ST16: Skewness & Outliers
Learn how to identify positive and negative skew, calculate Pearson's skewness coefficient, detect outliers using 1.5xIQR and 2 standard deviations, and understand the effect of outliers on averages.
Skewness Outliers
Skewness & Outliers
Learn how to identify positive and negative skew, calculate Pearson's skewness coefficient, detect outliers using 1.5xIQR and 2 standard deviations, and understand the effect of outliers on averages.
Key Fact: A symmetric distribution has its mean, median, and mode approximately equal; the left and right sides are mirror images.
Key Fact: Positive (right) skew has a long tail to the right; the mean is greater than the median, which is greater than the mode.
Key Fact: Negative (left) skew has a long tail to the left; the mean is less than the median, which is less than the mode.
Key Fact: Pearson's skewness coefficient = 3 × (mean – median) ÷ standard deviation; a positive value indicates positive skew, a negative value indicates negative skew.
Key Fact: Values of Pearson's skewness coefficient between –0.5 and 0.5 suggest approximately symmetric data; values beyond ±1 indicate substantial skew.
Key Fact: On a box plot, positive skew is shown by a longer right whisker and the median closer to Q1; negative skew by a longer left whisker and the median closer to Q3.
Key Fact: Outliers are extreme values that differ significantly from the rest of the data; they can arise from measurement error or genuinely unusual observations.
Key Fact: The 1.5 × IQR rule: any value below Q1 – 1.5 × IQR or above Q3 + 1.5 × IQR is classified as an outlier.
Key Fact: The 2 standard deviations rule: any value more than 2 standard deviations from the mean may be classified as an outlier, used when the distribution is approximately normal.
Key Fact: Outliers pull the mean towards them but do not affect the median; this is why the median is preferred for skewed data.
Key Fact: When deciding whether to include or exclude an outlier, check for data recording errors first; if the value is genuine, analyse the data both with and without the outlier.
Key Fact: In exam answers, always identify outliers by showing the calculation (e.g. Q1 – 1.5 × IQR = 10 – 18 = –8) and state whether the outlier should be kept or removed with justification.
📋 Key Vocabulary and Concepts
For Skewness & Outliers, you must know:
- Skewness: A measure of the asymmetry of a distribution; positive skew has a tail to the right, negative skew has a tail to the left.
- Pearson's skewness coefficient: A numerical measure of skewness calculated as 3 × (mean – median) ÷ standard deviation.
- Positive skew: A distribution where the tail extends to the right; the mean is pulled above the median by high-valued outliers.
- Negative skew: A distribution where the tail extends to the left; the mean is pulled below the median by low-valued outliers.
- Outlier: An extreme value that lies an abnormal distance from other values; detected using the 1.5 × IQR rule or the 2 standard deviations rule.
- 1.5 × IQR rule: A method for identifying outliers: any value below Q1 – 1.5 × IQR or above Q3 + 1.5 × IQR is an outlier.
❓ Practice Questions
Q: For a distribution with mean = 45, median = 40, and SD = 10, calculate Pearson's skewness coefficient and interpret it.
Q: A dataset has Q1 = 20, Q3 = 44. A value of 82 is recorded. Is it an outlier using the 1.5 × IQR rule?
Q: The data 5, 6, 7, 8, 100 has mean 25.2 and median 7. Why is the median more appropriate?
Q: On a box plot, the right whisker is much longer than the left and the median is near Q1. What does this indicate?
Q: A value of 12 is recorded in a dataset with mean 50 and SD 18. Is it a potential outlier using the 2 SD rule?
✅ Answers
- Skewness = 3 × (45 – 40) ÷ 10 = 1.5. This is a positive value indicating positive (right) skew.
- IQR = 44 – 20 = 24. Upper boundary = 44 + 1.5 × 24 = 80. Since 82 > 80, yes it is an outlier.
- The outlier 100 inflates the mean to 25.2, far from most values. The median of 7 is resistant to the outlier and better represents the centre.
- This indicates positive skew — the data is stretched towards higher values, with a long tail to the right.
- Distance from mean = |12 – 50| = 38. Since 38 > 2 × 18 = 36, the value 12 is more than 2 SDs from the mean and is a potential outlier.
🎯 Exam Tips
- When describing skewness, always state the direction (positive/negative) and support it with evidence from mean vs median or from the box plot shape.
- Show full working when using the 1.5 × IQR rule: calculate IQR, find the boundaries, then compare each suspected outlier to the boundaries.
- When an outlier is found, always comment on whether it should be included or excluded and give a reason (e.g. 'it may be a recording error' or 'it is a genuine extreme value').
- Pearson's skewness coefficient requires the standard deviation; if not given, you can still describe skewness qualitatively by comparing mean and median.
- Remember that the 2 SD rule assumes the data is approximately normally distributed; the 1.5 × IQR rule does not require this assumption.
📝 Exam Technique
GCSE Statistics Exam Tips — Skewness & Outliers:
1. For Skewness & Outliers questions, show every step of your working clearly — 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 Skewness & Outliers 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 Skewness & Outliers questions, show every step of your working clearly — 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 Skewness & Outliers 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
⚠️ Common Errors
✗ Saying a distribution is skewed just because it is not perfectly symmetric ✓ Small deviations from symmetry are normal; only describe a distribution as skewed when the asymmetry is clearly visible or the skewness coefficient is substantial.
✗ Confusing the direction of skew ✓ In positive skew, the tail goes to the RIGHT (towards higher values) and the mean is above the median. In negative skew, the tail goes to the LEFT and the mean is below the median.
✗ Forgetting to check whether an outlier is a data error before deciding to remove it ✓ Always investigate outliers first; if a value is a genuine measurement, it should not be removed without justification. Only remove values confirmed as recording errors.
✗ Applying the 2 SD rule to heavily skewed data ✓ The 2 SD rule relies on the assumption of approximate normality; for skewed data, the 1.5 × IQR rule is more appropriate.
✍️ Model Answer
Full-Mark Response
A dataset of 10 test scores has Q1 = 35, Q2 = 52, Q3 = 65, mean = 54, and SD = 18. The maximum value is 110. Determine whether the maximum is an outlier and calculate Pearson's skewness coefficient.
Outlier test using 1.5 × IQR: IQR = 65 – 35 = 30 Upper boundary = 65 + 1.5 × 30 = 65 + 45 = 110 Since 110 equals (not exceeds) the boundary, it is not classified as an outlier by the 1.5 × IQR rule. It is at the very edge. Pearson's skewness coefficient: Skewness = 3 × (mean – median) ÷ SD = 3 × (54 – 52) ÷ 18 = 6 ÷ 18 = 0.333 Since 0.333 is between –0.5 and 0.5, the distribution is approximately symmetric with a very slight positive skew.
📊 AO Deep Dive
Assessment Objective Analysis
AO1 (Knowledge & Understanding): Demonstrate knowledge and understanding of skewness & outliers, including data collection, presentation and calculation techniques relevant to Edexcel 1ST0 & AQA 8382.
AO2 (Application): Apply knowledge and understanding of skewness & outliers 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.
📝 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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📄 Past Papers for Statistics (AQA)
- AQA Past Papers & Mark Schemes
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- CCEA Past Papers & Mark Schemes
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