Measure position and spread
Calculate quartiles, percentiles, range, interquartile range and standard deviation, then compare distributions fairly.
Exam scope and assessment notes
This lesson maps the Measure position and spread section of Cambridge IGCSE Statistics 0479 (2027 and 2028–2030 syllabus versions). This is original StudyForA teaching and self-check practice, not Cambridge material, an official mark scheme, a timed mock or an official grade. Charts and written working are not automatically assessed; human subject-teacher review remains pending. Use the syllabus version for your exam year. Cambridge offers 0479 in the June series only and does not make it available in every administrative zone. Check the current Cambridge 0479 page and confirm your centre's entry eligibility with your exams officer.
Before you begin
- Use the earlier 0479 ideas where needed; write units, working and a contextual interpretation for each result.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
A dataset ranges from 8 to 31. Find its range.
What you will learn
- Find quartiles for small sets and ungrouped frequency distributions
- Estimate grouped quartiles and percentiles from curves or interpolation
- Estimate frequencies within parts of a grouped distribution
- Calculate range, interquartile range and standard deviation, including grouped estimates
- Explain and compare measures of dispersion
Quartiles and percentiles describe position
Q1 is the lower quartile, Q2 is the median and Q3 is the upper quartile. The interquartile range is Q3 − Q1 and covers the middle half of the distribution. A percentile states the position below which a stated percentage of observations lies.
For grouped data, locate the required cumulative count on the cumulative-frequency curve. Linear interpolation estimates a value between class boundaries; it assumes an even spread within the class. The reverse task—estimating the number between two values—subtracts the corresponding cumulative frequencies.
Range and standard deviation describe spread differently
Range = maximum − minimum and uses only the extremes. Standard deviation summarises typical distance from the mean and uses all observations. For data x with count n, σ = √(Σx²/n − (Σx/n)²). For a frequency table use Σfx and Σfx²; for grouped data use class midpoints, so the result is an estimate.
Compare both centre and variability
A larger standard deviation or interquartile range indicates more spread on that measure. Compare distributions in context: one may have a higher typical value but also less consistency. The range is sensitive to extreme values; interquartile range focuses on the middle half; standard deviation is affected by all values.
Pause and explain
What does a larger interquartile range mean?
Worked example
For values 2, 4 and 6, calculate the population standard deviation using σ = √(Σx²/n − (Σx/n)²).
Show the worked solution
- Σx = 12, so mean = 12/3 = 4.
- Σx² = 4 + 16 + 36 = 56.
- σ = √(56/3 − 4²) = √(8/3) ≈ 1.63.
Answer The standard deviation is √(8/3), approximately 1.63. The syllabus formula divides by n for this population calculation.
Try a topic test
Choose Easy, Normal or Difficult. Save your answers and open worked review after finishing.
Test this topic →From guided practice to a new situation
Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.
A dataset is 2, 4, 6, 8, 10. Find the median, Q1, Q3 and IQR using the median-of-halves convention.
Give me a hint
The overall median is 6; find the median of each half without including 6.
Compare my reasoning
- Lower half 2, 4 has median Q1 = 3.
- Upper half 8, 10 has median Q3 = 9.
- IQR = 9 − 3 = 6; overall median = 6.
Median 6, Q1 = 3, Q3 = 9 and IQR = 6, using the stated median-of-halves convention.
Look for these in your work
- Convention stated
- Quartiles correct
- IQR calculation correct
A grouped distribution has total frequency 80. From its cumulative-frequency curve, the 20th and 60th values are estimated as 14 and 26. Estimate the interquartile range.
Give me a hint
For 80 observations, the quartiles correspond approximately to cumulative counts 20 and 60.
Compare my reasoning
- Q1 estimate = 14.
- Q3 estimate = 26.
- IQR estimate = 26 − 14 = 12.
The estimated interquartile range is 12 units. It is approximate because the values were read from grouped cumulative data.
Look for these in your work
- Quartile positions correct
- Subtraction correct
- Estimate qualified
Two delivery routes have the same mean time. Route A has standard deviation 3 minutes and Route B has standard deviation 8 minutes. Which is more consistent, and what does the comparison not prove?
Give me a hint
Lower standard deviation means times are less spread around the mean.
Compare my reasoning
- Route A has the smaller standard deviation.
- Its delivery times are less variable around the shared mean on this measure.
- This does not show that every Route A delivery is faster or that its mean is lower.
Route A is more consistent because 3 minutes is less than 8 minutes. The means are equal, and standard deviation does not show that each Route A delivery is faster.
Look for these in your work
- Measure compared correctly
- Meaning in context
- Limit of conclusion stated
Common mistakes
- Confusing interquartile range with the upper quartile
- Reading cumulative frequency as a value rather than a count
- Using grouped midpoints while reporting an exact standard deviation
- Comparing spread without comparing the distributions' centres
What can you explain now?
If Q1 = 12 and Q3 = 27, what is the interquartile range?
Compare with the explanation
15
IQR = Q3 − Q1 = 27 − 12.
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Make it stick
What parts of a dataset do range, interquartile range and standard deviation each use?
If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.