Track how data transformations change a distribution
Predict the effect of shifts, scale changes, additions and removals, and combine or standardise summary statistics.
Exam scope and assessment notes
This lesson maps the Track how data transformations change a distribution 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 group of 10 has mean 6. Find its total.
What you will learn
- Describe effects on mean, standard deviation, median and IQR when constants are added or multiplied
- Assess how adding or removing observations affects measures
- Combine two datasets using their means and standard deviations
- Scale and standardise data using means and standard deviations
- Compare distributions using centre and spread
Adding a constant shifts the centre
If every x becomes x + a, mean and median increase by a; standard deviation and interquartile range stay the same. Adding a constant moves every value equally, so distances between values do not change.
Multiplying changes scale
If every x becomes bx, mean and median are multiplied by b. For positive b, standard deviation and IQR are multiplied by b; if b is negative, the measures of spread are multiplied by |b| because spread cannot be negative, and the order reverses. Check whether the question uses a positive scale factor.
Combine datasets from totals and sums of squares
For groups of sizes n₁ and n₂, the combined mean is the weighted mean: (n₁μ₁ + n₂μ₂)/(n₁+n₂). For the syllabus standard deviation formula, Σx² = n(σ² + μ²), so combine these sums of squares, combine totals, and then use σ = √(Σx²/n − μ²).
To standardise x, use z = (x − μ)/σ. Standardised values have mean 0 and standard deviation 1. Interpret the z-score as the number of standard deviations above or below the mean.
New or removed values can change each measure
The effect depends on the actual value and its position. Adding an observation above the current mean raises the mean, but the median can rise, fall or stay the same depending on the ordered values. Recalculate or reason from positions rather than assuming every measure moves together.
Pause and explain
For z = (x − μ)/σ, what does z = −2 mean?
Worked example
Every temperature x is converted to y = 1.8x + 32. The original mean is 20 and standard deviation is 4. Find the new mean and standard deviation.
Show the worked solution
- The +32 shifts the centre: 1.8×20 + 32 = 68.
- Multiplication by 1.8 scales spread by 1.8.
- New standard deviation = 1.8×4 = 7.2.
Answer The converted temperatures have mean 68 and standard deviation 7.2. Adding 32 changes the centre but not spread; multiplying by 1.8 scales both.
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.
Dataset A has 8 values with mean 12; dataset B has 12 values with mean 17. Find the combined mean.
Give me a hint
Combine the group totals before dividing by the combined number of values.
Compare my reasoning
- A total = 8×12 = 96.
- B total = 12×17 = 204; combined total = 300 for 20 values.
- Combined mean = 300/20 = 15.
The combined mean is 15: the 20 observations have a total of 300, so the weighted total divided by 20 gives 15.
Look for these in your work
- Group totals weighted
- Combined count correct
- Mean calculated
A set has mean 50 and standard deviation 6. Each value is transformed by y = 3x − 4. Find the new mean and standard deviation.
Give me a hint
Apply the shift and scale to the centre; only the scale affects spread.
Compare my reasoning
- New mean = 3×50 − 4 = 146.
- New standard deviation = |3|×6 = 18.
- The subtraction of 4 does not change spread.
Mean 146 and standard deviation 18.
Look for these in your work
- Mean transformation correct
- Spread scaled by magnitude
- Shift excluded from spread
A school combines classes of 20 and 30 students. Their means are 64 and 70, and their standard deviations are 8 and 10. Find the combined mean and standard deviation using the 0479 formula.
Give me a hint
Combine totals and Σx² = n(σ² + μ²), then calculate the overall variance.
Compare my reasoning
- Combined total = 20×64 + 30×70 = 3380; n = 50, so mean = 67.6.
- Combined Σx² = 20(8²+64²) + 30(10²+70²) = 233 200.
- Variance = 233200/50 − 67.6² = 94.24; standard deviation ≈ 9.71.
Combined mean = 67.6 and standard deviation ≈ 9.71. The combined spread includes within-class spread and the difference between the two class means.
Look for these in your work
- Weighted mean correct
- Sum of squares combined
- Variance and square root correct
Common mistakes
- Adding a constant to standard deviation
- Multiplying a standard deviation by a negative factor without using its magnitude
- Combining group means without weighting by group size
- Claiming the median always moves in the same direction as one added value
What can you explain now?
If every value is increased by 5, what happens to the IQR?
Compare with the explanation
It stays the same
All observations shift equally, so Q3 − Q1 is unchanged.
After trying it yourself, choose your next review. This is your self-assessment.
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Make it stick
Which summary measures change when a constant is added to every value, and which change when values are multiplied by a positive constant?
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.