learning representations; it cannot estimate a real educational effect or prove that another hour produces two extra marks.

Near transfer. For C=1, 2, 3, 4, 20, the total 30 gives mean 6, median 3, mode absent and range 19. Compare C with A: equal means again conceal a changed centre and spread. An independent frequency-count reconstruction checks the mean; ordered positions check the median by a different method.

First-error repair. A pupil averages distinct labels instead of observations; restore frequencies. Another declares the population mean equal to the sample mean; limit the conclusion and inspect selection. Another treats a high correlation or a far-outside line prediction as a causal certainty; return to the data range, scale and assumptions. Separate grouped-data, histogram, cumulative-frequency and box-plot progression from this core evidence and check the chosen specification before assigning tier requirements.

Interactive Tasks

Quiz: Test Your Knowledge

For 2, 3, 4, 4, 17, the mean is? (6) (!4) (!6.5) (!15)

For the same ordered data, the median is? (4) (!6) (!3) (!17)

For 4, 5, 6, 6, 9, the range is? (5) (!6) (!9) (!30)

In the exact frequency table, value 4 with frequency 2 contributes? (8) (!4) (!2) (!16)

Which summary can describe a most common transport category? (Mode) (!Arithmetic mean of category names) (!Numerical median of names without an order) (!Range measured in minutes)

A proportionate 40-person sample from 400 includes how many of 120 Year 8 pupils? (12) (!120) (!40) (!30)

What does a mean computed from the forty respondents describe exactly? (Those respondents) (!Every pupil in the school automatically) (!Every pupil in England) (!Only the sampling frame without observations)

For y=2 x+3, prediction at x 3.5 is? (10) (!7) (!6.5) (!12)

Observed x range 1 to 5: prediction at x 12 is? (Extrapolation) (!Interpolation) (!A directly observed value) (!Proof of causation)

What follows from positive correlation alone? (An association in the observed data) (!Proof that x causes y) (!A guaranteed individual prediction) (!Permission to ignore the sample method)

Memory Game

Mean	Total divided by total frequency
Median	Middle of an ordered numerical list
Mode	Most frequent value or category
Range	Maximum minus minimum
Population	Whole group targeted by the question
Interpolation	Prediction inside the observed input range

Drag and Drop

Exact frequency table	Preserve repetitions when calculating a mean
Proportionate stratification	Allocate sample counts using stratum population shares
Non-response	Selected people do not all provide observations
Extrapolation	Prediction beyond observed input values
Causal claim	Requires more evidence than correlation alone

Crossword Puzzle

MEDIAN	Middle observation in an ordered odd-length list
FREQUENCY	Number of observations belonging to a value
POPULATION	Whole group named by a research question
SAMPLE	Subset whose observations are collected
CORRELATION	Association between paired numerical variables
INTERPOLATION	Prediction within the observed input 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

Calculate the source summaries: Find total, mean, median, mode and range for distributionA.
Expand an exact frequency table: Restore every repeated observation.
Name the target group: Distinguish a fictional population from respondents.
Read the five scatter points: Use horizontal-first coordinates and the stated units.

Standard

Compare both distributions: Use centre and spread and explain the influence of 17.
Plan the forty-person sample: Calculate all five allocations and specify random selection within strata.
Evaluate the football-club survey: Identify frame and selection limitations without inventing actual responses.
Interpolate with a stated guide: Calculate the prediction at 3.5 hours and explain its uncertainty.

Advanced

Audit the far-outside prediction: Calculate at 12 hours and compare with the bounded score scale.
Separate association from cause: Propose plausible alternative explanations for a fictional relationship.
Solve the changed distribution: Find all summaries for 1, 2, 3, 4, 20 and compare withA.
Design a clear reporting note: Name planned versus achieved samples, non-response and what the evidence can support.

Learning Check

Two-distribution comparison: Use exact summaries and interpret them.
Frequency-method audit: Use both weighted calculation and reconstruction.
Proportionate sample design: Calculate allocations and specify limitations.
Scatter and predictions: Distinguish the observed data from a guide.
Changed-data transfer: Calculate independently and check the median.
Claims and evidence: Evaluate a sampling or causal overclaim.

Worked checks and assessment guidance

A has total 30, mean 6, median 4, mode 4, range 15. B has total 30, mean 6, median 6, mode 6, range 5. Equal means conceal a lower median and wider range inA. Value 17 influences the mean; no formal outlier classification is assumed.
ForA, frequencies 1, 1, 2, 1 sum to 5 and products 2, 3, 8, 17 sum to 30. Mean 30/5=6. Expanding restores 2, 3, 4, 4, 17. Averaging the four distinct labels gives 6.5 and wrongly loses a repeated observation.
Population 80, 120, 100, 60, 40 sums to 400. A 40-person proportionate sample uses 8, 12, 10, 6, 4, each one tenth. Random selection within the strata does not eliminate incomplete frames, non-response or ambiguous questions.
Points are(1, 5), (2, 8), (3, 8), (4, 12), (5, 13). The illustrative line y=2 x+3 predicts 10 at 3.5 within the observed 1 to 5 range. At 12 it predicts 27, outside both the observed range and maximum score 20. It is not a proved causal law or a claimed least-squares fit.
For 1, 2, 3, 4, 20, total 30 and count 5 give mean 6. The ordered middle is 3, all values are distinct so there is no mode, and range 19. Its equal mean withAdoes not imply the same shape, centre or spread.
A respondent mean describes those respondents; an exact whole-population claim needs more evidence. A larger convenience sample can retain selection bias. Correlation alone does not establish cause, and uncompleted surveys or later retrieval must not be reported as observed evidence.

Learning evidence

Keep three complete distribution summaries, frequency reconstruction, all five proportional sample allocations, the actual five-point plot, predictions with range limits, and a justified critique of sampling and causal claims.

OER and Wikipedia

Department for Education:GCSE Mathematics subject content and assessment objectives. Source area:GCSE Mathematics subject content > Scope of study > Statistics items 1 to 6. Year10 allocation is editorial within the GCSE pathway;Higher-only extensions are explicitly labelled. Original examples,teaching and feedback;Commons images retain their actual attribution,and videos identify original publishers. Reviewed12November2026.

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.