ST15: Measures of Spread
Learn how to calculate and interpret range, interquartile range, percentiles, interpercentile range, standard deviation, and variance for GCSE Statistics.
Measures Spread
Measures of Spread
Learn how to calculate and interpret range, interquartile range, percentiles, interpercentile range, standard deviation, and variance for GCSE Statistics.
Key Fact: The range is the difference between the largest and smallest values: range = maximum β minimum; it is affected by outliers.
Key Fact: The interquartile range (IQR) = Q3 β Q1; it measures the spread of the middle 50% of the data and is not affected by outliers.
Key Fact: The IQR is preferred over the range when the data contains outliers or is skewed, because it ignores extreme values.
Key Fact: The pth percentile is the value below which p% of the data falls; it can be read from a cumulative frequency curve.
Key Fact: The 10th percentile (P10) is the value at 10% of the total cumulative frequency; the 90th percentile (P90) is at 90%.
Key Fact: The interpercentile range is the difference between two percentiles, e.g. P90 β P10, which captures the middle 80% of the data.
Key Fact: The interpercentile range is more stable than the range because it excludes extreme values, similar to how the IQR works.
Key Fact: Variance measures the average squared deviation from the mean: ΟΒ² = Ξ£(x β xΜ)Β² Γ· n (population) or sΒ² = Ξ£(x β xΜ)Β² Γ· (n β 1) (sample).
Key Fact: Standard deviation (SD) is the square root of the variance; it measures spread in the same units as the original data.
Key Fact: A larger standard deviation means the data is more spread out from the mean; a smaller SD means the data is closer to the mean.
Key Fact: For grouped data, the estimated standard deviation uses midpoints: s = β[Ξ£f(x β xΜ)Β² Γ· (Ξ£f β 1)] where x is the midpoint of each class.
Key Fact: When comparing two distributions, compare both a measure of central tendency (e.g. median) and a measure of spread (e.g. IQR or SD).
π Key Vocabulary and Concepts
For Measures of Spread, you must know:
- Range: The difference between the maximum and minimum values in a dataset: range = max β min.
- Interquartile range (IQR): The difference between the upper quartile and lower quartile (Q3 β Q1), measuring the spread of the middle 50% of the data.
- Percentile: A value that divides ordered data into 100 equal parts; the pth percentile is the value below which p% of the data falls.
- Interpercentile range: The difference between two stated percentiles (e.g. P90 β P10), measuring the spread of a specified central portion of the data.
- Standard deviation: The square root of the variance; a measure of spread that uses all data values and is in the same units as the data.
- Variance: The average of the squared deviations from the mean; it measures spread but in squared units, so standard deviation is preferred for interpretation.
β Practice Questions
Q: Data: 5, 8, 12, 15, 20. Find the range.
Q: For a dataset of 40 values, at what positions are Q1 and Q3?
Q: From a cumulative frequency curve with total frequency 200, at what cumulative frequency do you read P10 and P90?
Q: State one advantage of the interpercentile range over the range.
Q: The standard deviation of dataset A is 2.3 and dataset B is 5.1. What does this tell you?
β Answers
- Range = 20 β 5 = 15.
- Q1 is at position 0.25 Γ 40 = 10 (the 10th value). Q3 is at position 0.75 Γ 40 = 30 (the 30th value).
- P10: 0.10 Γ 200 = 20. P90: 0.90 Γ 200 = 180.
- The interpercentile range excludes extreme values and outliers, giving a more reliable measure of the typical spread of the data.
- Dataset B is more spread out around its mean than dataset A; the values in B vary more from the average.
π― Exam Tips
- When calculating standard deviation, show each step: find the mean, calculate deviations, square them, sum, divide, and square root.
- Use (n β 1) as the divisor for sample standard deviation; use n for population standard deviation β check which the question specifies.
- When reading percentiles from a cumulative frequency curve, use a ruler for accuracy and show dotted lines from the y-axis to the curve and down to the x-axis.
- Always state which measure of spread you are using and why β e.g. 'I prefer the IQR because the data contains outliers'.
- In comparison questions, write full sentences: 'The IQR for group A (12) is smaller than the IQR for group B (18), so group A's data is less spread out in the middle 50%.'
π Exam Technique
GCSE Statistics Exam Tips β Measures of Spread:
1. For Measures of Spread questions, show every step of your working clearly β 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 Measures of Spread 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 Measures of Spread questions, show every step of your working clearly β 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 Measures of Spread 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
β οΈ Common Errors
β Using n instead of (n β 1) when calculating sample standard deviation β For a sample, divide by (n β 1) to get an unbiased estimate; dividing by n underestimates the true standard deviation of the population.
β Calculating range as maximum value minus minimum value including outliers without comment β The range is heavily affected by outliers; if outliers exist, use the IQR or interpercentile range instead, or state that the range is unreliable.
β Forgetting to square root the variance to get the standard deviation β Standard deviation = βvariance; if variance is 49, then SD = 7, not 49.
β Confusing percentile position with percentile value β The 25th percentile position is at 25% of the total frequency on the cumulative frequency axis; the percentile value is the corresponding data value read from the x-axis.
βοΈ Model Answer
Full-Mark Response
Calculate the standard deviation of the following sample data: 4, 7, 8, 10, 11.
Step 1: Find the mean. xΜ = (4 + 7 + 8 + 10 + 11) Γ· 5 = 40 Γ· 5 = 8 Step 2: Calculate deviations from the mean and square them: x | x β xΜ | (x β xΜ)Β² 4 | β4 | 16 7 | β1 | 1 8 | 0 | 0 10| 2 | 4 11| 3 | 9 Sum | 30 Step 3: Variance = Ξ£(x β xΜ)Β² Γ· (n β 1) = 30 Γ· 4 = 7.5 Step 4: Standard deviation = β7.5 β 2.74 (to 2 d.p.)
π AO Deep Dive
Assessment Objective Analysis
AO1 (Knowledge & Understanding): Demonstrate knowledge and understanding of measures of spread, including data collection, presentation and calculation techniques relevant to Edexcel 1ST0 & AQA 8382.
AO2 (Application): Apply knowledge and understanding of measures of spread 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.
π 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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π Past Papers for Statistics (AQA)
- AQA Past Papers & Mark Schemes
- Edexcel Past Papers & Mark Schemes
- OCR Past Papers & Mark Schemes
- Eduqas Past Papers & Mark Schemes
- CCEA Past Papers & Mark Schemes
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