ST17: Correlation & Regression
Learn how to interpret scatter graphs, draw lines of best fit, calculate Spearman's rank and Pearson's PMCC, and understand interpolation and extrapolation for GCSE Statistics.
Correlation Regression
Correlation & Regression
Learn how to interpret scatter graphs, draw lines of best fit, calculate Spearman's rank and Pearson's PMCC, and understand interpolation and extrapolation for GCSE Statistics.
Key Fact: Correlation measures the strength and direction of the linear relationship between two variables; it does not imply causation.
Key Fact: Positive correlation: as one variable increases, the other tends to increase. Negative correlation: as one increases, the other tends to decrease.
Key Fact: A scatter graph is the first step in assessing correlation; the pattern of points reveals the type and strength of the relationship.
Key Fact: A line of best fit is drawn through the data points on a scatter graph so that roughly equal numbers of points lie above and below the line.
Key Fact: The line of best fit should pass through the point (xฬ, ฬ
) and can be used to estimate one variable given a value of the other.
Key Fact: Interpolation is using the line of best fit to estimate within the range of observed data; it is generally reliable.
Key Fact: Extrapolation is using the line of best fit to estimate outside the range of observed data; it is unreliable because the relationship may not continue.
Key Fact: Spearman's rank correlation coefficient (rโ) measures the strength of monotonic (rank-based) relationships: rโ = 1 โ 6ฮฃdยฒ รท (n(nยฒ โ 1)), where d is the difference in ranks.
Key Fact: Spearman's rโ ranges from โ1 to +1; +1 means perfect increasing monotonic relationship, โ1 means perfect decreasing, and 0 means no monotonic relationship.
Key Fact: Pearson's product-moment correlation coefficient (r) measures the strength of linear correlation: it also ranges from โ1 to +1.
Key Fact: Pearson's r is suitable only for linear relationships between two quantitative variables; Spearman's rโ can handle non-linear monotonic relationships.
Key Fact: Correlation does not imply causation: two variables may be correlated because of a confounding variable, coincidence, or both being effects of a common cause.
๐ Key Vocabulary and Concepts
For Correlation & Regression, you must know:
- Correlation: A numerical measure of the strength and direction of the relationship between two variables, ranging from โ1 to +1.
- Line of best fit: A straight line drawn through a scatter graph so that approximately equal numbers of points lie above and below it; it must pass through (xฬ, ฬ ).
- Spearman's rank correlation coefficient: A measure of monotonic correlation based on the ranks of data values: rโ = 1 โ 6ฮฃdยฒ รท (n(nยฒ โ 1)).
- Pearson's product-moment correlation coefficient (PMCC): A measure of the strength of linear correlation between two quantitative variables, denoted r and ranging from โ1 to +1.
- Interpolation: Using a line of best fit to predict a value within the range of the observed data, which is generally reliable.
- Extrapolation: Using a line of best fit to predict a value outside the range of the observed data, which is unreliable as the relationship may not hold.
โ Practice Questions
Q: Two judges rank 5 contestants. The rank differences are: 0, 1, 1, 2, 0. Calculate Spearman's rโ.
Q: Pearson's r for two variables is โ0.85. Describe the correlation.
Q: A line of best fit is drawn for data where x ranges from 10 to 50. Is predicting y when x = 30 interpolation or extrapolation?
Q: Why is it unreliable to use a line of best fit to predict y when x = 90 if the observed x values range from 10 to 50?
Q: Give a reason why correlation does not imply causation.
โ Answers
- ฮฃdยฒ = 0 + 1 + 1 + 4 + 0 = 6. rโ = 1 โ (6 ร 6) รท (5 ร (25 โ 1)) = 1 โ 36 รท 120 = 1 โ 0.3 = 0.7. Moderate positive agreement.
- Strong negative linear correlation: as one variable increases, the other tends to decrease, and the points lie close to a straight line with negative slope.
- Interpolation, because x = 30 is within the observed range of 10 to 50.
- This is extrapolation; we cannot assume the linear relationship continues beyond the observed data range.
- Two variables may be correlated due to a third confounding variable; e.g. ice cream sales and drowning rates are both linked to hot weather, not to each other.
๐ฏ Exam Tips
- When calculating Spearman's rank, if two or more values are tied, assign each the mean of the ranks they would have occupied.
- Always state the type and strength of correlation (e.g. 'strong positive') when describing a scatter graph โ do not just say 'there is a correlation'.
- When using a line of best fit for predictions, always state whether you are interpolating or extrapolating and comment on reliability.
- In Spearman's rank questions, show a table of data values, ranks, rank differences (d), and dยฒ to gain full method marks.
- When asked 'does correlation imply causation?', the answer is always no โ explain that other factors (confounding variables) could cause both variables to change.
๐ Exam Technique
GCSE Statistics Exam Tips โ Correlation & Regression:
1. For Correlation & Regression questions, show every step of your working clearly โ method marks count even if the final answer is wrong
2. Check your answer makes sense in context (estimation, units, reasonableness)
3. Use correct mathematical notation and state formulae before substituting values
4. If a Correlation & Regression question asks you to 'prove' or 'show', write a logical chain of reasoning with a conclusion line
5. For problem-solving, identify the topic first, then recall the relevant method
1. For Correlation & Regression questions, show every step of your working clearly โ method marks count even if the final answer is wrong
2. Check your answer makes sense in context (estimation, units, reasonableness)
3. Use correct mathematical notation and state formulae before substituting values
4. If a Correlation & Regression question asks you to 'prove' or 'show', write a logical chain of reasoning with a conclusion line
5. For problem-solving, identify the topic first, then recall the relevant method
โ ๏ธ Common Errors
โ Forgetting to rank the data before calculating Spearman's rโ โ Spearman's uses RANKS, not the raw data values. Assign ranks from 1 (smallest) to n (largest) for each variable separately.
โ Confusing interpolation with extrapolation โ Interpolation is WITHIN the data range (reliable); extrapolation is OUTSIDE the data range (unreliable). Always check the x-value against the observed range.
โ Saying 'correlation proves causation' โ Correlation shows association only; causation requires controlled experiments or further evidence. Always state that correlation does not imply causation.
โ Not drawing the line of best fit through the mean point (xฬ, ฬ ) โ The line of best fit must pass through (xฬ, ฬ ); draw this point first, then position the line so roughly half the data points are above and half below.
โ๏ธ Model Answer
Full-Mark Response
Two judges rank 6 films. Calculate Spearman's rank correlation coefficient. Film: A, B, C, D, E, F Judge 1 ranks: 1, 2, 3, 4, 5, 6 Judge 2 ranks: 2, 1, 4, 3, 6, 5
Film | Rank 1 | Rank 2 | d (R1โR2) | dยฒ A | 1 | 2 | โ1 | 1 B | 2 | 1 | 1 | 1 C | 3 | 4 | โ1 | 1 D | 4 | 3 | 1 | 1 E | 5 | 6 | โ1 | 1 F | 6 | 5 | 1 | 1 ฮฃdยฒ = 6 rโ = 1 โ 6ฮฃdยฒ รท (n(nยฒ โ 1)) = 1 โ (6 ร 6) รท (6 ร 35) = 1 โ 36 รท 210 = 1 โ 0.171 = 0.829 rโ โ 0.83, indicating strong positive agreement between the two judges. They rank the films in a very similar order.
๐ AO Deep Dive
Assessment Objective Analysis
AO1 (Knowledge & Understanding): Demonstrate knowledge and understanding of correlation & regression, including data collection, presentation and calculation techniques relevant to Edexcel 1ST0 & AQA 8382.
AO2 (Application): Apply knowledge and understanding of correlation & regression to interpret data, reason statistically and draw conclusions in context.
AO3 (Evaluation): Evaluate statistical methods and conclusions, assessing appropriateness, reliability, validity and bias through the statistical enquiry cycle.
๐ Exam Questions by Topic
- AQA Past Papers - Topic Questions -- AQA
- Edexcel Past Papers - Topic Questions -- Edexcel
- CCEA Past Papers - Topic Questions -- CCEA
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