Scatter Graphs and Sampling

A population is the entire group being studied. A sample is a subset used when it is impractical to study the whole population.

Good sample requirements:

• Representative: reflects the full variety of the population
• Random: every member has an equal chance of selection
• Adequate size: larger samples are more reliable

Sampling methods:

Method	Description	Limitation
Simple random	Every member equally likely	Costly if population is large
Systematic	Every $k$th member (e.g. every 10th person)	May miss patterns with period $k$
Stratified	Sample from each subgroup in proportion to size	Requires knowledge of subgroup sizes
Opportunity/convenience	Those who are available	Likely biased — not representative

Worked example — a school of 600 students: 200 in Year 10, 400 in Year 11. A stratified sample of 30 is required.

Year 10: $\frac{200}{600}\times 30=10$ students. Year 11: $\frac{400}{600}\times 30=20$ students. ✓

Limitations: sample results may not perfectly represent the population due to random variation, sampling bias, or non-response.

Data collected from a sample is used to infer properties of the population. The reliability of inference depends on:

1. Sample size: larger samples → less random variation → more reliable estimates
2. Sample representativeness: a biased sample produces biased inferences regardless of size
3. Variability in the population: high variability requires a larger sample to estimate reliably

Worked example — a random sample of 50 light bulbs from a batch shows 3 defective. Estimate the number of defective bulbs in a batch of 2000.

$\frac{3}{50}\times 2000=120$ defective bulbs (expected). ✓

Caution: this is an estimate — the actual number may differ. A larger sample would give a more reliable estimate.

A scatter graph (scatter diagram) plots two variables for each data item — one on each axis.

Correlation describes the relationship between the variables:

Type	Description	Example
Strong positive	As $x$ increases, $y$ increases; points close to a line	Height and weight
Weak positive	General upward trend but scattered	Shoe size and income
No correlation	No clear pattern	Hair length and intelligence
Negative correlation	As $x$ increases, $y$ decreases	Speed and journey time

Correlation does not imply causation. Two variables may be correlated because of a hidden third factor, or simply by coincidence. Stating that one causes the other requires further evidence beyond the scatter graph.

Worked example — a scatter graph shows a strong positive correlation between hours of revision and exam score. This means they are associated — it does not prove that more revision causes higher scores (though it suggests it).

A line of best fit is a straight line drawn through the scatter plot to represent the trend. It does not have to pass through any data point.

Rules for drawing:

• Roughly equal numbers of points above and below the line
• The line should pass through (or very near) the mean point $(\bar{x},\bar{y})$ — this is the balancing point of the data
• Do not extend beyond the range of the data unless asked

Making predictions (interpolation): read off the $y$-value for a given $x$-value from the line, where the $x$-value is within the range of the data. Interpolation is reliable.

Extrapolation: predicting beyond the range of the data using the trend line. Extrapolation is unreliable because the trend may not continue.

Worked example — a line of best fit passes through $(5,40)$ and $(15,80)$. Estimate the $y$-value when $x=10$.

Gradient: $(80-40)/(15-5)=4$. Equation: $y=4x+20$.

At $x=10$: $y=60$ ✓ (interpolation — $x=10$ is within the data range)

At $x=30$: $y=140$ (extrapolation — may not be reliable)

Exam questions often ask you to describe and interpret scatter graphs. Use this structure:

1. Type of correlation: positive/negative/no correlation, strong/weak
2. Context: state what this means in the real-world context of the variables
3. Causation caveat: note that correlation does not prove causation if relevant

Worked example — a scatter graph shows the age of a car (years) against its value (£).

"There is a strong negative correlation between the age of a car and its value. As the car gets older, its value decreases. This suggests that older cars tend to be worth less, though other factors such as mileage and condition also affect value."

1. Describing correlation — confusing strength with direction

"Strong" and "positive/negative" are independent descriptions. A weak negative correlation means a general downward trend but with a lot of scatter. State both.

2. Causation claim from correlation

Stating "more revision causes higher exam scores" based solely on a scatter graph oversteps the data. Say "there is a positive correlation" or "there is an association" — not "causes."

3. Line of best fit — forcing it through the origin

The line of best fit should fit the data, not pass through the origin unless the context genuinely requires it (e.g. a conversion graph). Draw it where it fits best.

4. Extrapolation — treating it as equally reliable as interpolation

The further beyond the data range a prediction extends, the less reliable it is. Identify when a prediction is extrapolation and state the limitation.

Mistake	Correction
"Strong correlation proves causation"	Correlation shows association; proving causation requires controlled experiments
"Line of best fit must pass through as many points as possible"	It should balance above and below equally — most points will not lie on it
"Predicting outside the data range is fine"	Extrapolation may be unreliable; trends may not continue